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0705.1209.pdf | 11 | militarised conflict. The unit of analysis is a dyad-year. After the omission, a total of 27737 with
26845 peace and 892 conflict dyad-years were filtered out.
The dyadic data was classified into two sets, which are training and testing set. In their study
(Lagazio and Russett, 2003), have given a detailed discussion on how the training set should be
chosen. They have found out that a balanced set, equal number of conflict and peace dyads, gives
best results as training set for the neural network. The same principle was adhered to in this study.
The training set contains 1000 randomly chosen dyads, 500 from each group. The test set contains
26737 dyads of which 392 are conflict and 26345 non-conflict dyads.
# IV. RESULTS AND DISCUSSION
Neural network and support vector machine were employed to classify the MID data. The main
focus of the result is to look at the percentage of correct MID prediction of the test data set by each
technique. Table I depicts the confusion matrix of the results. Although NN performed as good as
SVM in predicting true conflicts (true positives), this is achieved at the expense of reducing the
number of correct peace (true negatives) prediction. SVM picked up the true conflicts (true
positives) better than NN without effectively minimising the number of true peace (true negatives).
SVM is able to pick 1450 more cases of the true peace (true negatives) than NN, which makes it a
better choice than NN. Over all, SVM is able to predict peace and conflict with $$79\%$$ and $$75\%$$ ,
respectively. The corresponding results for NN are $$74\%$$ and $$76\%$$ , respectively for peace and
conflict. The combined results of correct predictions are $$79\%$$ for SVM and $$74\%$$ for NN.
| <p>militarised conflict. The unit of analysis is a dyad-year. After the omission, a total of 27737 with
26845 peace and 892 conflict dyad-years were filtered out.</p>
<p>The dyadic data was classified into two sets, which are training and testing set. In their study
(Lagazio and Russett, 2003), have given a detailed discussion on how the training set should be
chosen. They have found out that a balanced set, equal number of conflict and peace dyads, gives
best results as training set for the neural network. The same principle was adhered to in this study.
The training set contains 1000 randomly chosen dyads, 500 from each group. The test set contains
26737 dyads of which 392 are conflict and 26345 non-conflict dyads.</p>
<h1>IV. RESULTS AND DISCUSSION</h1>
<p>Neural network and support vector machine were employed to classify the MID data. The main
focus of the result is to look at the percentage of correct MID prediction of the test data set by each
technique. Table I depicts the confusion matrix of the results. Although NN performed as good as
SVM in predicting true conflicts (true positives), this is achieved at the expense of reducing the
number of correct peace (true negatives) prediction. SVM picked up the true conflicts (true
positives) better than NN without effectively minimising the number of true peace (true negatives).
SVM is able to pick 1450 more cases of the true peace (true negatives) than NN, which makes it a
better choice than NN. Over all, SVM is able to predict peace and conflict with $$79\%$$ and $$75\%$$ ,
respectively. The corresponding results for NN are $$74\%$$ and $$76\%$$ , respectively for peace and
conflict. The combined results of correct predictions are $$79\%$$ for SVM and $$74\%$$ for NN.</p>
| [{"type": "text", "coordinates": [65, 70, 549, 114], "content": "militarised conflict. The unit of analysis is a dyad-year. After the omission, a total of 27737 with\n26845 peace and 892 conflict dyad-years were filtered out.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [64, 153, 549, 308], "content": "The dyadic data was classified into two sets, which are training and testing set. In their study\n(Lagazio and Russett, 2003), have given a detailed discussion on how the training set should be\nchosen. They have found out that a balanced set, equal number of conflict and peace dyads, gives\nbest results as training set for the neural network. The same principle was adhered to in this study.\nThe training set contains 1000 randomly chosen dyads, 500 from each group. The test set contains\n26737 dyads of which 392 are conflict and 26345 non-conflict dyads.", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [227, 347, 402, 362], "content": "IV. RESULTS AND DISCUSSION", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [63, 372, 550, 641], "content": "Neural network and support vector machine were employed to classify the MID data. The main\nfocus of the result is to look at the percentage of correct MID prediction of the test data set by each\ntechnique. Table I depicts the confusion matrix of the results. Although NN performed as good as\nSVM in predicting true conflicts (true positives), this is achieved at the expense of reducing the\nnumber of correct peace (true negatives) prediction. SVM picked up the true conflicts (true\npositives) better than NN without effectively minimising the number of true peace (true negatives).\nSVM is able to pick 1450 more cases of the true peace (true negatives) than NN, which makes it a\nbetter choice than NN. Over all, SVM is able to predict peace and conflict with $$79\\%$$ and $$75\\%$$ ,\nrespectively. 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The corresponding results for NN are ", "type": "text"}, {"bbox": [329, 596, 353, 609], "score": 0.86, "content": "74\\%", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [354, 598, 379, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [380, 596, 403, 609], "score": 0.86, "content": "76\\%", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [404, 598, 546, 613], "score": 1.0, "content": ", respectively for peace and", "type": "text"}], "index": 17}, {"bbox": [66, 624, 495, 640], "spans": [{"bbox": [66, 625, 339, 640], "score": 1.0, "content": "conflict. The combined results of correct predictions are", "type": "text"}, {"bbox": [339, 624, 363, 636], "score": 0.87, "content": "79\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [363, 625, 430, 640], "score": 1.0, "content": " for SVM and", "type": "text"}, {"bbox": [431, 624, 455, 636], "score": 0.85, "content": "74\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [455, 625, 495, 640], "score": 1.0, "content": " for NN.", "type": "text"}], "index": 18}], "index": 13.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [65, 70, 549, 114], "lines": [], "index": 0.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [64, 72, 547, 115], "lines_deleted": true}, {"type": "text", "bbox": [64, 153, 549, 308], "lines": [{"bbox": [83, 155, 546, 171], "spans": [{"bbox": [83, 155, 546, 171], "score": 1.0, "content": "The dyadic data was classified into two sets, which are training and testing set. In their study", "type": "text"}], "index": 2}, {"bbox": [65, 184, 546, 198], "spans": [{"bbox": [65, 184, 546, 198], "score": 1.0, "content": "(Lagazio and Russett, 2003), have given a detailed discussion on how the training set should be", "type": "text"}], "index": 3}, {"bbox": [65, 210, 546, 227], "spans": [{"bbox": [65, 210, 546, 227], "score": 1.0, "content": "chosen. They have found out that a balanced set, equal number of conflict and peace dyads, gives", "type": "text"}], "index": 4}, {"bbox": [64, 239, 545, 254], "spans": [{"bbox": [64, 239, 545, 254], "score": 1.0, "content": "best results as training set for the neural network. The same principle was adhered to in this study.", "type": "text"}], "index": 5}, {"bbox": [63, 265, 548, 282], "spans": [{"bbox": [63, 265, 548, 282], "score": 1.0, "content": "The training set contains 1000 randomly chosen dyads, 500 from each group. The test set contains ", "type": "text"}], "index": 6}, {"bbox": [65, 293, 399, 309], "spans": [{"bbox": [65, 293, 399, 309], "score": 1.0, "content": "26737 dyads of which 392 are conflict and 26345 non-conflict dyads.", "type": "text"}], "index": 7}], "index": 4.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 155, 548, 309]}, {"type": "title", "bbox": [227, 347, 402, 362], "lines": [{"bbox": [228, 350, 401, 361], "spans": [{"bbox": [228, 350, 401, 361], "score": 1.0, "content": "IV. RESULTS AND DISCUSSION", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 372, 550, 641], "lines": [{"bbox": [82, 376, 547, 393], "spans": [{"bbox": [82, 376, 547, 393], "score": 1.0, "content": "Neural network and support vector machine were employed to classify the MID data. The main", "type": "text"}], "index": 9}, {"bbox": [64, 403, 547, 420], "spans": [{"bbox": [64, 403, 547, 420], "score": 1.0, "content": "focus of the result is to look at the percentage of correct MID prediction of the test data set by each", "type": "text"}], "index": 10}, {"bbox": [65, 432, 546, 448], "spans": [{"bbox": [65, 432, 546, 448], "score": 1.0, "content": "technique. Table I depicts the confusion matrix of the results. Although NN performed as good as", "type": "text"}], "index": 11}, {"bbox": [65, 459, 546, 476], "spans": [{"bbox": [65, 459, 546, 476], "score": 1.0, "content": "SVM in predicting true conflicts (true positives), this is achieved at the expense of reducing the", "type": "text"}], "index": 12}, {"bbox": [65, 487, 546, 502], "spans": [{"bbox": [65, 487, 546, 502], "score": 1.0, "content": "number of correct peace (true negatives) prediction. SVM picked up the true conflicts (true", "type": "text"}], "index": 13}, {"bbox": [65, 515, 545, 530], "spans": [{"bbox": [65, 515, 545, 530], "score": 1.0, "content": "positives) better than NN without effectively minimising the number of true peace (true negatives).", "type": "text"}], "index": 14}, {"bbox": [65, 543, 547, 558], "spans": [{"bbox": [65, 543, 547, 558], "score": 1.0, "content": "SVM is able to pick 1450 more cases of the true peace (true negatives) than NN, which makes it a", "type": "text"}], "index": 15}, {"bbox": [65, 569, 546, 585], "spans": [{"bbox": [65, 570, 470, 585], "score": 1.0, "content": "better choice than NN. Over all, SVM is able to predict peace and conflict with ", "type": "text"}, {"bbox": [471, 569, 495, 581], "score": 0.87, "content": "79\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 570, 519, 585], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [519, 569, 543, 581], "score": 0.87, "content": "75\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [543, 570, 546, 585], "score": 1.0, "content": ",", "type": "text"}], "index": 16}, {"bbox": [66, 596, 546, 613], "spans": [{"bbox": [66, 598, 329, 613], "score": 1.0, "content": "respectively. The corresponding results for NN are ", "type": "text"}, {"bbox": [329, 596, 353, 609], "score": 0.86, "content": "74\\%", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [354, 598, 379, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [380, 596, 403, 609], "score": 0.86, "content": "76\\%", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [404, 598, 546, 613], "score": 1.0, "content": ", respectively for peace and", "type": "text"}], "index": 17}, {"bbox": [66, 624, 495, 640], "spans": [{"bbox": [66, 625, 339, 640], "score": 1.0, "content": "conflict. The combined results of correct predictions are", "type": "text"}, {"bbox": [339, 624, 363, 636], "score": 0.87, "content": "79\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [363, 625, 430, 640], "score": 1.0, "content": " for SVM and", "type": "text"}, {"bbox": [431, 624, 455, 636], "score": 0.85, "content": "74\\%", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [455, 625, 495, 640], "score": 1.0, "content": " for NN.", "type": "text"}], "index": 18}], "index": 13.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [64, 376, 547, 640]}]} | ./arxiv_full_mineru_outputs_20/images/0705.1209_11.png | images/0705.1209_11.png |
|
0705.1209.pdf | 12 | $$\mathbf{T}\mathbf{C}=$$ true conflict (true positive), $$\mathrm{FC=}$$ false conflict (false positive), $$\mathrm{TP=}$$ true peace (true negative), and $$\mathrm{FP=}$$ false
peace (false negative)
# A. Receiver operating characteristic (ROC) curve
The receiver operating characteristic is a technique used to evaluate the prediction ability of a
binary classifier (Zweig and Campbell, 1993). In the context of MID classifiers, sensitivity is
defined as the probability of a classifier predicting conflict correctly and specificity is the
probability of a classifier predicting peace correctly (Westin, 2001). The ROC curve is then a
graph, which plots the sensitivity on the vertical-axis, and 1-specificity on the horizontal-axis,
which is, also called false positive rate. The area under curve (AUC) is used as a measure to
compare the performance of each classifier. The AUC for NN and SVM are 0.81 and 0.84 with
standard errors of 0.00998 and 0.01022, respectively. According to Hanley and McNeil (1983), the
normal distribution z value, which is used to compare if there is a significant difference between
AUCs of two classifiers that are derived from the same cases, is given by:
where $$A_{I},\,A_{2},$$ , $$S E_{2}$$ and $$S E_{2}$$ are the areas and standard errors of the respective curves. The value $$r$$
represents the estimated correlation between $$\mathbf{A}_{1}$$ and $$\mathbf{A}_{2}$$ (Hanley and McNeil, 1983). The value of $$z$$
| <p>$$\mathbf{T}\mathbf{C}=$$ true conflict (true positive), $$\mathrm{FC=}$$ false conflict (false positive), $$\mathrm{TP=}$$ true peace (true negative), and $$\mathrm{FP=}$$ false
peace (false negative)</p>
<h1>A. Receiver operating characteristic (ROC) curve</h1>
<p>The receiver operating characteristic is a technique used to evaluate the prediction ability of a
binary classifier (Zweig and Campbell, 1993). In the context of MID classifiers, sensitivity is
defined as the probability of a classifier predicting conflict correctly and specificity is the
probability of a classifier predicting peace correctly (Westin, 2001). The ROC curve is then a
graph, which plots the sensitivity on the vertical-axis, and 1-specificity on the horizontal-axis,
which is, also called false positive rate. The area under curve (AUC) is used as a measure to
compare the performance of each classifier. The AUC for NN and SVM are 0.81 and 0.84 with
standard errors of 0.00998 and 0.01022, respectively. According to Hanley and McNeil (1983), the
normal distribution z value, which is used to compare if there is a significant difference between
AUCs of two classifiers that are derived from the same cases, is given by:</p>
<p>where $$A_{I},\,A_{2},$$ , $$S E_{2}$$ and $$S E_{2}$$ are the areas and standard errors of the respective curves. The value $$r$$
represents the estimated correlation between $$\mathbf{A}_{1}$$ and $$\mathbf{A}_{2}$$ (Hanley and McNeil, 1983). The value of $$z$$</p>
| [{"type": "table", "coordinates": [174, 168, 451, 233], "content": "", "block_type": "table", "index": 1}, {"type": "text", "coordinates": [64, 249, 548, 280], "content": "$$\\mathbf{T}\\mathbf{C}=$$ true conflict (true positive), $$\\mathrm{FC=}$$ false conflict (false positive), $$\\mathrm{TP=}$$ true peace (true negative), and $$\\mathrm{FP=}$$ false\npeace (false negative)", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [82, 312, 325, 327], "content": "A. Receiver operating characteristic (ROC) curve", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [63, 338, 550, 605], "content": "The receiver operating characteristic is a technique used to evaluate the prediction ability of a\nbinary classifier (Zweig and Campbell, 1993). In the context of MID classifiers, sensitivity is\ndefined as the probability of a classifier predicting conflict correctly and specificity is the\nprobability of a classifier predicting peace correctly (Westin, 2001). The ROC curve is then a\ngraph, which plots the sensitivity on the vertical-axis, and 1-specificity on the horizontal-axis,\nwhich is, also called false positive rate. The area under curve (AUC) is used as a measure to\ncompare the performance of each classifier. The AUC for NN and SVM are 0.81 and 0.84 with\nstandard errors of 0.00998 and 0.01022, respectively. 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The area under curve (AUC) is used as a measure to", "type": "text"}], "index": 13}, {"bbox": [66, 506, 546, 522], "spans": [{"bbox": [66, 506, 546, 522], "score": 1.0, "content": "compare the performance of each classifier. The AUC for NN and SVM are 0.81 and 0.84 with", "type": "text"}], "index": 14}, {"bbox": [65, 534, 546, 550], "spans": [{"bbox": [65, 534, 546, 550], "score": 1.0, "content": "standard errors of 0.00998 and 0.01022, respectively. According to Hanley and McNeil (1983), the", "type": "text"}], "index": 15}, {"bbox": [64, 562, 547, 577], "spans": [{"bbox": [64, 562, 547, 577], "score": 1.0, "content": "normal distribution z value, which is used to compare if there is a significant difference between", "type": "text"}], "index": 16}, {"bbox": [65, 588, 421, 606], "spans": [{"bbox": [65, 588, 421, 606], "score": 1.0, "content": "AUCs of two classifiers that are derived from the same cases, is given by:", "type": "text"}], "index": 17}], "index": 12.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [64, 341, 548, 606]}, {"type": "interline_equation", "bbox": [211, 616, 355, 652], "lines": [{"bbox": [211, 616, 355, 652], "spans": [{"bbox": [211, 616, 355, 652], "score": 0.93, "content": "\\mathbf{Z}=\\frac{\\mathbf{A}_{1}-\\mathbf{A}_{2}}{\\sqrt{\\mathbf{S}\\mathbf{E}_{1}^{2}+\\mathbf{S}\\mathbf{E}_{2}^{2}-2\\mathbf{rS}\\mathbf{E}_{1}\\mathbf{S}\\mathbf{E}_{2}}}", "type": "interline_equation"}], "index": 18}], "index": 18, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [63, 665, 549, 711], "lines": [{"bbox": [65, 667, 548, 683], "spans": [{"bbox": [65, 668, 96, 683], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [97, 668, 131, 680], "score": 0.52, "content": "A_{I},\\,A_{2},", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [131, 668, 136, 683], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [137, 667, 156, 680], "score": 0.77, "content": "S E_{2}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [156, 668, 180, 683], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [180, 668, 199, 680], "score": 0.92, "content": "S E_{2}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [199, 668, 539, 683], "score": 1.0, "content": " are the areas and standard errors of the respective curves. The value ", "type": "text"}, {"bbox": [540, 670, 547, 678], "score": 0.58, "content": "r", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [547, 668, 548, 683], "score": 1.0, "content": " ", "type": "text"}], "index": 19}, {"bbox": [65, 695, 547, 710], "spans": [{"bbox": [65, 696, 282, 710], "score": 1.0, "content": "represents the estimated correlation between", "type": "text"}, {"bbox": [282, 695, 297, 708], "score": 0.89, "content": "\\mathbf{A}_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [297, 696, 319, 710], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 695, 334, 708], "score": 0.85, "content": "\\mathbf{A}_{2}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [334, 696, 539, 710], "score": 1.0, "content": " (Hanley and McNeil, 1983). The value of", "type": "text"}, {"bbox": [540, 697, 547, 707], "score": 0.36, "content": "z", "type": "inline_equation", "height": 10, "width": 7}], "index": 20}], "index": 19.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [65, 667, 548, 710]}]} | ./arxiv_full_mineru_outputs_20/images/0705.1209_12.png | images/0705.1209_12.png |
|
0705.1209.pdf | 13 | is 2.697, which gives significant difference in a $$95\%$$ confidence interval. The results of the SVM
are much better in predicting the conflicts without affecting the prediction of peace as it is clearly
shown in I. The ROC graphs of the NN and SVM results are given in figure 3.
# B. Influence of each variable on the MID outcome
In order to see the influence of each variable on the MID result, two separate sensitivity
analysis were done for NN and SVM. The two techniques agree in picking up the influences of
some of the variables while they differ on others.
1) Experiment one: This experiment looked at how assigning each variable to its possible
maximum value while keeping the rest at their possible minimum values and vise versa affect the
MID outcome. The results for NN show that democracy level and capability ratio are able to
deliver a peaceful outcome while all the other variables are kept minimal. This means, dyadic
preponderance has a deterring effect to conflict, as is joint democracy of the states involved. On the
| <p>is 2.697, which gives significant difference in a $$95\%$$ confidence interval. The results of the SVM
are much better in predicting the conflicts without affecting the prediction of peace as it is clearly
shown in I. The ROC graphs of the NN and SVM results are given in figure 3.</p>
<h1>B. Influence of each variable on the MID outcome</h1>
<p>In order to see the influence of each variable on the MID result, two separate sensitivity
analysis were done for NN and SVM. The two techniques agree in picking up the influences of
some of the variables while they differ on others.</p>
<p>1) Experiment one: This experiment looked at how assigning each variable to its possible
maximum value while keeping the rest at their possible minimum values and vise versa affect the
MID outcome. The results for NN show that democracy level and capability ratio are able to
deliver a peaceful outcome while all the other variables are kept minimal. This means, dyadic
preponderance has a deterring effect to conflict, as is joint democracy of the states involved. On the</p>
| [{"type": "text", "coordinates": [65, 70, 549, 142], "content": "is 2.697, which gives significant difference in a $$95\\%$$ confidence interval. The results of the SVM\nare much better in predicting the conflicts without affecting the prediction of peace as it is clearly\nshown in I. The ROC graphs of the NN and SVM results are given in figure 3.", "block_type": "text", "index": 1}, {"type": "image", "coordinates": [177, 162, 443, 381], "content": "", "block_type": "image", "index": 2}, {"type": "title", "coordinates": [82, 459, 327, 474], "content": "B. Influence of each variable on the MID outcome", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [65, 486, 549, 558], "content": "In order to see the influence of each variable on the MID result, two separate sensitivity\nanalysis were done for NN and SVM. 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On the", "block_type": "text", "index": 5}] | [{"type": "text", "coordinates": [64, 73, 302, 88], "content": "is 2.697, which gives significant difference in a ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [302, 72, 326, 84], "content": "95\\%", "score": 0.86, "index": 2}, {"type": "text", "coordinates": [326, 73, 547, 88], "content": " confidence interval. The results of the SVM", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [65, 101, 546, 117], "content": "are much better in predicting the conflicts without affecting the prediction of peace as it is clearly", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [66, 128, 442, 144], "content": "shown in I. The ROC graphs of the NN and SVM results are given in figure 3.", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [84, 461, 325, 475], "content": "B. Influence of each variable on the MID outcome", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [82, 488, 545, 505], "content": "In order to see the influence of each variable on the MID result, two separate sensitivity", "score": 1.0, "index": 7}, {"type": "text", "coordinates": [65, 515, 547, 532], "content": "analysis were done for NN and SVM. The two techniques agree in picking up the influences of", "score": 1.0, "index": 8}, {"type": "text", "coordinates": [65, 543, 301, 560], "content": "some of the variables while they differ on others.", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [85, 571, 547, 588], "content": " 1) Experiment one: This experiment looked at how assigning each variable to its possible", "score": 1.0, "index": 10}, {"type": "text", "coordinates": [64, 599, 547, 615], "content": "maximum value while keeping the rest at their possible minimum values and vise versa affect the", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [64, 626, 547, 643], "content": "MID outcome. The results for NN show that democracy level and capability ratio are able to", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [64, 653, 547, 671], "content": "deliver a peaceful outcome while all the other variables are kept minimal. This means, dyadic", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [64, 682, 547, 698], "content": "preponderance has a deterring effect to conflict, as is joint democracy of the states involved. On the", "score": 1.0, "index": 14}] | [{"coordinates": [177, 162, 443, 381], "index": 13.5, "caption": "and se-nn are their respective standard errors.", "caption_coordinates": [64, 396, 548, 428]}] | [{"type": "inline", "coordinates": [302, 72, 326, 84], "content": "95\\%", "caption": ""}] | [] | [612.0, 792.0] | [{"type": "text", "text": "is 2.697, which gives significant difference in a $95\\%$ confidence interval. The results of the SVM are much better in predicting the conflicts without affecting the prediction of peace as it is clearly shown in I. The ROC graphs of the NN and SVM results are given in figure 3. ", "page_idx": 13}, {"type": "image", "img_path": "images/72d9a0cee621068e03d88fd1fd3f198f74c342c6a2be57d85cfc8206dc23f772.jpg", "img_caption": ["Fig. 3. ROC curve for both NN and SVM. Area-svm and area-nn signify the areas under the curves while se-svm and se-nn are their respective standard errors. "], "img_footnote": [], "page_idx": 13}, {"type": "text", "text": "B. Influence of each variable on the MID outcome ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "In order to see the influence of each variable on the MID result, two separate sensitivity analysis were done for NN and SVM. The two techniques agree in picking up the influences of some of the variables while they differ on others. ", "page_idx": 13}, {"type": "text", "text": "1) Experiment one: This experiment looked at how assigning each variable to its possible maximum value while keeping the rest at their possible minimum values and vise versa affect the MID outcome. The results for NN show that democracy level and capability ratio are able to deliver a peaceful outcome while all the other variables are kept minimal. This means, dyadic preponderance has a deterring effect to conflict, as is joint democracy of the states involved. On the other hand, keeping all the variables to their maximum values while assigning one variable to its minimum value resulted in a peaceful outcome. In other words, no single variable is able to change the outcome if all the other variables are set to their possible maximum values for NN. A similar experiment conducted for SVM shows that it is not able to pick the influence of a variable using the same approach, as it is possible with NN. That is, whether the variables are set to their minimum or maximum gives a peace outcome. 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The results of the SVM", "type": "text"}], "index": 0}, {"bbox": [65, 101, 546, 117], "spans": [{"bbox": [65, 101, 546, 117], "score": 1.0, "content": "are much better in predicting the conflicts without affecting the prediction of peace as it is clearly", "type": "text"}], "index": 1}, {"bbox": [66, 128, 442, 144], "spans": [{"bbox": [66, 128, 442, 144], "score": 1.0, "content": "shown in I. 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ROC curve for both NN and SVM. Area-svm and area-nn signify the areas under the curves while se-svm", "type": "text"}], "index": 17}, {"bbox": [65, 417, 248, 428], "spans": [{"bbox": [65, 417, 248, 428], "score": 1.0, "content": "and se-nn are their respective standard errors.", "type": "text"}], "index": 18}], "index": 17.5}], "index": 13.5, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "title", "bbox": [82, 459, 327, 474], "lines": [{"bbox": [84, 461, 325, 475], "spans": [{"bbox": [84, 461, 325, 475], "score": 1.0, "content": "B. Influence of each variable on the MID outcome", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [65, 486, 549, 558], "lines": [{"bbox": [82, 488, 545, 505], "spans": [{"bbox": [82, 488, 545, 505], "score": 1.0, "content": "In order to see the influence of each variable on the MID result, two separate sensitivity", "type": "text"}], "index": 20}, {"bbox": [65, 515, 547, 532], "spans": [{"bbox": [65, 515, 547, 532], "score": 1.0, "content": "analysis were done for NN and SVM. The two techniques agree in picking up the influences of", "type": "text"}], "index": 21}, {"bbox": [65, 543, 301, 560], "spans": [{"bbox": [65, 543, 301, 560], "score": 1.0, "content": "some of the variables while they differ on others.", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [65, 488, 547, 560]}, {"type": "text", "bbox": [64, 570, 549, 698], "lines": [{"bbox": [85, 571, 547, 588], "spans": [{"bbox": [85, 571, 547, 588], "score": 1.0, "content": " 1) Experiment one: This experiment looked at how assigning each variable to its possible", "type": "text"}], "index": 23}, {"bbox": [64, 599, 547, 615], "spans": [{"bbox": [64, 599, 547, 615], "score": 1.0, "content": "maximum value while keeping the rest at their possible minimum values and vise versa affect the", "type": "text"}], "index": 24}, {"bbox": [64, 626, 547, 643], "spans": [{"bbox": [64, 626, 547, 643], "score": 1.0, "content": "MID outcome. 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In other words, no single variable is able to change", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [65, 129, 546, 142], "spans": [{"bbox": [65, 129, 546, 142], "score": 1.0, "content": "the outcome if all the other variables are set to their possible maximum values for NN. A similar", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [65, 156, 546, 171], "spans": [{"bbox": [65, 156, 546, 171], "score": 1.0, "content": "experiment conducted for SVM shows that it is not able to pick the influence of a variable using the", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [64, 184, 549, 199], "spans": [{"bbox": [64, 184, 549, 199], "score": 1.0, "content": "same approach, as it is possible with NN. That is, whether the variables are set to their minimum or ", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [65, 213, 228, 226], "spans": [{"bbox": [65, 213, 228, 226], "score": 1.0, "content": "maximum gives a peace outcome.", "type": "text", "cross_page": true}], "index": 5}], "index": 25, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [64, 571, 547, 698]}]} | ./arxiv_full_mineru_outputs_20/images/0705.1209_13.png | images/0705.1209_13.png |
|
0705.1209.pdf | 14 | other hand, keeping all the variables to their maximum values while assigning one variable to its
minimum value resulted in a peaceful outcome. In other words, no single variable is able to change
the outcome if all the other variables are set to their possible maximum values for NN. A similar
experiment conducted for SVM shows that it is not able to pick the influence of a variable using the
same approach, as it is possible with NN. That is, whether the variables are set to their minimum or
maximum gives a peace outcome.
2) Experiment two: This experiment was done to measure the sensitivity of the variables in the
spirit of partial derivatives as (Zeng, 1999) has put it. The idea is basically to see the change in the
output for a change in one of the input variables. The experiment looks at how the MID varies
when one variable is assigned to its possible maximum and minimum values while keeping all the
other variables constant. The results found for both NN and SVM are shown in II. Our test data set
has 26737 cases of peace and 392 cases of war. The first line of the table shows the correct number
of peace and war prediction when all variables are used. Assigning each variable to take its possible
maximum and minimum values while keeping the other variables fixed then generated different
testing data sets. Each subsequent line of the table depicts the number of correct prediction of peace
and war.
NN result: It shows democracy level has the maximum effect in reducing conflict while
capability ratio is second in conformance to the first experiment. Allowing democracy to have its
possible maximum value for the whole data set was able to avoid conflict totally. Capability ratio
reduced the occurrence of conflict by $$98\%$$ . Maximising alliance between the dyads reduced the
number of conflicts by $$20\%$$ . Maximising dependency has a $$6\%$$ effect in reducing possible
conflicts. Reducing major power was able to cut the number of conflicts by $$3\%$$ . Minimising the
contiguity of the dyads to their possible lower values and maximising the distance reduced the
| <p>other hand, keeping all the variables to their maximum values while assigning one variable to its
minimum value resulted in a peaceful outcome. In other words, no single variable is able to change
the outcome if all the other variables are set to their possible maximum values for NN. A similar
experiment conducted for SVM shows that it is not able to pick the influence of a variable using the
same approach, as it is possible with NN. That is, whether the variables are set to their minimum or
maximum gives a peace outcome.</p>
<p>2) Experiment two: This experiment was done to measure the sensitivity of the variables in the
spirit of partial derivatives as (Zeng, 1999) has put it. The idea is basically to see the change in the
output for a change in one of the input variables. The experiment looks at how the MID varies
when one variable is assigned to its possible maximum and minimum values while keeping all the
other variables constant. The results found for both NN and SVM are shown in II. Our test data set
has 26737 cases of peace and 392 cases of war. The first line of the table shows the correct number
of peace and war prediction when all variables are used. Assigning each variable to take its possible
maximum and minimum values while keeping the other variables fixed then generated different
testing data sets. Each subsequent line of the table depicts the number of correct prediction of peace
and war.</p>
<p>NN result: It shows democracy level has the maximum effect in reducing conflict while
capability ratio is second in conformance to the first experiment. Allowing democracy to have its
possible maximum value for the whole data set was able to avoid conflict totally. Capability ratio
reduced the occurrence of conflict by $$98\%$$ . Maximising alliance between the dyads reduced the
number of conflicts by $$20\%$$ . Maximising dependency has a $$6\%$$ effect in reducing possible
conflicts. Reducing major power was able to cut the number of conflicts by $$3\%$$ . Minimising the
contiguity of the dyads to their possible lower values and maximising the distance reduced the</p>
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The experiment looks at how the MID varies\nwhen one variable is assigned to its possible maximum and minimum values while keeping all the\nother variables constant. The results found for both NN and SVM are shown in II. Our test data set\nhas 26737 cases of peace and 392 cases of war. The first line of the table shows the correct number\nof peace and war prediction when all variables are used. Assigning each variable to take its possible\nmaximum and minimum values while keeping the other variables fixed then generated different\ntesting data sets. Each subsequent line of the table depicts the number of correct prediction of peace\nand war.", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [64, 512, 550, 696], "content": "NN result: It shows democracy level has the maximum effect in reducing conflict while\ncapability ratio is second in conformance to the first experiment. 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The experiment looks at how the MID varies when one variable is assigned to its possible maximum and minimum values while keeping all the other variables constant. The results found for both NN and SVM are shown in II. Our test data set has 26737 cases of peace and 392 cases of war. The first line of the table shows the correct number of peace and war prediction when all variables are used. Assigning each variable to take its possible maximum and minimum values while keeping the other variables fixed then generated different testing data sets. Each subsequent line of the table depicts the number of correct prediction of peace and war. ", "page_idx": 14}, {"type": "text", "text": "NN result: It shows democracy level has the maximum effect in reducing conflict while capability ratio is second in conformance to the first experiment. Allowing democracy to have its possible maximum value for the whole data set was able to avoid conflict totally. Capability ratio reduced the occurrence of conflict by $98\\%$ . Maximising alliance between the dyads reduced the number of conflicts by $20\\%$ . Maximising dependency has a $6\\%$ effect in reducing possible conflicts. Reducing major power was able to cut the number of conflicts by $3\\%$ . Minimising the contiguity of the dyads to their possible lower values and maximising the distance reduced the number of conflicts by $45\\%$ and $31\\%$ , respectively. This last result agrees with the realist theory that says far apart countries have less reasons to have conflicts (Oneal and Russett, 1999). 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This last result agrees with the realist theory", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [65, 100, 497, 115], "spans": [{"bbox": [65, 100, 497, 115], "score": 1.0, "content": "that says far apart countries have less reasons to have conflicts (Oneal and Russett, 1999).", "type": "text", "cross_page": true}], "index": 1}], "index": 19, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [64, 513, 547, 695]}]} | ./arxiv_full_mineru_outputs_20/images/0705.1209_14.png | images/0705.1209_14.png |
|
0705.1209.pdf | 15 | number of conflicts by $$45\%$$ and $$31\%$$ , respectively. This last result agrees with the realist theory
that says far apart countries have less reasons to have conflicts (Oneal and Russett, 1999).
SVM result: The results of the experiment show inconsistency on how the MID outcome is
affected when the variables are maximised and minimised. Further investigation is required to
understand more clearly the influence of each variable (e.g. exploring some other sensitivity
analysis techniques). Therefore, an alternative sensitivity analysis that involves using only one
explanatory variable to predict the MID and see the goodness of accuracy is used. The ROC curves
were drawn and the area under the curve (AUC) calculated for the purpose of ranking as is
| <p>number of conflicts by $$45\%$$ and $$31\%$$ , respectively. This last result agrees with the realist theory
that says far apart countries have less reasons to have conflicts (Oneal and Russett, 1999).</p>
<p>SVM result: The results of the experiment show inconsistency on how the MID outcome is
affected when the variables are maximised and minimised. Further investigation is required to
understand more clearly the influence of each variable (e.g. exploring some other sensitivity
analysis techniques). Therefore, an alternative sensitivity analysis that involves using only one
explanatory variable to predict the MID and see the goodness of accuracy is used. The ROC curves
were drawn and the area under the curve (AUC) calculated for the purpose of ranking as is</p>
| [{"type": "text", "coordinates": [63, 70, 548, 116], "content": "number of conflicts by $$45\\%$$ and $$31\\%$$ , respectively. This last result agrees with the realist theory\nthat says far apart countries have less reasons to have conflicts (Oneal and Russett, 1999).", "block_type": "text", "index": 1}, {"type": "table", "coordinates": [189, 204, 440, 528], "content": "", "block_type": "table", "index": 2}, {"type": "text", "coordinates": [64, 551, 549, 708], "content": "SVM result: The results of the experiment show inconsistency on how the MID outcome is\naffected when the variables are maximised and minimised. Further investigation is required to\nunderstand more clearly the influence of each variable (e.g. exploring some other sensitivity\nanalysis techniques). Therefore, an alternative sensitivity analysis that involves using only one\nexplanatory variable to predict the MID and see the goodness of accuracy is used. The ROC curves\nwere drawn and the area under the curve (AUC) calculated for the purpose of ranking as is", "block_type": "text", "index": 3}] | [{"type": "text", "coordinates": [64, 73, 181, 88], "content": "number of conflicts by ", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [181, 72, 205, 84], "content": "45\\%", "score": 0.86, "index": 2}, {"type": "text", "coordinates": [205, 73, 228, 88], "content": " and ", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [229, 72, 252, 84], "content": "31\\%", "score": 0.87, "index": 4}, {"type": "text", "coordinates": [253, 73, 547, 88], "content": ", respectively. 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|
0705.1209.pdf | 16 | suggested in (Guyon and Elisseeff, 2003). The rankings of the effects of variables on the MID for
NN and SVM vary as shown in table III. The reason for the variance may be because in the case of
NN the chain effect of changes in one variable to the other variables is accounted for (as the
variables are believed to be highly interdependent (Beck, King, and Zeng, 2000) as opposed to the
case of SVM where the effect of one variable is considered separately.
# TABLE III
# V. CONCLUSION
In this paper two artificial intelligence techniques, neural networks (NN) and support vector
machine (SVM), are used to predict militarised interstate disputes. The independent/input variables
are Democracy, Allies, Contingency, Distance, Capability, Dependency and Major Power while the
dependent/ output variable is MID result which is either peace or conflict. A neural network trained
with scaled conjugate gradient algorithm and an SVM with a radial basis kernel function together
with grid-search and cross-validation techniques were employed to find the optimal model.
| <p>suggested in (Guyon and Elisseeff, 2003). The rankings of the effects of variables on the MID for
NN and SVM vary as shown in table III. The reason for the variance may be because in the case of
NN the chain effect of changes in one variable to the other variables is accounted for (as the
variables are believed to be highly interdependent (Beck, King, and Zeng, 2000) as opposed to the
case of SVM where the effect of one variable is considered separately.</p>
<h1>TABLE III</h1>
<h1>V. CONCLUSION</h1>
<p>In this paper two artificial intelligence techniques, neural networks (NN) and support vector
machine (SVM), are used to predict militarised interstate disputes. The independent/input variables
are Democracy, Allies, Contingency, Distance, Capability, Dependency and Major Power while the
dependent/ output variable is MID result which is either peace or conflict. A neural network trained
with scaled conjugate gradient algorithm and an SVM with a radial basis kernel function together
with grid-search and cross-validation techniques were employed to find the optimal model.</p>
| [{"type": "text", "coordinates": [64, 70, 549, 197], "content": "suggested in (Guyon and Elisseeff, 2003). The rankings of the effects of variables on the MID for\nNN and SVM vary as shown in table III. The reason for the variance may be because in the case of\nNN the chain effect of changes in one variable to the other variables is accounted for (as the\nvariables are believed to be highly interdependent (Beck, King, and Zeng, 2000) as opposed to the\ncase of SVM where the effect of one variable is considered separately.", "block_type": "text", "index": 1}, {"type": "title", "coordinates": [291, 236, 338, 249], "content": "TABLE III", "block_type": "title", "index": 2}, {"type": "table", "coordinates": [221, 279, 417, 436], "content": "", "block_type": "table", "index": 3}, {"type": "title", "coordinates": [268, 516, 361, 531], "content": "V. 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|
0705.1209.pdf | 17 | The results found show that SVM has better capacity in forecasting conflicts without effectively
affecting the correct peace prediction than NN. Two separate experiments were conducted to see
the influence of each variable to the MID outcome. The first one assigns each variable to its
possible highest value while keeping the rest to their possible lowest values. The NN results show
that both democracy level and capability ratio are able to influence the outcome to be peace. On the
other hand, none of the variables was able to influence the MID outcome to be conflict when all the
other variables were maximum. SVM was not able to pick the effects of the variable for this
experiment.
The second experiment assigns each variable to its possible highest or lowest value while
keeping the other variables fixed to their original values. The results agree with the previous
experiment. If we group the variables in terms of their effect and rank them, Democracy level and
capability ratio are first, contiguity, distance and alliance second and dependency, major power are
ranked third using NN. Although SVM performs better than NN, the results of NN are easier to be
interpreted in relation to variable influence.
# REFERENCES
Beck, N., King, G.and Zeng, L. (2000), Improving Quantitative Studies of International Conflict: A
Conjecture. American Political Science Review, vol. 94, no. 1, pp. 21–33, March 2000.
Bishop, C. (1995), Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press
Burges, C. (1998), A Tutorial on Support Vector Machines for Pattern Recognition., Data Mining
and Knowledge Discovery, vol. 2, no. 2, pp. 121–167,
Chen, D and J. Odobez. (2002), Comparison of Support Vector Machine and Neural Network for
Text Texture Verification, IDIAP-RR-02 19, IDIAP, Martigny, April 2002.
Ftp://ftp.idiap.ch/pub/reports/2002/rr-02-19.pdf.
| <p>The results found show that SVM has better capacity in forecasting conflicts without effectively
affecting the correct peace prediction than NN. Two separate experiments were conducted to see
the influence of each variable to the MID outcome. The first one assigns each variable to its
possible highest value while keeping the rest to their possible lowest values. The NN results show
that both democracy level and capability ratio are able to influence the outcome to be peace. On the
other hand, none of the variables was able to influence the MID outcome to be conflict when all the
other variables were maximum. SVM was not able to pick the effects of the variable for this
experiment.</p>
<p>The second experiment assigns each variable to its possible highest or lowest value while
keeping the other variables fixed to their original values. The results agree with the previous
experiment. If we group the variables in terms of their effect and rank them, Democracy level and
capability ratio are first, contiguity, distance and alliance second and dependency, major power are
ranked third using NN. Although SVM performs better than NN, the results of NN are easier to be
interpreted in relation to variable influence.</p>
<h1>REFERENCES</h1>
<p>Beck, N., King, G.and Zeng, L. (2000), Improving Quantitative Studies of International Conflict: A
Conjecture. American Political Science Review, vol. 94, no. 1, pp. 21–33, March 2000.
Bishop, C. (1995), Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press
Burges, C. (1998), A Tutorial on Support Vector Machines for Pattern Recognition., Data Mining
and Knowledge Discovery, vol. 2, no. 2, pp. 121–167,
Chen, D and J. Odobez. (2002), Comparison of Support Vector Machine and Neural Network for
Text Texture Verification, IDIAP-RR-02 19, IDIAP, Martigny, April 2002.
Ftp://ftp.idiap.ch/pub/reports/2002/rr-02-19.pdf.</p>
| [{"type": "text", "coordinates": [65, 70, 548, 281], "content": "The results found show that SVM has better capacity in forecasting conflicts without effectively\naffecting the correct peace prediction than NN. Two separate experiments were conducted to see\nthe influence of each variable to the MID outcome. The first one assigns each variable to its\npossible highest value while keeping the rest to their possible lowest values. The NN results show\nthat both democracy level and capability ratio are able to influence the outcome to be peace. On the\nother hand, none of the variables was able to influence the MID outcome to be conflict when all the\nother variables were maximum. SVM was not able to pick the effects of the variable for this\nexperiment.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [65, 291, 548, 445], "content": "The second experiment assigns each variable to its possible highest or lowest value while\nkeeping the other variables fixed to their original values. The results agree with the previous\nexperiment. If we group the variables in terms of their effect and rank them, Democracy level and\ncapability ratio are first, contiguity, distance and alliance second and dependency, major power are\nranked third using NN. Although SVM performs better than NN, the results of NN are easier to be\ninterpreted in relation to variable influence.", "block_type": "text", "index": 2}, {"type": "title", "coordinates": [267, 513, 344, 527], "content": "REFERENCES", "block_type": "title", "index": 3}, {"type": "text", "coordinates": [64, 553, 547, 709], "content": "Beck, N., King, G.and Zeng, L. (2000), Improving Quantitative Studies of International Conflict: A\nConjecture. American Political Science Review, vol. 94, no. 1, pp. 21\u201333, March 2000.\nBishop, C. (1995), Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press\nBurges, C. (1998), A Tutorial on Support Vector Machines for Pattern Recognition., Data Mining\nand Knowledge Discovery, vol. 2, no. 2, pp. 121\u2013167,\nChen, D and J. Odobez. (2002), Comparison of Support Vector Machine and Neural Network for\nText Texture Verification, IDIAP-RR-02 19, IDIAP, Martigny, April 2002.\nFtp://ftp.idiap.ch/pub/reports/2002/rr-02-19.pdf.", "block_type": "text", "index": 4}] | [{"type": "text", "coordinates": [82, 73, 546, 89], "content": "The results found show that SVM has better capacity in forecasting conflicts without effectively", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [66, 101, 547, 116], "content": "affecting the correct peace prediction than NN. Two separate experiments were conducted to see", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [65, 127, 547, 144], "content": "the influence of each variable to the MID outcome. The first one assigns each variable to its", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [65, 156, 546, 172], "content": "possible highest value while keeping the rest to their possible lowest values. The NN results show", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [65, 184, 547, 199], "content": "that both democracy level and capability ratio are able to influence the outcome to be peace. On the", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [66, 211, 546, 225], "content": "other hand, none of the variables was able to influence the MID outcome to be conflict when all the", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [65, 238, 547, 254], "content": "other variables were maximum. 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If we group the variables in terms of their effect and rank them, Democracy level and", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [66, 377, 547, 392], "content": "capability ratio are first, contiguity, distance and alliance second and dependency, major power are", "score": 1.0, "index": 12}, {"type": "text", "coordinates": [65, 404, 547, 419], "content": "ranked third using NN. Although SVM performs better than NN, the results of NN are easier to be", "score": 1.0, "index": 13}, {"type": "text", "coordinates": [64, 431, 275, 447], "content": "interpreted in relation to variable influence.", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [267, 514, 344, 526], "content": "REFERENCES", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [64, 555, 547, 572], "content": "Beck, N., King, G.and Zeng, L. (2000), Improving Quantitative Studies of International Conflict: A", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [65, 570, 486, 585], "content": "Conjecture. American Political Science Review, vol. 94, no. 1, pp. 21\u201333, March 2000.", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [65, 598, 543, 613], "content": "Bishop, C. (1995), Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [64, 623, 547, 642], "content": "Burges, C. (1998), A Tutorial on Support Vector Machines for Pattern Recognition., Data Mining", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [66, 639, 326, 654], "content": "and Knowledge Discovery, vol. 2, no. 2, pp. 121\u2013167,", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [64, 666, 547, 681], "content": "Chen, D and J. Odobez. (2002), Comparison of Support Vector Machine and Neural Network for", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [64, 680, 309, 695], "content": "Text Texture Verification, IDIAP-RR-02", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [312, 680, 545, 697], "content": "19, IDIAP, Martigny, April 2002.", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [65, 695, 298, 710], "content": "Ftp://ftp.idiap.ch/pub/reports/2002/rr-02-19.pdf.", "score": 1.0, "index": 24}] | [] | [] | [] | [612.0, 792.0] | [{"type": "text", "text": "The results found show that SVM has better capacity in forecasting conflicts without effectively affecting the correct peace prediction than NN. Two separate experiments were conducted to see the influence of each variable to the MID outcome. The first one assigns each variable to its possible highest value while keeping the rest to their possible lowest values. The NN results show that both democracy level and capability ratio are able to influence the outcome to be peace. On the other hand, none of the variables was able to influence the MID outcome to be conflict when all the other variables were maximum. SVM was not able to pick the effects of the variable for this experiment. ", "page_idx": 17}, {"type": "text", "text": "The second experiment assigns each variable to its possible highest or lowest value while keeping the other variables fixed to their original values. The results agree with the previous experiment. If we group the variables in terms of their effect and rank them, Democracy level and capability ratio are first, contiguity, distance and alliance second and dependency, major power are ranked third using NN. Although SVM performs better than NN, the results of NN are easier to be interpreted in relation to variable influence. ", "page_idx": 17}, {"type": "text", "text": "REFERENCES ", "text_level": 1, "page_idx": 17}, {"type": "text", "text": "Beck, N., King, G.and Zeng, L. (2000), Improving Quantitative Studies of International Conflict: A Conjecture. American Political Science Review, vol. 94, no. 1, pp. 21\u201333, March 2000. \nBishop, C. (1995), Neural Networks for Pattern Recognition. Oxford, UK: Oxford University Press Burges, C. (1998), A Tutorial on Support Vector Machines for Pattern Recognition., Data Mining and Knowledge Discovery, vol. 2, no. 2, pp. 121\u2013167, \nChen, D and J. Odobez. (2002), Comparison of Support Vector Machine and Neural Network for Text Texture Verification, IDIAP-RR-02 19, IDIAP, Martigny, April 2002. Ftp://ftp.idiap.ch/pub/reports/2002/rr-02-19.pdf. COW, (2004), Correlates of War Poject, http://www.umich.edu/ cowproj/index.html, Last date accessed: Sept, 2004. \nGochman, C and Z. Maoz. (1984), Militarized Interstate Disputes 1816-1976, Journal of Conflict Resolution, vol. 28, no. 4, pp. 585\u2013615 \nGuyon, I. and A. Elisseeff, (2003), An introduction to variable and feature selection, Journal of Machine Learning Research, vol. 3, pp. 1157\u2013 1182, 2003. \nHanley, J. and B. McNeil, (1983), A Method of Comparing the Areas under the Receiver Operating Characteristic Curves Derived from the Same Cases, Radiology, vol. 148, no. 3 \nHaykin, S. (1999), Neural Networks: A Comprehensive Foundation. Upper Saddle River, New Jersey 07458: Prentice Hall International, Inc, second edition. \nLagazio, M and B. Russett, (2003), A Neural Network Analysis of Militarised Disputes, 1885- 1992: Temporal Stability and Causal Complexity, University of Michigan Press \nMarwala, T. and M. Lagazio, (2004), Modelling and Controlling Interstate Conflict, Budapest, Hungary: IEEE International Joint Conference on Neural networks, 2004. \nMoller, M. (1993), A scaled Conjugate Gradient Algorithm for Fast Supervised Learning, Neural Networks, vol. 6, no. 4, pp. 525\u2013533, \nMuller K.R, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, (2001), An Introduction to KernelBased Learning Algorithms, IEEE Transactions on Neural Networks, vol. 12, no. 2, March 2001 Oneal, J. and B. Russett. (2001), Clear and Cean: The Fixed Effects of Liberal Peace, International Organization, vol 52, no. 2, pp. 469\u201385, \nOneal, J. and B. Russett, (1999), The Kantian Peace: The Pacific Benefits of Democracy, Interdependence, and International Organization, World Politics, vol. 52, no. 1, pp. 1\u201337 \nPires, M. and T. Marwala, (2004), Option pricing using neural networks and support vector machines, In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics. The Hague, Holland: IEEE Computer Society TCC, pp. 161-166. \nRussett, B. and J. Oneal, (2001), Triangulating peace: Democracy, Interdependence, and International Organizations. New York: W.W. Norton. \nSch\u00f6lkopf, B and A. J. Smola, (2003), A short introduction to learning with kernels. pp. 41\u201364. Vapnik, V. and A. 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|
0705.1209.pdf | 18 | COW, (2004), Correlates of War Poject, http://www.umich.edu/ cowproj/index.html, Last date
accessed: Sept, 2004.
Gochman, C and Z. Maoz. (1984), Militarized Interstate Disputes 1816-1976, Journal of Conflict
Resolution, vol. 28, no. 4, pp. 585–615
Guyon, I. and A. Elisseeff, (2003), An introduction to variable and feature selection, Journal of
Machine Learning Research, vol. 3, pp. 1157– 1182, 2003.
Hanley, J. and B. McNeil, (1983), A Method of Comparing the Areas under the Receiver Operating
Characteristic Curves Derived from the Same Cases, Radiology, vol. 148, no. 3
Haykin, S. (1999), Neural Networks: A Comprehensive Foundation. Upper Saddle River, New
Jersey 07458: Prentice Hall International, Inc, second edition.
Lagazio, M and B. Russett, (2003), A Neural Network Analysis of Militarised Disputes, 1885-
1992: Temporal Stability and Causal Complexity, University of Michigan Press
Marwala, T. and M. Lagazio, (2004), Modelling and Controlling Interstate Conflict, Budapest,
Hungary: IEEE International Joint Conference on Neural networks, 2004.
Moller, M. (1993), A scaled Conjugate Gradient Algorithm for Fast Supervised Learning, Neural
Networks, vol. 6, no. 4, pp. 525–533,
Muller K.R, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, (2001), An Introduction to Kernel-
Based Learning Algorithms, IEEE Transactions on Neural Networks, vol. 12, no. 2, March 2001
Oneal, J. and B. Russett. (2001), Clear and Cean: The Fixed Effects of Liberal Peace, International
Organization, vol 52, no. 2, pp. 469–85,
Oneal, J. and B. Russett, (1999), The Kantian Peace: The Pacific Benefits of Democracy,
Interdependence, and International Organization, World Politics, vol. 52, no. 1, pp. 1–37
Pires, M. and T. Marwala, (2004), Option pricing using neural networks and support vector
machines, In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics.
The Hague, Holland: IEEE Computer Society TCC, pp. 161-166.
Russett, B. and J. Oneal, (2001), Triangulating peace: Democracy, Interdependence, and
International Organizations. New York: W.W. Norton.
Schölkopf, B and A. J. Smola, (2003), A short introduction to learning with kernels. pp. 41–64.
Vapnik, V. and A. Lerner, (1963), Pattern recognition using generalized portrait method,
Automation and Remote Control, vol. 24, pp. 774–780, 1963.
| <p>COW, (2004), Correlates of War Poject, http://www.umich.edu/ cowproj/index.html, Last date
accessed: Sept, 2004.
Gochman, C and Z. Maoz. (1984), Militarized Interstate Disputes 1816-1976, Journal of Conflict
Resolution, vol. 28, no. 4, pp. 585–615
Guyon, I. and A. Elisseeff, (2003), An introduction to variable and feature selection, Journal of
Machine Learning Research, vol. 3, pp. 1157– 1182, 2003.
Hanley, J. and B. McNeil, (1983), A Method of Comparing the Areas under the Receiver Operating
Characteristic Curves Derived from the Same Cases, Radiology, vol. 148, no. 3
Haykin, S. (1999), Neural Networks: A Comprehensive Foundation. Upper Saddle River, New
Jersey 07458: Prentice Hall International, Inc, second edition.
Lagazio, M and B. Russett, (2003), A Neural Network Analysis of Militarised Disputes, 1885-
1992: Temporal Stability and Causal Complexity, University of Michigan Press
Marwala, T. and M. Lagazio, (2004), Modelling and Controlling Interstate Conflict, Budapest,
Hungary: IEEE International Joint Conference on Neural networks, 2004.
Moller, M. (1993), A scaled Conjugate Gradient Algorithm for Fast Supervised Learning, Neural
Networks, vol. 6, no. 4, pp. 525–533,
Muller K.R, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, (2001), An Introduction to Kernel-
Based Learning Algorithms, IEEE Transactions on Neural Networks, vol. 12, no. 2, March 2001
Oneal, J. and B. Russett. (2001), Clear and Cean: The Fixed Effects of Liberal Peace, International
Organization, vol 52, no. 2, pp. 469–85,
Oneal, J. and B. Russett, (1999), The Kantian Peace: The Pacific Benefits of Democracy,
Interdependence, and International Organization, World Politics, vol. 52, no. 1, pp. 1–37
Pires, M. and T. Marwala, (2004), Option pricing using neural networks and support vector
machines, In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics.
The Hague, Holland: IEEE Computer Society TCC, pp. 161-166.
Russett, B. and J. Oneal, (2001), Triangulating peace: Democracy, Interdependence, and
International Organizations. New York: W.W. Norton.
Schölkopf, B and A. J. Smola, (2003), A short introduction to learning with kernels. pp. 41–64.
Vapnik, V. and A. Lerner, (1963), Pattern recognition using generalized portrait method,</p>
<p>Automation and Remote Control, vol. 24, pp. 774–780, 1963.</p>
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|
0705.1209.pdf | 19 | Vapnik, V (1995), The Nature of Statistical Learning Theory. New York: Springer Verlag
Westin, L (2001), Receiver operating characteristic (ROC) analysis: Evaluating discriminance
effects among decision support systems, Tech. Rep. UMINF 01.18, ISSN-0348-0542, Department
of Computer Science, Umea University, SE-90187, Umea, Sweden,
http://www.cs.umu.se/research/reports/2001/018/part1.pdf.
Zeng, L. (1999), Prediction and Classification With Neural Network Models, Sociological Methods
& Research, vol. 27, no. 4, pp. 499–524
Zweig, M.H and G. Campbell, (1993), Receiver-operating characteristic (ROC) plots: a
fundamental evaluation tool in clinical medicine, Clinical Chemistry, vol. 39, no. 4, pp. 561–577.
# Biographies
Eyasu Hayemariam is an MSc in Electrical Engineering student at the University of the
Witwatersrand in South Africa. He graduated with a BSc Degree in Statistics at the University of
Osmara in Eritrea
Tshilidzi Marwala received a BS in Mechanical Engineering at Case Western Reserve University in
Ohio, an MSc in Mechanical Engineering from University of Pretoria and a PhD in Artificial
Intelligence from the University of Cambridge. Previously he was a post-doctoral fellow at
Imperial College (London). He is currently a professor at the University of the Witwatersrand in
South Africa.
Monica Lagazio holds a PhD in Politics and Artificial Intelligence from Nottingham University and
an MA in Politics from the University of London. Before joining the University of Kent at
Canterbury in 2004, she was Lecturer in Politics at the University of the Witwatersrand and
Research Fellow at Yale University. She also held a position of senior consultant in the economic
and financial service of one of the leading global consulting companies in London.
| <p>Vapnik, V (1995), The Nature of Statistical Learning Theory. New York: Springer Verlag</p>
<p>Westin, L (2001), Receiver operating characteristic (ROC) analysis: Evaluating discriminance
effects among decision support systems, Tech. Rep. UMINF 01.18, ISSN-0348-0542, Department
of Computer Science, Umea University, SE-90187, Umea, Sweden,
http://www.cs.umu.se/research/reports/2001/018/part1.pdf.</p>
<p>Zeng, L. (1999), Prediction and Classification With Neural Network Models, Sociological Methods
& Research, vol. 27, no. 4, pp. 499–524</p>
<p>Zweig, M.H and G. Campbell, (1993), Receiver-operating characteristic (ROC) plots: a
fundamental evaluation tool in clinical medicine, Clinical Chemistry, vol. 39, no. 4, pp. 561–577.</p>
<h1>Biographies</h1>
<p>Eyasu Hayemariam is an MSc in Electrical Engineering student at the University of the
Witwatersrand in South Africa. He graduated with a BSc Degree in Statistics at the University of
Osmara in Eritrea</p>
<p>Tshilidzi Marwala received a BS in Mechanical Engineering at Case Western Reserve University in
Ohio, an MSc in Mechanical Engineering from University of Pretoria and a PhD in Artificial
Intelligence from the University of Cambridge. Previously he was a post-doctoral fellow at
Imperial College (London). He is currently a professor at the University of the Witwatersrand in
South Africa.</p>
<p>Monica Lagazio holds a PhD in Politics and Artificial Intelligence from Nottingham University and
an MA in Politics from the University of London. Before joining the University of Kent at
Canterbury in 2004, she was Lecturer in Politics at the University of the Witwatersrand and
Research Fellow at Yale University. She also held a position of senior consultant in the economic
and financial service of one of the leading global consulting companies in London.</p>
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|
0704.1394.pdf | 0 | # Calculating Valid Domains for BDD-Based Interactive
Configuration
Tarik Hadzic, Rune Moller Jensen, Henrik Reif Andersen
Computational Logic and Algorithms Group,
IT University of Copenhagen, Denmark
[email protected],[email protected],[email protected]
Abstract. In these notes we formally describe the functionality of Calculating
Valid Domains from the BDD representing the solution space of valid configu-
rations. The formalization is largely based on the CLab [1] configuration frame-
work.
# 1 Introduction
Interactive configuration problems are special applications of Constraint Satisfaction
Problems (CSP) where a user is assisted in interactively assigning values to variables by
a software tool. This software, called a configurator, assists the user by calculating and
displaying the available, valid choices for each unassigned variable in what are called
valid domains computations. Application areas include customising physical products
(such as PC’s and cars) and services (such as airplane tickets and insurances).
Three important features are required of a tool that implements interactive configu-
ration: it should be complete (all valid configurations should be reachable through user
interaction), backtrack-free (a user is never forced to change an earlier choice due to
incompleteness in the logical deductions), and it should provide real-time performance
(feedback should be fast enough to allow real-time interactions). The requirement of
obtaining backtrack-freeness while maintaining completeness makes the problem of
calculating valid domains NP-hard. The real-time performance requirement enforces
further that runtime calculations are bounded in polynomial time. According to user-
interface design criteria, for a user to perceive interaction as being real-time, system
response needs to be within about 250 milliseconds in practice [2]. Therefore, the cur-
rent approaches that meet all three conditions use off-line precomputation to generate
an efficient runtime data structure representing the solution space [3,4,5,6]. The chal-
lenge with this data structure is that the solution space is almost always exponentially
large and it is NP-hard to find. Despite the bad worst-case bounds, it has nevertheless
turned out in real industrial applications that the data structures can often be kept small
[7,5,4].
# 2 Interactive Configuration
The input model to an interactive configuration problem is a special kind of Constraint
Satisfaction Problem (CSP) [8,9] where constraints are represented as propositional
formulas:
| <h1>Calculating Valid Domains for BDD-Based Interactive
Configuration</h1>
<p>Tarik Hadzic, Rune Moller Jensen, Henrik Reif Andersen</p>
<p>Computational Logic and Algorithms Group,
IT University of Copenhagen, Denmark
[email protected],[email protected],[email protected]</p>
<p>Abstract. In these notes we formally describe the functionality of Calculating
Valid Domains from the BDD representing the solution space of valid configu-
rations. The formalization is largely based on the CLab [1] configuration frame-
work.</p>
<h1>1 Introduction</h1>
<p>Interactive configuration problems are special applications of Constraint Satisfaction
Problems (CSP) where a user is assisted in interactively assigning values to variables by
a software tool. This software, called a configurator, assists the user by calculating and
displaying the available, valid choices for each unassigned variable in what are called
valid domains computations. Application areas include customising physical products
(such as PC’s and cars) and services (such as airplane tickets and insurances).</p>
<p>Three important features are required of a tool that implements interactive configu-
ration: it should be complete (all valid configurations should be reachable through user
interaction), backtrack-free (a user is never forced to change an earlier choice due to
incompleteness in the logical deductions), and it should provide real-time performance
(feedback should be fast enough to allow real-time interactions). The requirement of
obtaining backtrack-freeness while maintaining completeness makes the problem of
calculating valid domains NP-hard. The real-time performance requirement enforces
further that runtime calculations are bounded in polynomial time. According to user-
interface design criteria, for a user to perceive interaction as being real-time, system
response needs to be within about 250 milliseconds in practice [2]. Therefore, the cur-
rent approaches that meet all three conditions use off-line precomputation to generate
an efficient runtime data structure representing the solution space [3,4,5,6]. The chal-
lenge with this data structure is that the solution space is almost always exponentially
large and it is NP-hard to find. Despite the bad worst-case bounds, it has nevertheless
turned out in real industrial applications that the data structures can often be kept small
[7,5,4].</p>
<h1>2 Interactive Configuration</h1>
<p>The input model to an interactive configuration problem is a special kind of Constraint
Satisfaction Problem (CSP) [8,9] where constraints are represented as propositional
formulas:</p>
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In these notes we formally describe the functionality of Calculating Valid Domains from the BDD representing the solution space of valid configurations. The formalization is largely based on the CLab [1] configuration framework. ", "page_idx": 0}, {"type": "text", "text": "1 Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Interactive configuration problems are special applications of Constraint Satisfaction Problems (CSP) where a user is assisted in interactively assigning values to variables by a software tool. This software, called a configurator, assists the user by calculating and displaying the available, valid choices for each unassigned variable in what are called valid domains computations. Application areas include customising physical products (such as PC\u2019s and cars) and services (such as airplane tickets and insurances). ", "page_idx": 0}, {"type": "text", "text": "Three important features are required of a tool that implements interactive configuration: it should be complete (all valid configurations should be reachable through user interaction), backtrack-free (a user is never forced to change an earlier choice due to incompleteness in the logical deductions), and it should provide real-time performance (feedback should be fast enough to allow real-time interactions). The requirement of obtaining backtrack-freeness while maintaining completeness makes the problem of calculating valid domains NP-hard. The real-time performance requirement enforces further that runtime calculations are bounded in polynomial time. According to userinterface design criteria, for a user to perceive interaction as being real-time, system response needs to be within about 250 milliseconds in practice [2]. Therefore, the current approaches that meet all three conditions use off-line precomputation to generate an efficient runtime data structure representing the solution space [3,4,5,6]. The challenge with this data structure is that the solution space is almost always exponentially large and it is NP-hard to find. Despite the bad worst-case bounds, it has nevertheless turned out in real industrial applications that the data structures can often be kept small [7,5,4]. 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In these notes we formally describe the functionality of Calculating", "type": "text"}], "index": 6}, {"bbox": [162, 257, 451, 268], "spans": [{"bbox": [162, 257, 451, 268], "score": 1.0, "content": "Valid Domains from the BDD representing the solution space of valid configu-", "type": "text"}], "index": 7}, {"bbox": [161, 268, 452, 280], "spans": [{"bbox": [161, 268, 452, 280], "score": 1.0, "content": "rations. The formalization is largely based on the CLab [1] configuration frame-", "type": "text"}], "index": 8}, {"bbox": [162, 280, 183, 289], "spans": [{"bbox": [162, 280, 183, 289], "score": 1.0, "content": "work.", "type": "text"}], "index": 9}], "index": 7.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [161, 245, 452, 289]}, {"type": "title", "bbox": [134, 305, 218, 320], "lines": [{"bbox": [134, 307, 218, 319], "spans": [{"bbox": [134, 308, 142, 317], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [150, 307, 218, 319], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [133, 328, 482, 400], "lines": [{"bbox": [133, 330, 480, 342], "spans": [{"bbox": [133, 330, 480, 342], "score": 1.0, "content": "Interactive configuration problems are special applications of Constraint Satisfaction", "type": "text"}], "index": 11}, {"bbox": [133, 341, 480, 354], "spans": [{"bbox": [133, 341, 480, 354], "score": 1.0, "content": "Problems (CSP) where a user is assisted in interactively assigning values to variables by", "type": "text"}], "index": 12}, {"bbox": [134, 354, 481, 365], "spans": [{"bbox": [134, 354, 481, 365], "score": 1.0, "content": "a software tool. 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|
0704.1394.pdf | 1 | Definition 1. $$A$$ configuration model $$C$$ is a triple $$(X,D,F)$$ where $$X$$ is a set of vari-
ables $$\{x_{0},\ldots,x_{n-1}\}$$ , $$D=D_{0}\times...\times D_{n-1}$$ is the Cartesian product of their finite
domains $$D_{0},\ldots,D_{n-1}$$ and $$F=\{f_{0},...,f_{m-1}\}$$ is a set of propositional formulae over
atomic propositions $$x_{i}=v$$ , where $$v\,\in\,D_{i}$$ , specifying conditions on the values of the
variables.
Concretely, every domain can be defined as $$D_{i}\,=\,\{0,\dots,|D_{i}|\,-\,1\}$$ . An assign-
ment of values $$v_{0},\ldots,v_{n-1}$$ to variables $$x_{0},\ldots,x_{n-1}$$ is denoted as an assignment
$$\rho\,=\,\left\{\left(x_{0},v_{0}\right),\ldots,\left(x_{n-1},v_{n-1}\right)\right\}$$ . Domain of assignment $$d o m(\rho)$$ is the set of vari-
ables which are assigned: $$d o m(\rho)=\{x_{i}\mid\exists v\in D_{i}.(x_{i},v)\in\rho\}$$ and if $$d o m(\rho)=X$$
we refer to $$\rho$$ as a total assignment. We say that a total assignment $$\rho$$ is valid, if it satisfies
all the rules which is denoted as $${\boldsymbol\rho}\vDash F$$ .
A partial assignment $$\rho^{\prime},d o m(\rho^{\prime})\subseteq X$$ is valid if there is at least one total assign-
ment $$\rho\supseteq\rho^{\prime}$$ that is valid $${\boldsymbol{\rho}}\vDash F$$ , i.e. if there is at least one way to successfully finish
the existing configuration process.
Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or
blue), the size (small, medium, or large) and the print (”Men In Black” - MIB or ”Save
The Whales” - STW). There are two rules that we have to observe: if we choose the
MIB print then the color black has to be chosen as well, and if we choose the small size
then the STW print (including a big picture of a whale) cannot be selected as the large
whale does not fit on the small shirt. The configuration problem $$(X,D,F)$$ of the $$\scriptstyle\mathrm{\mathrm{~T~}}$$
shirt example consists of variables $$X=\{x_{1},x_{2},x_{3}\}$$ representing color, size and print.
Variable domains are $$D_{1}=\{b l a c k$$ , white, red, blue}, $$D_{2}=\{s m a l l,m e d i u m,l a r g e\}$$ ,
and $${\cal D}_{3}~=~\{M I B,S T W\}$$ . The two rules translate to $$F\;=\;\{f_{1},f_{2}\}$$ , where $$f_{1}~=$$
$$(x_{3}\,=\,M I B)\,\Rightarrow\,(x_{1}\,=\,b l a c k)$$ and $$f_{2}\,=\,(x_{3}\,=\,S T W)\,\Rightarrow\,(x_{2}\,\neq\,s m a l l).$$ There
are $$|D_{1}||D_{2}||D_{3}|\,=\,24$$ possible assignments. Eleven of these assignments are valid
configurations and they form the solution space shown in Fig. 1. $$\diamondsuit$$
# 2.1 User Interaction
Configurator assists a user interactively to reach a valid product specification, i.e. to
reach total valid assignment. The key operation in this interaction is that of computing,
for each unassigned variable $$x_{i}\in X\backslash d o m(\rho)$$ , the valid domain $$D_{i}^{\rho}\subseteq D_{i}$$ . The domain
is valid if it contains those and only those values with which $$\rho$$ can be extended to be-
come a total valid assignment, i.e. $$D_{i}^{\rho}=\{v\in D_{i}\mid\exists\rho^{\prime}:\rho^{\prime}\models F\land\rho\cup\{(x_{i},v)\}\subseteq\rho^{\prime}\}$$ .
| <p>Definition 1. $$A$$ configuration model $$C$$ is a triple $$(X,D,F)$$ where $$X$$ is a set of vari-
ables $$\{x_{0},\ldots,x_{n-1}\}$$ , $$D=D_{0}\times...\times D_{n-1}$$ is the Cartesian product of their finite
domains $$D_{0},\ldots,D_{n-1}$$ and $$F=\{f_{0},...,f_{m-1}\}$$ is a set of propositional formulae over
atomic propositions $$x_{i}=v$$ , where $$v\,\in\,D_{i}$$ , specifying conditions on the values of the
variables.</p>
<p>Concretely, every domain can be defined as $$D_{i}\,=\,\{0,\dots,|D_{i}|\,-\,1\}$$ . An assign-
ment of values $$v_{0},\ldots,v_{n-1}$$ to variables $$x_{0},\ldots,x_{n-1}$$ is denoted as an assignment
$$\rho\,=\,\left\{\left(x_{0},v_{0}\right),\ldots,\left(x_{n-1},v_{n-1}\right)\right\}$$ . Domain of assignment $$d o m(\rho)$$ is the set of vari-
ables which are assigned: $$d o m(\rho)=\{x_{i}\mid\exists v\in D_{i}.(x_{i},v)\in\rho\}$$ and if $$d o m(\rho)=X$$
we refer to $$\rho$$ as a total assignment. We say that a total assignment $$\rho$$ is valid, if it satisfies
all the rules which is denoted as $${\boldsymbol\rho}\vDash F$$ .</p>
<p>A partial assignment $$\rho^{\prime},d o m(\rho^{\prime})\subseteq X$$ is valid if there is at least one total assign-
ment $$\rho\supseteq\rho^{\prime}$$ that is valid $${\boldsymbol{\rho}}\vDash F$$ , i.e. if there is at least one way to successfully finish
the existing configuration process.</p>
<p>Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or
blue), the size (small, medium, or large) and the print (”Men In Black” - MIB or ”Save
The Whales” - STW). There are two rules that we have to observe: if we choose the
MIB print then the color black has to be chosen as well, and if we choose the small size
then the STW print (including a big picture of a whale) cannot be selected as the large
whale does not fit on the small shirt. The configuration problem $$(X,D,F)$$ of the $$\scriptstyle\mathrm{\mathrm{~T~}}$$
shirt example consists of variables $$X=\{x_{1},x_{2},x_{3}\}$$ representing color, size and print.
Variable domains are $$D_{1}=\{b l a c k$$ , white, red, blue}, $$D_{2}=\{s m a l l,m e d i u m,l a r g e\}$$ ,
and $${\cal D}_{3}~=~\{M I B,S T W\}$$ . The two rules translate to $$F\;=\;\{f_{1},f_{2}\}$$ , where $$f_{1}~=$$
$$(x_{3}\,=\,M I B)\,\Rightarrow\,(x_{1}\,=\,b l a c k)$$ and $$f_{2}\,=\,(x_{3}\,=\,S T W)\,\Rightarrow\,(x_{2}\,\neq\,s m a l l).$$ There
are $$|D_{1}||D_{2}||D_{3}|\,=\,24$$ possible assignments. Eleven of these assignments are valid
configurations and they form the solution space shown in Fig. 1. $$\diamondsuit$$</p>
<h1>2.1 User Interaction</h1>
<p>Configurator assists a user interactively to reach a valid product specification, i.e. to
reach total valid assignment. The key operation in this interaction is that of computing,
for each unassigned variable $$x_{i}\in X\backslash d o m(\rho)$$ , the valid domain $$D_{i}^{\rho}\subseteq D_{i}$$ . The domain
is valid if it contains those and only those values with which $$\rho$$ can be extended to be-
come a total valid assignment, i.e. $$D_{i}^{\rho}=\{v\in D_{i}\mid\exists\rho^{\prime}:\rho^{\prime}\models F\land\rho\cup\{(x_{i},v)\}\subseteq\rho^{\prime}\}$$ .</p>
| [{"type": "text", "coordinates": [133, 115, 482, 175], "content": "Definition 1. $$A$$ configuration model $$C$$ is a triple $$(X,D,F)$$ where $$X$$ is a set of vari-\nables $$\\{x_{0},\\ldots,x_{n-1}\\}$$ , $$D=D_{0}\\times...\\times D_{n-1}$$ is the Cartesian product of their finite\ndomains $$D_{0},\\ldots,D_{n-1}$$ and $$F=\\{f_{0},...,f_{m-1}\\}$$ is a set of propositional formulae over\natomic propositions $$x_{i}=v$$ , where $$v\\,\\in\\,D_{i}$$ , specifying conditions on the values of the\nvariables.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [133, 184, 481, 256], "content": "Concretely, every domain can be defined as $$D_{i}\\,=\\,\\{0,\\dots,|D_{i}|\\,-\\,1\\}$$ . An assign-\nment of values $$v_{0},\\ldots,v_{n-1}$$ to variables $$x_{0},\\ldots,x_{n-1}$$ is denoted as an assignment\n$$\\rho\\,=\\,\\left\\{\\left(x_{0},v_{0}\\right),\\ldots,\\left(x_{n-1},v_{n-1}\\right)\\right\\}$$ . Domain of assignment $$d o m(\\rho)$$ is the set of vari-\nables which are assigned: $$d o m(\\rho)=\\{x_{i}\\mid\\exists v\\in D_{i}.(x_{i},v)\\in\\rho\\}$$ and if $$d o m(\\rho)=X$$\nwe refer to $$\\rho$$ as a total assignment. We say that a total assignment $$\\rho$$ is valid, if it satisfies\nall the rules which is denoted as $${\\boldsymbol\\rho}\\vDash F$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [134, 257, 481, 292], "content": "A partial assignment $$\\rho^{\\prime},d o m(\\rho^{\\prime})\\subseteq X$$ is valid if there is at least one total assign-\nment $$\\rho\\supseteq\\rho^{\\prime}$$ that is valid $${\\boldsymbol{\\rho}}\\vDash F$$ , i.e. if there is at least one way to successfully finish\nthe existing configuration process.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [133, 300, 482, 444], "content": "Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or\nblue), the size (small, medium, or large) and the print (\u201dMen In Black\u201d - MIB or \u201dSave\nThe Whales\u201d - STW). There are two rules that we have to observe: if we choose the\nMIB print then the color black has to be chosen as well, and if we choose the small size\nthen the STW print (including a big picture of a whale) cannot be selected as the large\nwhale does not fit on the small shirt. The configuration problem $$(X,D,F)$$ of the $$\\scriptstyle\\mathrm{\\mathrm{~T~}}$$\nshirt example consists of variables $$X=\\{x_{1},x_{2},x_{3}\\}$$ representing color, size and print.\nVariable domains are $$D_{1}=\\{b l a c k$$ , white, red, blue}, $$D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}$$ ,\nand $${\\cal D}_{3}~=~\\{M I B,S T W\\}$$ . The two rules translate to $$F\\;=\\;\\{f_{1},f_{2}\\}$$ , where $$f_{1}~=$$\n$$(x_{3}\\,=\\,M I B)\\,\\Rightarrow\\,(x_{1}\\,=\\,b l a c k)$$ and $$f_{2}\\,=\\,(x_{3}\\,=\\,S T W)\\,\\Rightarrow\\,(x_{2}\\,\\neq\\,s m a l l).$$ There\nare $$|D_{1}||D_{2}||D_{3}|\\,=\\,24$$ possible assignments. Eleven of these assignments are valid\nconfigurations and they form the solution space shown in Fig. 1. $$\\diamondsuit$$", "block_type": "text", "index": 4}, {"type": "interline_equation", "coordinates": [150, 470, 464, 517], "content": "", "block_type": "interline_equation", "index": 5}, {"type": "image", "coordinates": [], "content": "", "block_type": "image", "index": 6}, {"type": "title", "coordinates": [134, 587, 228, 598], "content": "2.1 User Interaction", "block_type": "title", "index": 7}, {"type": "text", "coordinates": [134, 606, 481, 666], "content": "Configurator assists a user interactively to reach a valid product specification, i.e. to\nreach total valid assignment. The key operation in this interaction is that of computing,\nfor each unassigned variable $$x_{i}\\in X\\backslash d o m(\\rho)$$ , the valid domain $$D_{i}^{\\rho}\\subseteq D_{i}$$ . The domain\nis valid if it contains those and only those values with which $$\\rho$$ can be extended to be-\ncome a total valid assignment, i.e. $$D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}$$ .", "block_type": "text", "index": 8}] | [{"type": "text", "coordinates": [133, 117, 189, 129], "content": "Definition 1.", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [190, 117, 198, 126], "content": "A", "score": 0.28, "index": 2}, {"type": "text", "coordinates": [198, 117, 283, 129], "content": " configuration model", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [284, 117, 293, 127], "content": "C", "score": 0.82, "index": 4}, {"type": "text", "coordinates": [293, 117, 337, 129], "content": " is a triple ", "score": 1.0, "index": 5}, {"type": "inline_equation", "coordinates": [337, 117, 379, 128], "content": "(X,D,F)", "score": 0.93, "index": 6}, {"type": "text", "coordinates": [379, 117, 407, 129], "content": " where", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [408, 117, 416, 126], "content": "X", "score": 0.56, "index": 8}, {"type": "text", "coordinates": [417, 117, 480, 129], "content": " is a set of vari-", "score": 1.0, "index": 9}, {"type": "text", "coordinates": [133, 129, 158, 142], "content": "ables ", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [158, 128, 221, 141], "content": "\\{x_{0},\\ldots,x_{n-1}\\}", "score": 0.88, "index": 11}, {"type": "text", "coordinates": [222, 129, 226, 142], "content": ",", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [226, 129, 325, 140], "content": "D=D_{0}\\times...\\times D_{n-1}", "score": 0.89, "index": 13}, {"type": "text", "coordinates": [325, 129, 481, 142], "content": " is the Cartesian product of their finite", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [133, 141, 169, 154], "content": "domains", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [170, 141, 229, 152], "content": "D_{0},\\ldots,D_{n-1}", "score": 0.91, "index": 16}, {"type": "text", "coordinates": [229, 141, 247, 154], "content": " and", "score": 1.0, "index": 17}, {"type": "inline_equation", "coordinates": [248, 141, 328, 153], "content": "F=\\{f_{0},...,f_{m-1}\\}", "score": 0.9, "index": 18}, {"type": "text", "coordinates": [329, 141, 482, 154], "content": " is a set of propositional formulae over", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [133, 153, 215, 165], "content": "atomic propositions", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [216, 154, 245, 163], "content": "x_{i}=v", "score": 0.88, "index": 21}, {"type": "text", "coordinates": [246, 153, 276, 165], "content": ", where", "score": 1.0, "index": 22}, {"type": "inline_equation", "coordinates": [277, 153, 308, 163], "content": "v\\,\\in\\,D_{i}", "score": 0.88, "index": 23}, {"type": "text", "coordinates": [308, 153, 481, 165], "content": ", specifying conditions on the values of the", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [133, 165, 174, 177], "content": "variables.", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [149, 185, 329, 198], "content": "Concretely, every domain can be defined as", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [329, 184, 431, 196], "content": "D_{i}\\,=\\,\\{0,\\dots,|D_{i}|\\,-\\,1\\}", "score": 0.94, "index": 27}, {"type": "text", "coordinates": [431, 185, 480, 198], "content": ". An assign-", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [133, 196, 198, 210], "content": "ment of values ", "score": 1.0, "index": 29}, {"type": "inline_equation", "coordinates": [198, 198, 251, 208], "content": "v_{0},\\ldots,v_{n-1}", "score": 0.86, "index": 30}, {"type": "text", "coordinates": [252, 196, 305, 210], "content": " to variables ", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [306, 198, 360, 208], "content": "x_{0},\\ldots,x_{n-1}", "score": 0.9, "index": 32}, {"type": "text", "coordinates": [360, 196, 482, 210], "content": " is denoted as an assignment", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [133, 208, 274, 220], "content": "\\rho\\,=\\,\\left\\{\\left(x_{0},v_{0}\\right),\\ldots,\\left(x_{n-1},v_{n-1}\\right)\\right\\}", "score": 0.89, "index": 34}, {"type": "text", "coordinates": [274, 209, 374, 221], "content": ". Domain of assignment ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [374, 208, 407, 220], "content": "d o m(\\rho)", "score": 0.81, "index": 36}, {"type": "text", "coordinates": [407, 209, 481, 221], "content": " is the set of vari-", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [134, 221, 238, 232], "content": "ables which are assigned:", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [239, 220, 397, 232], "content": "d o m(\\rho)=\\{x_{i}\\mid\\exists v\\in D_{i}.(x_{i},v)\\in\\rho\\}", "score": 0.92, "index": 39}, {"type": "text", "coordinates": [397, 221, 424, 232], "content": " and if", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [425, 219, 481, 232], "content": "d o m(\\rho)=X", "score": 0.91, "index": 41}, {"type": "text", "coordinates": [133, 232, 177, 245], "content": "we refer to", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [178, 234, 184, 243], "content": "\\rho", "score": 0.81, "index": 43}, {"type": "text", "coordinates": [185, 232, 391, 245], "content": " as a total assignment. We say that a total assignment", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [392, 234, 398, 243], "content": "\\rho", "score": 0.83, "index": 45}, {"type": "text", "coordinates": [398, 232, 481, 245], "content": " is valid, if it satisfies", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [133, 245, 263, 257], "content": "all the rules which is denoted as", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [263, 244, 291, 256], "content": "{\\boldsymbol\\rho}\\vDash F", "score": 0.91, "index": 48}, {"type": "text", "coordinates": [292, 245, 295, 257], "content": ".", "score": 1.0, "index": 49}, {"type": "text", "coordinates": [149, 257, 235, 269], "content": "A partial assignment ", "score": 1.0, "index": 50}, {"type": "inline_equation", "coordinates": [235, 256, 306, 268], "content": "\\rho^{\\prime},d o m(\\rho^{\\prime})\\subseteq X", "score": 0.91, "index": 51}, {"type": "text", "coordinates": [307, 257, 480, 269], "content": " is valid if there is at least one total assign-", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [133, 268, 156, 281], "content": "ment", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [156, 268, 185, 280], "content": "\\rho\\supseteq\\rho^{\\prime}", "score": 0.92, "index": 54}, {"type": "text", "coordinates": [185, 268, 236, 281], "content": " that is valid", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [237, 268, 266, 280], "content": "{\\boldsymbol{\\rho}}\\vDash F", "score": 0.91, "index": 56}, {"type": "text", "coordinates": [267, 268, 480, 281], "content": ", i.e. if there is at least one way to successfully finish", "score": 1.0, "index": 57}, {"type": "text", "coordinates": [134, 280, 271, 293], "content": "the existing configuration process.", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [134, 301, 482, 312], "content": "Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or", "score": 1.0, "index": 59}, {"type": "text", "coordinates": [133, 312, 481, 324], "content": "blue), the size (small, medium, or large) and the print (\u201dMen In Black\u201d - MIB or \u201dSave", "score": 1.0, "index": 60}, {"type": "text", "coordinates": [133, 325, 481, 336], "content": "The Whales\u201d - STW). There are two rules that we have to observe: if we choose the", "score": 1.0, "index": 61}, {"type": "text", "coordinates": [133, 336, 481, 348], "content": "MIB print then the color black has to be chosen as well, and if we choose the small size", "score": 1.0, "index": 62}, {"type": "text", "coordinates": [133, 348, 481, 361], "content": "then the STW print (including a big picture of a whale) cannot be selected as the large", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [134, 361, 399, 372], "content": "whale does not fit on the small shirt. The configuration problem ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [399, 360, 441, 372], "content": "(X,D,F)", "score": 0.93, "index": 65}, {"type": "text", "coordinates": [442, 361, 470, 372], "content": " of the", "score": 1.0, "index": 66}, {"type": "inline_equation", "coordinates": [470, 360, 480, 370], "content": "\\scriptstyle\\mathrm{\\mathrm{~T~}}", "score": 0.32, "index": 67}, {"type": "text", "coordinates": [133, 372, 272, 385], "content": "shirt example consists of variables", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [273, 371, 345, 384], "content": "X=\\{x_{1},x_{2},x_{3}\\}", "score": 0.93, "index": 69}, {"type": "text", "coordinates": [345, 372, 481, 385], "content": " representing color, size and print.", "score": 1.0, "index": 70}, {"type": "text", "coordinates": [134, 385, 220, 396], "content": "Variable domains are", "score": 1.0, "index": 71}, {"type": "inline_equation", "coordinates": [220, 383, 275, 395], "content": "D_{1}=\\{b l a c k", "score": 0.91, "index": 72}, {"type": "text", "coordinates": [275, 385, 352, 396], "content": ", white, red, blue},", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [352, 383, 477, 395], "content": "D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}", "score": 0.71, "index": 74}, {"type": "text", "coordinates": [478, 385, 481, 396], "content": ",", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [133, 396, 151, 408], "content": "and", "score": 1.0, "index": 76}, {"type": "inline_equation", "coordinates": [152, 395, 245, 407], "content": "{\\cal D}_{3}~=~\\{M I B,S T W\\}", "score": 0.89, "index": 77}, {"type": "text", "coordinates": [245, 396, 361, 408], "content": ". The two rules translate to ", "score": 1.0, "index": 78}, {"type": "inline_equation", "coordinates": [362, 396, 423, 407], "content": "F\\;=\\;\\{f_{1},f_{2}\\}", "score": 0.92, "index": 79}, {"type": "text", "coordinates": [423, 396, 456, 408], "content": ", where ", "score": 1.0, "index": 80}, {"type": "inline_equation", "coordinates": [456, 396, 481, 407], "content": "f_{1}~=", "score": 0.89, "index": 81}, {"type": "inline_equation", "coordinates": [135, 408, 266, 419], "content": "(x_{3}\\,=\\,M I B)\\,\\Rightarrow\\,(x_{1}\\,=\\,b l a c k)", "score": 0.75, "index": 82}, {"type": "text", "coordinates": [267, 408, 286, 420], "content": " and", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [287, 407, 452, 419], "content": "f_{2}\\,=\\,(x_{3}\\,=\\,S T W)\\,\\Rightarrow\\,(x_{2}\\,\\neq\\,s m a l l).", "score": 0.84, "index": 84}, {"type": "text", "coordinates": [453, 408, 482, 420], "content": " There", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [133, 420, 149, 433], "content": "are ", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [149, 419, 232, 432], "content": "|D_{1}||D_{2}||D_{3}|\\,=\\,24", "score": 0.92, "index": 87}, {"type": "text", "coordinates": [232, 420, 482, 433], "content": " possible assignments. Eleven of these assignments are valid", "score": 1.0, "index": 88}, {"type": "text", "coordinates": [134, 433, 392, 444], "content": "configurations and they form the solution space shown in Fig. 1.", "score": 1.0, "index": 89}, {"type": "inline_equation", "coordinates": [473, 431, 481, 442], "content": "\\diamondsuit", "score": 0.68, "index": 90}, {"type": "interline_equation", "coordinates": [150, 470, 464, 517], "content": "{\\begin{array}{l l l l}{\\left(b l a c k,s m a l l,M I B\\right)}&&{\\left(b l a c k,l a r g e,S T W\\right)}&&{\\left(r e d,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,M I B\\right)}&&{\\left(w h i t e,m e d i u m,S T W\\right)}&&{\\left(b l u e,m e d i u m,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,S T W\\right)}&&{\\left(w h i t e,l a r g e,S T W\\right)}&&{\\left(b l u e,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,l a r g e,M I B\\right)}&&{\\left(r e d,m e d i u m,S T W\\right)}\\end{array}}", "score": 0.49, "index": 91}, {"type": "text", "coordinates": [133, 587, 228, 597], "content": "2.1 User Interaction", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [133, 606, 481, 618], "content": "Configurator assists a user interactively to reach a valid product specification, i.e. to", "score": 1.0, "index": 93}, {"type": "text", "coordinates": [133, 618, 481, 632], "content": "reach total valid assignment. The key operation in this interaction is that of computing,", "score": 1.0, "index": 94}, {"type": "text", "coordinates": [133, 631, 248, 642], "content": "for each unassigned variable", "score": 1.0, "index": 95}, {"type": "inline_equation", "coordinates": [248, 630, 317, 642], "content": "x_{i}\\in X\\backslash d o m(\\rho)", "score": 0.93, "index": 96}, {"type": "text", "coordinates": [318, 631, 389, 642], "content": ", the valid domain", "score": 1.0, "index": 97}, {"type": "inline_equation", "coordinates": [389, 629, 428, 642], "content": "D_{i}^{\\rho}\\subseteq D_{i}", "score": 0.93, "index": 98}, {"type": "text", "coordinates": [429, 631, 482, 642], "content": ". The domain", "score": 1.0, "index": 99}, {"type": "text", "coordinates": [133, 642, 380, 654], "content": "is valid if it contains those and only those values with which", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [380, 644, 387, 653], "content": "\\rho", "score": 0.8, "index": 101}, {"type": "text", "coordinates": [388, 642, 480, 654], "content": " can be extended to be-", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [133, 655, 268, 666], "content": "come a total valid assignment, i.e.", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [269, 653, 478, 666], "content": "D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}", "score": 0.88, "index": 104}, {"type": "text", "coordinates": [479, 655, 481, 666], "content": ".", "score": 1.0, "index": 105}] | [{"coordinates": [], "index": 27, "caption": "Fig. 1. Solution space for the T-shirt example", "caption_coordinates": [216, 526, 399, 538]}] | [{"type": "block", "coordinates": [150, 470, 464, 517], "content": "", "caption": ""}, {"type": "inline", "coordinates": [190, 117, 198, 126], "content": "A", "caption": ""}, {"type": "inline", "coordinates": [284, 117, 293, 127], "content": "C", "caption": ""}, {"type": "inline", "coordinates": [337, 117, 379, 128], "content": "(X,D,F)", "caption": ""}, {"type": "inline", "coordinates": [408, 117, 416, 126], "content": "X", "caption": ""}, {"type": "inline", "coordinates": [158, 128, 221, 141], "content": "\\{x_{0},\\ldots,x_{n-1}\\}", "caption": ""}, {"type": "inline", "coordinates": [226, 129, 325, 140], "content": "D=D_{0}\\times...\\times D_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [170, 141, 229, 152], "content": "D_{0},\\ldots,D_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [248, 141, 328, 153], "content": "F=\\{f_{0},...,f_{m-1}\\}", "caption": ""}, {"type": "inline", "coordinates": [216, 154, 245, 163], "content": "x_{i}=v", "caption": ""}, {"type": "inline", "coordinates": [277, 153, 308, 163], "content": "v\\,\\in\\,D_{i}", "caption": ""}, {"type": "inline", "coordinates": [329, 184, 431, 196], "content": "D_{i}\\,=\\,\\{0,\\dots,|D_{i}|\\,-\\,1\\}", "caption": ""}, {"type": "inline", "coordinates": [198, 198, 251, 208], "content": "v_{0},\\ldots,v_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [306, 198, 360, 208], "content": "x_{0},\\ldots,x_{n-1}", "caption": ""}, {"type": "inline", "coordinates": [133, 208, 274, 220], "content": "\\rho\\,=\\,\\left\\{\\left(x_{0},v_{0}\\right),\\ldots,\\left(x_{n-1},v_{n-1}\\right)\\right\\}", "caption": ""}, {"type": "inline", "coordinates": [374, 208, 407, 220], "content": "d o m(\\rho)", "caption": ""}, {"type": "inline", "coordinates": [239, 220, 397, 232], "content": "d o m(\\rho)=\\{x_{i}\\mid\\exists v\\in D_{i}.(x_{i},v)\\in\\rho\\}", "caption": ""}, {"type": "inline", "coordinates": [425, 219, 481, 232], "content": "d o m(\\rho)=X", "caption": ""}, {"type": "inline", "coordinates": [178, 234, 184, 243], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [392, 234, 398, 243], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [263, 244, 291, 256], "content": "{\\boldsymbol\\rho}\\vDash F", "caption": ""}, {"type": "inline", "coordinates": [235, 256, 306, 268], "content": "\\rho^{\\prime},d o m(\\rho^{\\prime})\\subseteq X", "caption": ""}, {"type": "inline", "coordinates": [156, 268, 185, 280], "content": "\\rho\\supseteq\\rho^{\\prime}", "caption": ""}, {"type": "inline", "coordinates": [237, 268, 266, 280], "content": "{\\boldsymbol{\\rho}}\\vDash F", "caption": ""}, {"type": "inline", "coordinates": [399, 360, 441, 372], "content": "(X,D,F)", "caption": ""}, {"type": "inline", "coordinates": [470, 360, 480, 370], "content": "\\scriptstyle\\mathrm{\\mathrm{~T~}}", "caption": ""}, {"type": "inline", "coordinates": [273, 371, 345, 384], "content": "X=\\{x_{1},x_{2},x_{3}\\}", "caption": ""}, {"type": "inline", "coordinates": [220, 383, 275, 395], "content": "D_{1}=\\{b l a c k", "caption": ""}, {"type": "inline", "coordinates": [352, 383, 477, 395], "content": "D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}", "caption": ""}, {"type": "inline", "coordinates": [152, 395, 245, 407], "content": "{\\cal D}_{3}~=~\\{M I B,S T W\\}", "caption": ""}, {"type": "inline", "coordinates": [362, 396, 423, 407], "content": "F\\;=\\;\\{f_{1},f_{2}\\}", "caption": ""}, {"type": "inline", "coordinates": [456, 396, 481, 407], "content": "f_{1}~=", "caption": ""}, {"type": "inline", "coordinates": [135, 408, 266, 419], "content": "(x_{3}\\,=\\,M I B)\\,\\Rightarrow\\,(x_{1}\\,=\\,b l a c k)", "caption": ""}, {"type": "inline", "coordinates": [287, 407, 452, 419], "content": "f_{2}\\,=\\,(x_{3}\\,=\\,S T W)\\,\\Rightarrow\\,(x_{2}\\,\\neq\\,s m a l l).", "caption": ""}, {"type": "inline", "coordinates": [149, 419, 232, 432], "content": "|D_{1}||D_{2}||D_{3}|\\,=\\,24", "caption": ""}, {"type": "inline", "coordinates": [473, 431, 481, 442], "content": "\\diamondsuit", "caption": ""}, {"type": "inline", "coordinates": [248, 630, 317, 642], "content": "x_{i}\\in X\\backslash d o m(\\rho)", "caption": ""}, {"type": "inline", "coordinates": [389, 629, 428, 642], "content": "D_{i}^{\\rho}\\subseteq D_{i}", "caption": ""}, {"type": "inline", "coordinates": [380, 644, 387, 653], "content": "\\rho", "caption": ""}, {"type": "inline", "coordinates": [269, 653, 478, 666], "content": "D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}", "caption": ""}] | [] | [612.0, 792.0] | [{"type": "text", "text": "Definition 1. $A$ configuration model $C$ is a triple $(X,D,F)$ where $X$ is a set of variables $\\{x_{0},\\ldots,x_{n-1}\\}$ , $D=D_{0}\\times...\\times D_{n-1}$ is the Cartesian product of their finite domains $D_{0},\\ldots,D_{n-1}$ and $F=\\{f_{0},...,f_{m-1}\\}$ is a set of propositional formulae over atomic propositions $x_{i}=v$ , where $v\\,\\in\\,D_{i}$ , specifying conditions on the values of the variables. ", "page_idx": 1}, {"type": "text", "text": "Concretely, every domain can be defined as $D_{i}\\,=\\,\\{0,\\dots,|D_{i}|\\,-\\,1\\}$ . An assignment of values $v_{0},\\ldots,v_{n-1}$ to variables $x_{0},\\ldots,x_{n-1}$ is denoted as an assignment $\\rho\\,=\\,\\left\\{\\left(x_{0},v_{0}\\right),\\ldots,\\left(x_{n-1},v_{n-1}\\right)\\right\\}$ . Domain of assignment $d o m(\\rho)$ is the set of variables which are assigned: $d o m(\\rho)=\\{x_{i}\\mid\\exists v\\in D_{i}.(x_{i},v)\\in\\rho\\}$ and if $d o m(\\rho)=X$ we refer to $\\rho$ as a total assignment. We say that a total assignment $\\rho$ is valid, if it satisfies all the rules which is denoted as ${\\boldsymbol\\rho}\\vDash F$ . ", "page_idx": 1}, {"type": "text", "text": "A partial assignment $\\rho^{\\prime},d o m(\\rho^{\\prime})\\subseteq X$ is valid if there is at least one total assignment $\\rho\\supseteq\\rho^{\\prime}$ that is valid ${\\boldsymbol{\\rho}}\\vDash F$ , i.e. if there is at least one way to successfully finish the existing configuration process. ", "page_idx": 1}, {"type": "text", "text": "Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or blue), the size (small, medium, or large) and the print (\u201dMen In Black\u201d - MIB or \u201dSave The Whales\u201d - STW). There are two rules that we have to observe: if we choose the MIB print then the color black has to be chosen as well, and if we choose the small size then the STW print (including a big picture of a whale) cannot be selected as the large whale does not fit on the small shirt. The configuration problem $(X,D,F)$ of the $\\scriptstyle\\mathrm{\\mathrm{~T~}}$ shirt example consists of variables $X=\\{x_{1},x_{2},x_{3}\\}$ representing color, size and print. Variable domains are $D_{1}=\\{b l a c k$ , white, red, blue}, $D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}$ , and ${\\cal D}_{3}~=~\\{M I B,S T W\\}$ . The two rules translate to $F\\;=\\;\\{f_{1},f_{2}\\}$ , where $f_{1}~=$ $(x_{3}\\,=\\,M I B)\\,\\Rightarrow\\,(x_{1}\\,=\\,b l a c k)$ and $f_{2}\\,=\\,(x_{3}\\,=\\,S T W)\\,\\Rightarrow\\,(x_{2}\\,\\neq\\,s m a l l).$ There are $|D_{1}||D_{2}||D_{3}|\\,=\\,24$ possible assignments. Eleven of these assignments are valid configurations and they form the solution space shown in Fig. 1. $\\diamondsuit$ ", "page_idx": 1}, {"type": "equation", "text": "$$\n{\\begin{array}{l l l l}{\\left(b l a c k,s m a l l,M I B\\right)}&&{\\left(b l a c k,l a r g e,S T W\\right)}&&{\\left(r e d,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,M I B\\right)}&&{\\left(w h i t e,m e d i u m,S T W\\right)}&&{\\left(b l u e,m e d i u m,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,S T W\\right)}&&{\\left(w h i t e,l a r g e,S T W\\right)}&&{\\left(b l u e,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,l a r g e,M I B\\right)}&&{\\left(r e d,m e d i u m,S T W\\right)}\\end{array}}\n$$", "text_format": "latex", "page_idx": 1}, {"type": "image", "img_path": "", "img_caption": ["Fig. 1. Solution space for the T-shirt example "], "img_footnote": [], "page_idx": 1}, {"type": "text", "text": "2.1 User Interaction ", "text_level": 1, "page_idx": 1}, {"type": "text", "text": "Configurator assists a user interactively to reach a valid product specification, i.e. to reach total valid assignment. The key operation in this interaction is that of computing, for each unassigned variable $x_{i}\\in X\\backslash d o m(\\rho)$ , the valid domain $D_{i}^{\\rho}\\subseteq D_{i}$ . The domain is valid if it contains those and only those values with which $\\rho$ can be extended to become a total valid assignment, i.e. $D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}$ . 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The configuration problem ", "type": "text"}, {"bbox": [399, 360, 441, 372], "score": 0.93, "content": "(X,D,F)", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [442, 361, 470, 372], "score": 1.0, "content": " of the", "type": "text"}, {"bbox": [470, 360, 480, 370], "score": 0.32, "content": "\\scriptstyle\\mathrm{\\mathrm{~T~}}", "type": "inline_equation", "height": 10, "width": 10}], "index": 19}, {"bbox": [133, 371, 481, 385], "spans": [{"bbox": [133, 372, 272, 385], "score": 1.0, "content": "shirt example consists of variables", "type": "text"}, {"bbox": [273, 371, 345, 384], "score": 0.93, "content": "X=\\{x_{1},x_{2},x_{3}\\}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [345, 372, 481, 385], "score": 1.0, "content": " representing color, size and print.", "type": "text"}], "index": 20}, {"bbox": [134, 383, 481, 396], "spans": [{"bbox": [134, 385, 220, 396], "score": 1.0, "content": "Variable domains are", "type": "text"}, {"bbox": [220, 383, 275, 395], "score": 0.91, "content": "D_{1}=\\{b l a c k", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [275, 385, 352, 396], "score": 1.0, "content": ", white, red, blue},", "type": "text"}, {"bbox": [352, 383, 477, 395], "score": 0.71, "content": "D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}", "type": "inline_equation", "height": 12, "width": 125}, {"bbox": [478, 385, 481, 396], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [133, 395, 481, 408], "spans": [{"bbox": [133, 396, 151, 408], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [152, 395, 245, 407], "score": 0.89, "content": "{\\cal D}_{3}~=~\\{M I B,S T W\\}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [245, 396, 361, 408], "score": 1.0, "content": ". 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Eleven of these assignments are valid", "type": "text"}], "index": 24}, {"bbox": [134, 431, 481, 444], "spans": [{"bbox": [134, 433, 392, 444], "score": 1.0, "content": "configurations and they form the solution space shown in Fig. 1.", "type": "text"}, {"bbox": [473, 431, 481, 442], "score": 0.68, "content": "\\diamondsuit", "type": "inline_equation", "height": 11, "width": 8}], "index": 25}], "index": 19.5}, {"type": "interline_equation", "bbox": [150, 470, 464, 517], "lines": [{"bbox": [150, 470, 464, 517], "spans": [{"bbox": [150, 470, 464, 517], "score": 0.49, "content": "{\\begin{array}{l l l l}{\\left(b l a c k,s m a l l,M I B\\right)}&&{\\left(b l a c k,l a r g e,S T W\\right)}&&{\\left(r e d,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,M I B\\right)}&&{\\left(w h i t e,m e d i u m,S T W\\right)}&&{\\left(b l u e,m e d i u m,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,S T W\\right)}&&{\\left(w h i t e,l a r g e,S T W\\right)}&&{\\left(b l u e,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,l a r g e,M I B\\right)}&&{\\left(r e d,m e d i u m,S T W\\right)}\\end{array}}", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "image", "bbox": [], "blocks": [{"type": "image_caption", "bbox": [216, 526, 399, 538], "group_id": 0, "lines": [{"bbox": [216, 526, 398, 539], "spans": [{"bbox": [216, 526, 398, 539], "score": 1.0, "content": "Fig. 1. Solution space for the T-shirt example", "type": "text"}], "index": 27}], "index": 27}], "index": 27}, {"type": "title", "bbox": [134, 587, 228, 598], "lines": [{"bbox": [133, 587, 228, 597], "spans": [{"bbox": [133, 587, 228, 597], "score": 1.0, "content": "2.1 User Interaction", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [134, 606, 481, 666], "lines": [{"bbox": [133, 606, 481, 618], "spans": [{"bbox": [133, 606, 481, 618], "score": 1.0, "content": "Configurator assists a user interactively to reach a valid product specification, i.e. to", "type": "text"}], "index": 29}, {"bbox": [133, 618, 481, 632], "spans": [{"bbox": [133, 618, 481, 632], "score": 1.0, "content": "reach total valid assignment. The key operation in this interaction is that of computing,", "type": "text"}], "index": 30}, {"bbox": [133, 629, 482, 642], "spans": [{"bbox": [133, 631, 248, 642], "score": 1.0, "content": "for each unassigned variable", "type": "text"}, {"bbox": [248, 630, 317, 642], "score": 0.93, "content": "x_{i}\\in X\\backslash d o m(\\rho)", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [318, 631, 389, 642], "score": 1.0, "content": ", the valid domain", "type": "text"}, {"bbox": [389, 629, 428, 642], "score": 0.93, "content": "D_{i}^{\\rho}\\subseteq D_{i}", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [429, 631, 482, 642], "score": 1.0, "content": ". The domain", "type": "text"}], "index": 31}, {"bbox": [133, 642, 480, 654], "spans": [{"bbox": [133, 642, 380, 654], "score": 1.0, "content": "is valid if it contains those and only those values with which", "type": "text"}, {"bbox": [380, 644, 387, 653], "score": 0.8, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [388, 642, 480, 654], "score": 1.0, "content": " can be extended to be-", "type": "text"}], "index": 32}, {"bbox": [133, 653, 481, 666], "spans": [{"bbox": [133, 655, 268, 666], "score": 1.0, "content": "come a total valid assignment, i.e.", "type": "text"}, {"bbox": [269, 653, 478, 666], "score": 0.88, "content": "D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 209}, {"bbox": [479, 655, 481, 666], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [], "blocks": [{"type": "image_caption", "bbox": [216, 526, 399, 538], "group_id": 0, "lines": [{"bbox": [216, 526, 398, 539], "spans": [{"bbox": [216, 526, 398, 539], "score": 1.0, "content": "Fig. 1. Solution space for the T-shirt example", "type": "text"}], "index": 27}], "index": 27}], "index": 27}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [150, 470, 464, 517], "lines": [{"bbox": [150, 470, 464, 517], "spans": [{"bbox": [150, 470, 464, 517], "score": 0.49, "content": "{\\begin{array}{l l l l}{\\left(b l a c k,s m a l l,M I B\\right)}&&{\\left(b l a c k,l a r g e,S T W\\right)}&&{\\left(r e d,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,M I B\\right)}&&{\\left(w h i t e,m e d i u m,S T W\\right)}&&{\\left(b l u e,m e d i u m,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,S T W\\right)}&&{\\left(w h i t e,l a r g e,S T W\\right)}&&{\\left(b l u e,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,l a r g e,M I B\\right)}&&{\\left(r e d,m e d i u m,S T W\\right)}\\end{array}}", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [303, 684, 311, 693], "lines": [{"bbox": [303, 685, 311, 695], "spans": [{"bbox": [303, 685, 311, 695], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [133, 115, 482, 175], "lines": [{"bbox": [133, 117, 480, 129], "spans": [{"bbox": [133, 117, 189, 129], "score": 1.0, "content": "Definition 1.", "type": "text"}, {"bbox": [190, 117, 198, 126], "score": 0.28, "content": "A", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 117, 283, 129], "score": 1.0, "content": " configuration model", "type": "text"}, {"bbox": [284, 117, 293, 127], "score": 0.82, "content": "C", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [293, 117, 337, 129], "score": 1.0, "content": " is a triple ", "type": "text"}, {"bbox": [337, 117, 379, 128], "score": 0.93, "content": "(X,D,F)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [379, 117, 407, 129], "score": 1.0, "content": " where", "type": "text"}, {"bbox": [408, 117, 416, 126], "score": 0.56, "content": "X", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [417, 117, 480, 129], "score": 1.0, "content": " is a set of vari-", "type": "text"}], "index": 0}, {"bbox": [133, 128, 481, 142], "spans": [{"bbox": [133, 129, 158, 142], "score": 1.0, "content": "ables ", "type": "text"}, {"bbox": [158, 128, 221, 141], "score": 0.88, "content": "\\{x_{0},\\ldots,x_{n-1}\\}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [222, 129, 226, 142], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [226, 129, 325, 140], "score": 0.89, "content": "D=D_{0}\\times...\\times D_{n-1}", "type": "inline_equation", "height": 11, "width": 99}, {"bbox": [325, 129, 481, 142], "score": 1.0, "content": " is the Cartesian product of their finite", "type": "text"}], "index": 1}, {"bbox": [133, 141, 482, 154], "spans": [{"bbox": [133, 141, 169, 154], "score": 1.0, "content": "domains", "type": "text"}, {"bbox": [170, 141, 229, 152], "score": 0.91, "content": "D_{0},\\ldots,D_{n-1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [229, 141, 247, 154], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 141, 328, 153], "score": 0.9, "content": "F=\\{f_{0},...,f_{m-1}\\}", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [329, 141, 482, 154], "score": 1.0, "content": " is a set of propositional formulae over", "type": "text"}], "index": 2}, {"bbox": [133, 153, 481, 165], "spans": [{"bbox": [133, 153, 215, 165], "score": 1.0, "content": "atomic propositions", "type": "text"}, {"bbox": [216, 154, 245, 163], "score": 0.88, "content": "x_{i}=v", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [246, 153, 276, 165], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [277, 153, 308, 163], "score": 0.88, "content": "v\\,\\in\\,D_{i}", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [308, 153, 481, 165], "score": 1.0, "content": ", specifying conditions on the values of the", "type": "text"}], "index": 3}, {"bbox": [133, 165, 174, 177], "spans": [{"bbox": [133, 165, 174, 177], "score": 1.0, "content": "variables.", "type": "text"}], "index": 4}], "index": 2, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [133, 117, 482, 177]}, {"type": "text", "bbox": [133, 184, 481, 256], "lines": [{"bbox": [149, 184, 480, 198], "spans": [{"bbox": [149, 185, 329, 198], "score": 1.0, "content": "Concretely, every domain can be defined as", "type": "text"}, {"bbox": [329, 184, 431, 196], "score": 0.94, "content": "D_{i}\\,=\\,\\{0,\\dots,|D_{i}|\\,-\\,1\\}", "type": "inline_equation", "height": 12, "width": 102}, {"bbox": [431, 185, 480, 198], "score": 1.0, "content": ". An assign-", "type": "text"}], "index": 5}, {"bbox": [133, 196, 482, 210], "spans": [{"bbox": [133, 196, 198, 210], "score": 1.0, "content": "ment of values ", "type": "text"}, {"bbox": [198, 198, 251, 208], "score": 0.86, "content": "v_{0},\\ldots,v_{n-1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [252, 196, 305, 210], "score": 1.0, "content": " to variables ", "type": "text"}, {"bbox": [306, 198, 360, 208], "score": 0.9, "content": "x_{0},\\ldots,x_{n-1}", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [360, 196, 482, 210], "score": 1.0, "content": " is denoted as an assignment", "type": "text"}], "index": 6}, {"bbox": [133, 208, 481, 221], "spans": [{"bbox": [133, 208, 274, 220], "score": 0.89, "content": "\\rho\\,=\\,\\left\\{\\left(x_{0},v_{0}\\right),\\ldots,\\left(x_{n-1},v_{n-1}\\right)\\right\\}", "type": "inline_equation", "height": 12, "width": 141}, {"bbox": [274, 209, 374, 221], "score": 1.0, "content": ". Domain of assignment ", "type": "text"}, {"bbox": [374, 208, 407, 220], "score": 0.81, "content": "d o m(\\rho)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [407, 209, 481, 221], "score": 1.0, "content": " is the set of vari-", "type": "text"}], "index": 7}, {"bbox": [134, 219, 481, 232], "spans": [{"bbox": [134, 221, 238, 232], "score": 1.0, "content": "ables which are assigned:", "type": "text"}, {"bbox": [239, 220, 397, 232], "score": 0.92, "content": "d o m(\\rho)=\\{x_{i}\\mid\\exists v\\in D_{i}.(x_{i},v)\\in\\rho\\}", "type": "inline_equation", "height": 12, "width": 158}, {"bbox": [397, 221, 424, 232], "score": 1.0, "content": " and if", "type": "text"}, {"bbox": [425, 219, 481, 232], "score": 0.91, "content": "d o m(\\rho)=X", "type": "inline_equation", "height": 13, "width": 56}], "index": 8}, {"bbox": [133, 232, 481, 245], "spans": [{"bbox": [133, 232, 177, 245], "score": 1.0, "content": "we refer to", "type": "text"}, {"bbox": [178, 234, 184, 243], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [185, 232, 391, 245], "score": 1.0, "content": " as a total assignment. We say that a total assignment", "type": "text"}, {"bbox": [392, 234, 398, 243], "score": 0.83, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [398, 232, 481, 245], "score": 1.0, "content": " is valid, if it satisfies", "type": "text"}], "index": 9}, {"bbox": [133, 244, 295, 257], "spans": [{"bbox": [133, 245, 263, 257], "score": 1.0, "content": "all the rules which is denoted as", "type": "text"}, {"bbox": [263, 244, 291, 256], "score": 0.91, "content": "{\\boldsymbol\\rho}\\vDash F", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [292, 245, 295, 257], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 7.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [133, 184, 482, 257]}, {"type": "text", "bbox": [134, 257, 481, 292], "lines": [{"bbox": [149, 256, 480, 269], "spans": [{"bbox": [149, 257, 235, 269], "score": 1.0, "content": "A partial assignment ", "type": "text"}, {"bbox": [235, 256, 306, 268], "score": 0.91, "content": "\\rho^{\\prime},d o m(\\rho^{\\prime})\\subseteq X", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [307, 257, 480, 269], "score": 1.0, "content": " is valid if there is at least one total assign-", "type": "text"}], "index": 11}, {"bbox": [133, 268, 480, 281], "spans": [{"bbox": [133, 268, 156, 281], "score": 1.0, "content": "ment", "type": "text"}, {"bbox": [156, 268, 185, 280], "score": 0.92, "content": "\\rho\\supseteq\\rho^{\\prime}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [185, 268, 236, 281], "score": 1.0, "content": " that is valid", "type": "text"}, {"bbox": [237, 268, 266, 280], "score": 0.91, "content": "{\\boldsymbol{\\rho}}\\vDash F", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [267, 268, 480, 281], "score": 1.0, "content": ", i.e. if there is at least one way to successfully finish", "type": "text"}], "index": 12}, {"bbox": [134, 280, 271, 293], "spans": [{"bbox": [134, 280, 271, 293], "score": 1.0, "content": "the existing configuration process.", "type": "text"}], "index": 13}], "index": 12, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [133, 256, 480, 293]}, {"type": "text", "bbox": [133, 300, 482, 444], "lines": [{"bbox": [134, 301, 482, 312], "spans": [{"bbox": [134, 301, 482, 312], "score": 1.0, "content": "Example 1. Consider specifying a T-shirt by choosing the color (black, white, red, or", "type": "text"}], "index": 14}, {"bbox": [133, 312, 481, 324], "spans": [{"bbox": [133, 312, 481, 324], "score": 1.0, "content": "blue), the size (small, medium, or large) and the print (\u201dMen In Black\u201d - MIB or \u201dSave", "type": "text"}], "index": 15}, {"bbox": [133, 325, 481, 336], "spans": [{"bbox": [133, 325, 481, 336], "score": 1.0, "content": "The Whales\u201d - STW). There are two rules that we have to observe: if we choose the", "type": "text"}], "index": 16}, {"bbox": [133, 336, 481, 348], "spans": [{"bbox": [133, 336, 481, 348], "score": 1.0, "content": "MIB print then the color black has to be chosen as well, and if we choose the small size", "type": "text"}], "index": 17}, {"bbox": [133, 348, 481, 361], "spans": [{"bbox": [133, 348, 481, 361], "score": 1.0, "content": "then the STW print (including a big picture of a whale) cannot be selected as the large", "type": "text"}], "index": 18}, {"bbox": [134, 360, 480, 372], "spans": [{"bbox": [134, 361, 399, 372], "score": 1.0, "content": "whale does not fit on the small shirt. The configuration problem ", "type": "text"}, {"bbox": [399, 360, 441, 372], "score": 0.93, "content": "(X,D,F)", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [442, 361, 470, 372], "score": 1.0, "content": " of the", "type": "text"}, {"bbox": [470, 360, 480, 370], "score": 0.32, "content": "\\scriptstyle\\mathrm{\\mathrm{~T~}}", "type": "inline_equation", "height": 10, "width": 10}], "index": 19}, {"bbox": [133, 371, 481, 385], "spans": [{"bbox": [133, 372, 272, 385], "score": 1.0, "content": "shirt example consists of variables", "type": "text"}, {"bbox": [273, 371, 345, 384], "score": 0.93, "content": "X=\\{x_{1},x_{2},x_{3}\\}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [345, 372, 481, 385], "score": 1.0, "content": " representing color, size and print.", "type": "text"}], "index": 20}, {"bbox": [134, 383, 481, 396], "spans": [{"bbox": [134, 385, 220, 396], "score": 1.0, "content": "Variable domains are", "type": "text"}, {"bbox": [220, 383, 275, 395], "score": 0.91, "content": "D_{1}=\\{b l a c k", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [275, 385, 352, 396], "score": 1.0, "content": ", white, red, blue},", "type": "text"}, {"bbox": [352, 383, 477, 395], "score": 0.71, "content": "D_{2}=\\{s m a l l,m e d i u m,l a r g e\\}", "type": "inline_equation", "height": 12, "width": 125}, {"bbox": [478, 385, 481, 396], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [133, 395, 481, 408], "spans": [{"bbox": [133, 396, 151, 408], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [152, 395, 245, 407], "score": 0.89, "content": "{\\cal D}_{3}~=~\\{M I B,S T W\\}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [245, 396, 361, 408], "score": 1.0, "content": ". The two rules translate to ", "type": "text"}, {"bbox": [362, 396, 423, 407], "score": 0.92, "content": "F\\;=\\;\\{f_{1},f_{2}\\}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [423, 396, 456, 408], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [456, 396, 481, 407], "score": 0.89, "content": "f_{1}~=", "type": "inline_equation", "height": 11, "width": 25}], "index": 22}, {"bbox": [135, 407, 482, 420], "spans": [{"bbox": [135, 408, 266, 419], "score": 0.75, "content": "(x_{3}\\,=\\,M I B)\\,\\Rightarrow\\,(x_{1}\\,=\\,b l a c k)", "type": "inline_equation", "height": 11, "width": 131}, {"bbox": [267, 408, 286, 420], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [287, 407, 452, 419], "score": 0.84, "content": "f_{2}\\,=\\,(x_{3}\\,=\\,S T W)\\,\\Rightarrow\\,(x_{2}\\,\\neq\\,s m a l l).", "type": "inline_equation", "height": 12, "width": 165}, {"bbox": [453, 408, 482, 420], "score": 1.0, "content": " There", "type": "text"}], "index": 23}, {"bbox": [133, 419, 482, 433], "spans": [{"bbox": [133, 420, 149, 433], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [149, 419, 232, 432], "score": 0.92, "content": "|D_{1}||D_{2}||D_{3}|\\,=\\,24", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [232, 420, 482, 433], "score": 1.0, "content": " possible assignments. Eleven of these assignments are valid", "type": "text"}], "index": 24}, {"bbox": [134, 431, 481, 444], "spans": [{"bbox": [134, 433, 392, 444], "score": 1.0, "content": "configurations and they form the solution space shown in Fig. 1.", "type": "text"}, {"bbox": [473, 431, 481, 442], "score": 0.68, "content": "\\diamondsuit", "type": "inline_equation", "height": 11, "width": 8}], "index": 25}], "index": 19.5, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [133, 301, 482, 444]}, {"type": "interline_equation", "bbox": [150, 470, 464, 517], "lines": [{"bbox": [150, 470, 464, 517], "spans": [{"bbox": [150, 470, 464, 517], "score": 0.49, "content": "{\\begin{array}{l l l l}{\\left(b l a c k,s m a l l,M I B\\right)}&&{\\left(b l a c k,l a r g e,S T W\\right)}&&{\\left(r e d,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,M I B\\right)}&&{\\left(w h i t e,m e d i u m,S T W\\right)}&&{\\left(b l u e,m e d i u m,S T W\\right)}\\\\ {\\left(b l a c k,m e d i u m,S T W\\right)}&&{\\left(w h i t e,l a r g e,S T W\\right)}&&{\\left(b l u e,l a r g e,S T W\\right)}\\\\ {\\left(b l a c k,l a r g e,M I B\\right)}&&{\\left(r e d,m e d i u m,S T W\\right)}\\end{array}}", "type": "interline_equation"}], "index": 26}], "index": 26, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "image", "bbox": [], "blocks": [{"type": "image_caption", "bbox": [216, 526, 399, 538], "group_id": 0, "lines": [{"bbox": [216, 526, 398, 539], "spans": [{"bbox": [216, 526, 398, 539], "score": 1.0, "content": "Fig. 1. Solution space for the T-shirt example", "type": "text"}], "index": 27}], "index": 27}], "index": 27, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "title", "bbox": [134, 587, 228, 598], "lines": [{"bbox": [133, 587, 228, 597], "spans": [{"bbox": [133, 587, 228, 597], "score": 1.0, "content": "2.1 User Interaction", "type": "text"}], "index": 28}], "index": 28, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [134, 606, 481, 666], "lines": [{"bbox": [133, 606, 481, 618], "spans": [{"bbox": [133, 606, 481, 618], "score": 1.0, "content": "Configurator assists a user interactively to reach a valid product specification, i.e. to", "type": "text"}], "index": 29}, {"bbox": [133, 618, 481, 632], "spans": [{"bbox": [133, 618, 481, 632], "score": 1.0, "content": "reach total valid assignment. The key operation in this interaction is that of computing,", "type": "text"}], "index": 30}, {"bbox": [133, 629, 482, 642], "spans": [{"bbox": [133, 631, 248, 642], "score": 1.0, "content": "for each unassigned variable", "type": "text"}, {"bbox": [248, 630, 317, 642], "score": 0.93, "content": "x_{i}\\in X\\backslash d o m(\\rho)", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [318, 631, 389, 642], "score": 1.0, "content": ", the valid domain", "type": "text"}, {"bbox": [389, 629, 428, 642], "score": 0.93, "content": "D_{i}^{\\rho}\\subseteq D_{i}", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [429, 631, 482, 642], "score": 1.0, "content": ". The domain", "type": "text"}], "index": 31}, {"bbox": [133, 642, 480, 654], "spans": [{"bbox": [133, 642, 380, 654], "score": 1.0, "content": "is valid if it contains those and only those values with which", "type": "text"}, {"bbox": [380, 644, 387, 653], "score": 0.8, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [388, 642, 480, 654], "score": 1.0, "content": " can be extended to be-", "type": "text"}], "index": 32}, {"bbox": [133, 653, 481, 666], "spans": [{"bbox": [133, 655, 268, 666], "score": 1.0, "content": "come a total valid assignment, i.e.", "type": "text"}, {"bbox": [269, 653, 478, 666], "score": 0.88, "content": "D_{i}^{\\rho}=\\{v\\in D_{i}\\mid\\exists\\rho^{\\prime}:\\rho^{\\prime}\\models F\\land\\rho\\cup\\{(x_{i},v)\\}\\subseteq\\rho^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 209}, {"bbox": [479, 655, 481, 666], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 31, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [133, 606, 482, 666]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_1.png | images/0704.1394_1.png |
|
0704.1394.pdf | 2 | The significance of this demand is that it guarantees the user backtrack-free assignment
to variables as long as he selects values from valid domains. This reduces cognitive
effort during the interaction and increases usability.
At each step of the interaction, the configurator reports the valid domains to the
user, based on the current partial assignment $$\rho$$ resulting from his earlier choices. The
user then picks an unassigned variable $$x_{j}\;\in\;X\;\backslash\;d o m(\rho)$$ and selects a value from
the calculated valid domain $$v_{j}\,\in{\cal D}_{j}^{\rho}$$ . The partial assignment is then extended to $$\rho\cup$$
$$\{(x_{j},v_{j})\}$$ and a new interaction step is initiated.
# 3 BDD Based Configuration
In [5,10] the interactive configuration was delivered by dividing the computational ef-
fort into an offline and online phase. First, in the offline phase, the authors compiled a
BDD representing the solution space of all valid configurations $$S o l\,=\,\{\rho\mid\,\rho\,\left\vert\,=\,F\right\}$$ .
Then, the functionality of calculating valid domains $$\left(C V D\right)$$ was delivered online, by
efficient algorithms executing during the interaction with a user. The benefit of this ap-
proach is that the BDD needs to be compiled only once, and can be reused for multiple
user sessions. The user interaction process is illustrated in Fig. 2.
$$\begin{array}{r l}&{I n C o(S o l,\rho)}\\ &{1:\qquad\mathrm{whi\,i\,e}\ \ |S o l^{\rho}|>1}\\ &{2:\qquad\qquad\mathrm{compute}\ \ D^{\rho}=C V D(S o l,\rho)}\\ &{3:\qquad\quad\mathrm{report}\ \ D^{\rho}\ \mathrm{to}\ \ \mathrm{the}\ \ \mathrm{user}}\\ &{4:\qquad\quad\mathrm{the}\ \ \mathrm{user}\ \ \mathrm{chooses}\ \ (x_{i},v)}\\ &{5:\qquad\quad\rho\leftarrow\rho\cup\{(x_{i},v)\}}\\ &{6:\qquad\quad\mathrm{return}\ \rho}\end{array}$$ for some $$x_{i}\not\in\mathrm{dom}(\rho)$$ , $$v\in D_{i}^{\rho}$$
Fig. 2. Interactive configuration algorithm working on a BDD representation of the so-
lutions Sol reaches a valid total configuration as an extension of the argument $$\rho$$ .
Important requirement for online user-interaction is the guaranteed real-time expe-
rience of user-configurator interaction. Therefore, the algorithms that are executing in
the online phase must be provably efficient in the size of the BDD representation. This
is what we call the real-time guarantee. As the $$C V D$$ functionality is NP-hard, and the
online algorithms are polynomial in the size of generated BDD, there is no hope of pro-
viding polynomial size guarantees for the worst-case BDD representation. However, it
suffices that the BDD size is small enough for all the configuration instances occurring
in practice [10].
# 3.1 Binary Decision Diagrams
A reduced ordered Binary Decision Diagram (BDD) is a rooted directed acyclic graph
representing a Boolean function on a set of linearly ordered Boolean variables. It has
one or two terminal nodes labeled 1 or 0 and a set of variable nodes. Each variable node
| <p>The significance of this demand is that it guarantees the user backtrack-free assignment
to variables as long as he selects values from valid domains. This reduces cognitive
effort during the interaction and increases usability.</p>
<p>At each step of the interaction, the configurator reports the valid domains to the
user, based on the current partial assignment $$\rho$$ resulting from his earlier choices. The
user then picks an unassigned variable $$x_{j}\;\in\;X\;\backslash\;d o m(\rho)$$ and selects a value from
the calculated valid domain $$v_{j}\,\in{\cal D}_{j}^{\rho}$$ . The partial assignment is then extended to $$\rho\cup$$
$$\{(x_{j},v_{j})\}$$ and a new interaction step is initiated.</p>
<h1>3 BDD Based Configuration</h1>
<p>In [5,10] the interactive configuration was delivered by dividing the computational ef-
fort into an offline and online phase. First, in the offline phase, the authors compiled a
BDD representing the solution space of all valid configurations $$S o l\,=\,\{\rho\mid\,\rho\,\left\vert\,=\,F\right\}$$ .
Then, the functionality of calculating valid domains $$\left(C V D\right)$$ was delivered online, by
efficient algorithms executing during the interaction with a user. The benefit of this ap-
proach is that the BDD needs to be compiled only once, and can be reused for multiple
user sessions. The user interaction process is illustrated in Fig. 2.</p>
<p>$$\begin{array}{r l}&{I n C o(S o l,\rho)}\\ &{1:\qquad\mathrm{whi\,i\,e}\ \ |S o l^{\rho}|>1}\\ &{2:\qquad\qquad\mathrm{compute}\ \ D^{\rho}=C V D(S o l,\rho)}\\ &{3:\qquad\quad\mathrm{report}\ \ D^{\rho}\ \mathrm{to}\ \ \mathrm{the}\ \ \mathrm{user}}\\ &{4:\qquad\quad\mathrm{the}\ \ \mathrm{user}\ \ \mathrm{chooses}\ \ (x_{i},v)}\\ &{5:\qquad\quad\rho\leftarrow\rho\cup\{(x_{i},v)\}}\\ &{6:\qquad\quad\mathrm{return}\ \rho}\end{array}$$ for some $$x_{i}\not\in\mathrm{dom}(\rho)$$ , $$v\in D_{i}^{\rho}$$</p>
<p>Fig. 2. Interactive configuration algorithm working on a BDD representation of the so-
lutions Sol reaches a valid total configuration as an extension of the argument $$\rho$$ .</p>
<p>Important requirement for online user-interaction is the guaranteed real-time expe-
rience of user-configurator interaction. Therefore, the algorithms that are executing in
the online phase must be provably efficient in the size of the BDD representation. This
is what we call the real-time guarantee. As the $$C V D$$ functionality is NP-hard, and the
online algorithms are polynomial in the size of generated BDD, there is no hope of pro-
viding polynomial size guarantees for the worst-case BDD representation. However, it
suffices that the BDD size is small enough for all the configuration instances occurring
in practice [10].</p>
<h1>3.1 Binary Decision Diagrams</h1>
<p>A reduced ordered Binary Decision Diagram (BDD) is a rooted directed acyclic graph
representing a Boolean function on a set of linearly ordered Boolean variables. It has
one or two terminal nodes labeled 1 or 0 and a set of variable nodes. Each variable node</p>
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This reduces cognitive effort during the interaction and increases usability. ", "page_idx": 2}, {"type": "text", "text": "At each step of the interaction, the configurator reports the valid domains to the user, based on the current partial assignment $\\rho$ resulting from his earlier choices. The user then picks an unassigned variable $x_{j}\\;\\in\\;X\\;\\backslash\\;d o m(\\rho)$ and selects a value from the calculated valid domain $v_{j}\\,\\in{\\cal D}_{j}^{\\rho}$ . The partial assignment is then extended to $\\rho\\cup$ $\\{(x_{j},v_{j})\\}$ and a new interaction step is initiated. ", "page_idx": 2}, {"type": "text", "text": "3 BDD Based Configuration ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "In [5,10] the interactive configuration was delivered by dividing the computational effort into an offline and online phase. First, in the offline phase, the authors compiled a BDD representing the solution space of all valid configurations $S o l\\,=\\,\\{\\rho\\mid\\,\\rho\\,\\left\\vert\\,=\\,F\\right\\}$ . Then, the functionality of calculating valid domains $\\left(C V D\\right)$ was delivered online, by efficient algorithms executing during the interaction with a user. The benefit of this approach is that the BDD needs to be compiled only once, and can be reused for multiple user sessions. The user interaction process is illustrated in Fig. 2. ", "page_idx": 2}, {"type": "text", "text": "$\\begin{array}{r l}&{I n C o(S o l,\\rho)}\\\\ &{1:\\qquad\\mathrm{whi\\,i\\,e}\\ \\ |S o l^{\\rho}|>1}\\\\ &{2:\\qquad\\qquad\\mathrm{compute}\\ \\ D^{\\rho}=C V D(S o l,\\rho)}\\\\ &{3:\\qquad\\quad\\mathrm{report}\\ \\ D^{\\rho}\\ \\mathrm{to}\\ \\ \\mathrm{the}\\ \\ \\mathrm{user}}\\\\ &{4:\\qquad\\quad\\mathrm{the}\\ \\ \\mathrm{user}\\ \\ \\mathrm{chooses}\\ \\ (x_{i},v)}\\\\ &{5:\\qquad\\quad\\rho\\leftarrow\\rho\\cup\\{(x_{i},v)\\}}\\\\ &{6:\\qquad\\quad\\mathrm{return}\\ \\rho}\\end{array}$ for some $x_{i}\\not\\in\\mathrm{dom}(\\rho)$ , $v\\in D_{i}^{\\rho}$ ", "page_idx": 2}, {"type": "text", "text": "Fig. 2. Interactive configuration algorithm working on a BDD representation of the solutions Sol reaches a valid total configuration as an extension of the argument $\\rho$ . 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The user interaction process is illustrated in Fig. 2.", "type": "text"}], "index": 15}], "index": 12}, {"type": "text", "bbox": [148, 360, 468, 439], "lines": [{"bbox": [148, 360, 466, 439], "spans": [{"bbox": [148, 360, 325, 439], "score": 0.43, "content": "\\begin{array}{r l}&{I n C o(S o l,\\rho)}\\\\ &{1:\\qquad\\mathrm{whi\\,i\\,e}\\ \\ |S o l^{\\rho}|>1}\\\\ &{2:\\qquad\\qquad\\mathrm{compute}\\ \\ D^{\\rho}=C V D(S o l,\\rho)}\\\\ &{3:\\qquad\\quad\\mathrm{report}\\ \\ D^{\\rho}\\ \\mathrm{to}\\ \\ \\mathrm{the}\\ \\ \\mathrm{user}}\\\\ &{4:\\qquad\\quad\\mathrm{the}\\ \\ \\mathrm{user}\\ \\ \\mathrm{chooses}\\ \\ (x_{i},v)}\\\\ &{5:\\qquad\\quad\\rho\\leftarrow\\rho\\cup\\{(x_{i},v)\\}}\\\\ &{6:\\qquad\\quad\\mathrm{return}\\ \\rho}\\end{array}", "type": "inline_equation"}, {"bbox": [326, 405, 378, 417], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [379, 405, 426, 416], "score": 0.6, "content": "x_{i}\\not\\in\\mathrm{dom}(\\rho)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [427, 405, 436, 417], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [436, 405, 466, 416], "score": 0.82, "content": "v\\in D_{i}^{\\rho}", "type": "inline_equation", "height": 11, "width": 30}], "index": 16}], "index": 16}, {"type": "text", "bbox": [133, 448, 481, 473], "lines": [{"bbox": [134, 449, 481, 462], "spans": [{"bbox": [134, 449, 481, 462], "score": 1.0, "content": "Fig. 2. 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First, in the offline phase, the authors compiled a", "type": "text"}], "index": 10}, {"bbox": [133, 281, 481, 295], "spans": [{"bbox": [133, 282, 392, 295], "score": 1.0, "content": "BDD representing the solution space of all valid configurations", "type": "text"}, {"bbox": [393, 281, 478, 294], "score": 0.93, "content": "S o l\\,=\\,\\{\\rho\\mid\\,\\rho\\,\\left\\vert\\,=\\,F\\right\\}", "type": "inline_equation", "height": 13, "width": 85}, {"bbox": [478, 282, 481, 295], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [134, 293, 481, 306], "spans": [{"bbox": [134, 294, 348, 306], "score": 1.0, "content": "Then, the functionality of calculating valid domains ", "type": "text"}, {"bbox": [348, 293, 379, 304], "score": 0.67, "content": "\\left(C V D\\right)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [379, 294, 481, 306], "score": 1.0, "content": " was delivered online, by", "type": "text"}], "index": 12}, {"bbox": [133, 305, 481, 318], "spans": [{"bbox": [133, 305, 481, 318], "score": 1.0, "content": "efficient algorithms executing during the interaction with a user. The benefit of this ap-", "type": "text"}], "index": 13}, {"bbox": [133, 317, 481, 330], "spans": [{"bbox": [133, 317, 481, 330], "score": 1.0, "content": "proach is that the BDD needs to be compiled only once, and can be reused for multiple", "type": "text"}], "index": 14}, {"bbox": [133, 330, 394, 342], "spans": [{"bbox": [133, 330, 394, 342], "score": 1.0, "content": "user sessions. The user interaction process is illustrated in Fig. 2.", "type": "text"}], "index": 15}], "index": 12, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [133, 257, 481, 342]}, {"type": "text", "bbox": [148, 360, 468, 439], "lines": [{"bbox": [148, 360, 466, 439], "spans": [{"bbox": [148, 360, 325, 439], "score": 0.43, "content": "\\begin{array}{r l}&{I n C o(S o l,\\rho)}\\\\ &{1:\\qquad\\mathrm{whi\\,i\\,e}\\ \\ |S o l^{\\rho}|>1}\\\\ &{2:\\qquad\\qquad\\mathrm{compute}\\ \\ D^{\\rho}=C V D(S o l,\\rho)}\\\\ &{3:\\qquad\\quad\\mathrm{report}\\ \\ D^{\\rho}\\ \\mathrm{to}\\ \\ \\mathrm{the}\\ \\ \\mathrm{user}}\\\\ &{4:\\qquad\\quad\\mathrm{the}\\ \\ \\mathrm{user}\\ \\ \\mathrm{chooses}\\ \\ (x_{i},v)}\\\\ &{5:\\qquad\\quad\\rho\\leftarrow\\rho\\cup\\{(x_{i},v)\\}}\\\\ &{6:\\qquad\\quad\\mathrm{return}\\ \\rho}\\end{array}", "type": "inline_equation"}, {"bbox": [326, 405, 378, 417], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [379, 405, 426, 416], "score": 0.6, "content": "x_{i}\\not\\in\\mathrm{dom}(\\rho)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [427, 405, 436, 417], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [436, 405, 466, 416], "score": 0.82, "content": "v\\in D_{i}^{\\rho}", "type": "inline_equation", "height": 11, "width": 30}], "index": 16}], "index": 16, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [148, 360, 466, 439]}, {"type": "text", "bbox": [133, 448, 481, 473], "lines": [{"bbox": [134, 449, 481, 462], "spans": [{"bbox": [134, 449, 481, 462], "score": 1.0, "content": "Fig. 2. Interactive configuration algorithm working on a BDD representation of the so-", "type": "text"}], "index": 17}, {"bbox": [134, 461, 452, 473], "spans": [{"bbox": [134, 461, 444, 473], "score": 1.0, "content": "lutions Sol reaches a valid total configuration as an extension of the argument", "type": "text"}, {"bbox": [444, 462, 450, 471], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [451, 461, 452, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [134, 449, 481, 473]}, {"type": "text", "bbox": [133, 495, 482, 591], "lines": [{"bbox": [149, 497, 479, 507], "spans": [{"bbox": [149, 497, 479, 507], "score": 1.0, "content": "Important requirement for online user-interaction is the guaranteed real-time expe-", "type": "text"}], "index": 19}, {"bbox": [133, 507, 481, 520], "spans": [{"bbox": [133, 507, 481, 520], "score": 1.0, "content": "rience of user-configurator interaction. 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It has", "type": "text"}], "index": 29}, {"bbox": [133, 655, 481, 665], "spans": [{"bbox": [133, 655, 481, 665], "score": 1.0, "content": "one or two terminal nodes labeled 1 or 0 and a set of variable nodes. Each variable node", "type": "text"}], "index": 30}, {"bbox": [133, 117, 480, 129], "spans": [{"bbox": [133, 117, 480, 129], "score": 1.0, "content": "is associated with a Boolean variable and has two outgoing edges low and high. Given", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [133, 130, 481, 142], "spans": [{"bbox": [133, 130, 481, 142], "score": 1.0, "content": "an assignment of the variables, the value of the Boolean function is determined by a", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [133, 142, 482, 153], "spans": [{"bbox": [133, 142, 482, 153], "score": 1.0, "content": "path starting at the root node and recursively following the high edge, if the associated", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [134, 154, 481, 164], "spans": [{"bbox": [134, 154, 481, 164], "score": 1.0, "content": "variable is true, and the low edge, if the associated variable is false. The function value", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [133, 165, 481, 177], "spans": [{"bbox": [133, 165, 481, 177], "score": 1.0, "content": "is true, if the label of the reached terminal node is 1; otherwise it is false. The graph is", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [133, 177, 387, 189], "spans": [{"bbox": [133, 177, 387, 189], "score": 1.0, "content": "ordered such that all paths respect the ordering of the variables.", "type": "text", "cross_page": true}], "index": 5}], "index": 29, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [133, 630, 481, 665]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_2.png | images/0704.1394_2.png |
|
0704.1394.pdf | 3 | is associated with a Boolean variable and has two outgoing edges low and high. Given
an assignment of the variables, the value of the Boolean function is determined by a
path starting at the root node and recursively following the high edge, if the associated
variable is true, and the low edge, if the associated variable is false. The function value
is true, if the label of the reached terminal node is 1; otherwise it is false. The graph is
ordered such that all paths respect the ordering of the variables.
A BDD is reduced such that no pair of distinct nodes $$u$$ and $$v$$ are associated with the
same variable and low and high successors (Fig. 3a), and no variable node $$u$$ has iden-
tical low and high successors (Fig. 3b). Due to these reductions, the number of nodes
Fig. 3. (a) nodes associated to the same variable with equal low and high successors
will be converted to a single node. (b) nodes causing redundant tests on a variable are
eliminated. High and low edges are drawn with solid and dashed lines, respectively
in a BDD for many functions encountered in practice is often much smaller than the
number of truth assignments of the function. Another advantage is that the reductions
make BDDs canonical [11]. Large space savings can be obtained by representing a col-
lection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are
shared. Due to the canonicity, two BDDs are identical if and only if they have the same
root. Consequently, when using this representation, equivalence checking between two
BDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any
Boolean operation on two BDDs can be carried out in time proportional to the product
of their size. The size of a BDD can depend critically on the variable ordering. To find
an optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for
choosing an ordering is to locate dependent variables close to each other in the order-
ing. For a comprehensive introduction to BDDs and branching programs in general, we
refer the reader to Bryant’s original paper [11] and the books [12,13].
# 3.2 Compiling the Configuration Model
Each of the finite domain variables $$x_{i}$$ with domain $$D_{i}=\{0,...\,,|D_{i}|-1\}$$ is encoded
by $$k_{i}\,=\,\lceil l o g|D_{i}|\rceil$$ Boolean variables $$x_{0}^{i},\ldots,x_{k_{i}-1}^{i}$$ . Each $$j~\in~D_{i}$$ , corresponds to a
| <p>is associated with a Boolean variable and has two outgoing edges low and high. Given
an assignment of the variables, the value of the Boolean function is determined by a
path starting at the root node and recursively following the high edge, if the associated
variable is true, and the low edge, if the associated variable is false. The function value
is true, if the label of the reached terminal node is 1; otherwise it is false. The graph is
ordered such that all paths respect the ordering of the variables.</p>
<p>A BDD is reduced such that no pair of distinct nodes $$u$$ and $$v$$ are associated with the
same variable and low and high successors (Fig. 3a), and no variable node $$u$$ has iden-
tical low and high successors (Fig. 3b). Due to these reductions, the number of nodes</p>
<p>Fig. 3. (a) nodes associated to the same variable with equal low and high successors
will be converted to a single node. (b) nodes causing redundant tests on a variable are
eliminated. High and low edges are drawn with solid and dashed lines, respectively</p>
<p>in a BDD for many functions encountered in practice is often much smaller than the
number of truth assignments of the function. Another advantage is that the reductions
make BDDs canonical [11]. Large space savings can be obtained by representing a col-
lection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are
shared. Due to the canonicity, two BDDs are identical if and only if they have the same
root. Consequently, when using this representation, equivalence checking between two
BDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any
Boolean operation on two BDDs can be carried out in time proportional to the product
of their size. The size of a BDD can depend critically on the variable ordering. To find
an optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for
choosing an ordering is to locate dependent variables close to each other in the order-
ing. For a comprehensive introduction to BDDs and branching programs in general, we
refer the reader to Bryant’s original paper [11] and the books [12,13].</p>
<h1>3.2 Compiling the Configuration Model</h1>
<p>Each of the finite domain variables $$x_{i}$$ with domain $$D_{i}=\{0,...\,,|D_{i}|-1\}$$ is encoded
by $$k_{i}\,=\,\lceil l o g|D_{i}|\rceil$$ Boolean variables $$x_{0}^{i},\ldots,x_{k_{i}-1}^{i}$$ . Each $$j~\in~D_{i}$$ , corresponds to a</p>
| [{"type": "text", "coordinates": [133, 116, 482, 188], "content": "is associated with a Boolean variable and has two outgoing edges low and high. Given\nan assignment of the variables, the value of the Boolean function is determined by a\npath starting at the root node and recursively following the high edge, if the associated\nvariable is true, and the low edge, if the associated variable is false. The function value\nis true, if the label of the reached terminal node is 1; otherwise it is false. The graph is\nordered such that all paths respect the ordering of the variables.", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [133, 188, 482, 225], "content": "A BDD is reduced such that no pair of distinct nodes $$u$$ and $$v$$ are associated with the\nsame variable and low and high successors (Fig. 3a), and no variable node $$u$$ has iden-\ntical low and high successors (Fig. 3b). Due to these reductions, the number of nodes", "block_type": "text", "index": 2}, {"type": "image", "coordinates": [236, 245, 376, 314], "content": "", "block_type": "image", "index": 3}, {"type": "text", "coordinates": [133, 370, 481, 407], "content": "Fig. 3. (a) nodes associated to the same variable with equal low and high successors\nwill be converted to a single node. (b) nodes causing redundant tests on a variable are\neliminated. High and low edges are drawn with solid and dashed lines, respectively", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [133, 446, 482, 602], "content": "in a BDD for many functions encountered in practice is often much smaller than the\nnumber of truth assignments of the function. Another advantage is that the reductions\nmake BDDs canonical [11]. Large space savings can be obtained by representing a col-\nlection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are\nshared. Due to the canonicity, two BDDs are identical if and only if they have the same\nroot. Consequently, when using this representation, equivalence checking between two\nBDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any\nBoolean operation on two BDDs can be carried out in time proportional to the product\nof their size. The size of a BDD can depend critically on the variable ordering. To find\nan optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for\nchoosing an ordering is to locate dependent variables close to each other in the order-\ning. For a comprehensive introduction to BDDs and branching programs in general, we\nrefer the reader to Bryant\u2019s original paper [11] and the books [12,13].", "block_type": "text", "index": 5}, {"type": "title", "coordinates": [134, 620, 311, 633], "content": "3.2 Compiling the Configuration Model", "block_type": "title", "index": 6}, {"type": "text", "coordinates": [133, 642, 482, 667], "content": "Each of the finite domain variables $$x_{i}$$ with domain $$D_{i}=\\{0,...\\,,|D_{i}|-1\\}$$ is encoded\nby $$k_{i}\\,=\\,\\lceil l o g|D_{i}|\\rceil$$ Boolean variables $$x_{0}^{i},\\ldots,x_{k_{i}-1}^{i}$$ . Each $$j~\\in~D_{i}$$ , corresponds to a", "block_type": "text", "index": 7}] | [{"type": "text", "coordinates": [133, 117, 480, 129], "content": "is associated with a Boolean variable and has two outgoing edges low and high. Given", "score": 1.0, "index": 1}, {"type": "text", "coordinates": [133, 130, 481, 142], "content": "an assignment of the variables, the value of the Boolean function is determined by a", "score": 1.0, "index": 2}, {"type": "text", "coordinates": [133, 142, 482, 153], "content": "path starting at the root node and recursively following the high edge, if the associated", "score": 1.0, "index": 3}, {"type": "text", "coordinates": [134, 154, 481, 164], "content": "variable is true, and the low edge, if the associated variable is false. The function value", "score": 1.0, "index": 4}, {"type": "text", "coordinates": [133, 165, 481, 177], "content": "is true, if the label of the reached terminal node is 1; otherwise it is false. The graph is", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [133, 177, 387, 189], "content": "ordered such that all paths respect the ordering of the variables.", "score": 1.0, "index": 6}, {"type": "text", "coordinates": [148, 189, 358, 201], "content": "A BDD is reduced such that no pair of distinct nodes", "score": 1.0, "index": 7}, {"type": "inline_equation", "coordinates": [358, 190, 365, 198], "content": "u", "score": 0.77, "index": 8}, {"type": "text", "coordinates": [366, 189, 382, 201], "content": " and", "score": 1.0, "index": 9}, {"type": "inline_equation", "coordinates": [383, 190, 389, 198], "content": "v", "score": 0.76, "index": 10}, {"type": "text", "coordinates": [389, 189, 481, 201], "content": " are associated with the", "score": 1.0, "index": 11}, {"type": "text", "coordinates": [133, 202, 434, 212], "content": "same variable and low and high successors (Fig. 3a), and no variable node", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [434, 203, 442, 210], "content": "u", "score": 0.68, "index": 13}, {"type": "text", "coordinates": [442, 202, 480, 212], "content": " has iden-", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [134, 213, 480, 225], "content": "tical low and high successors (Fig. 3b). Due to these reductions, the number of nodes", "score": 1.0, "index": 15}, {"type": "text", "coordinates": [134, 372, 481, 383], "content": "Fig. 3. (a) nodes associated to the same variable with equal low and high successors", "score": 1.0, "index": 16}, {"type": "text", "coordinates": [134, 384, 482, 396], "content": "will be converted to a single node. (b) nodes causing redundant tests on a variable are", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [134, 396, 466, 407], "content": "eliminated. High and low edges are drawn with solid and dashed lines, respectively", "score": 1.0, "index": 18}, {"type": "text", "coordinates": [133, 448, 481, 458], "content": "in a BDD for many functions encountered in practice is often much smaller than the", "score": 1.0, "index": 19}, {"type": "text", "coordinates": [133, 459, 482, 470], "content": "number of truth assignments of the function. Another advantage is that the reductions", "score": 1.0, "index": 20}, {"type": "text", "coordinates": [134, 472, 481, 483], "content": "make BDDs canonical [11]. Large space savings can be obtained by representing a col-", "score": 1.0, "index": 21}, {"type": "text", "coordinates": [133, 483, 482, 495], "content": "lection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [133, 495, 481, 507], "content": "shared. Due to the canonicity, two BDDs are identical if and only if they have the same", "score": 1.0, "index": 23}, {"type": "text", "coordinates": [133, 507, 481, 519], "content": "root. Consequently, when using this representation, equivalence checking between two", "score": 1.0, "index": 24}, {"type": "text", "coordinates": [133, 519, 481, 531], "content": "BDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [134, 532, 482, 543], "content": "Boolean operation on two BDDs can be carried out in time proportional to the product", "score": 1.0, "index": 26}, {"type": "text", "coordinates": [134, 543, 481, 554], "content": "of their size. The size of a BDD can depend critically on the variable ordering. To find", "score": 1.0, "index": 27}, {"type": "text", "coordinates": [134, 555, 481, 567], "content": "an optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for", "score": 1.0, "index": 28}, {"type": "text", "coordinates": [133, 567, 482, 579], "content": "choosing an ordering is to locate dependent variables close to each other in the order-", "score": 1.0, "index": 29}, {"type": "text", "coordinates": [133, 579, 482, 591], "content": "ing. For a comprehensive introduction to BDDs and branching programs in general, we", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [133, 591, 411, 603], "content": "refer the reader to Bryant\u2019s original paper [11] and the books [12,13].", "score": 1.0, "index": 31}, {"type": "text", "coordinates": [133, 621, 310, 634], "content": "3.2 Compiling the Configuration Model", "score": 1.0, "index": 32}, {"type": "text", "coordinates": [133, 641, 275, 654], "content": "Each of the finite domain variables ", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [275, 645, 284, 653], "content": "x_{i}", "score": 0.73, "index": 34}, {"type": "text", "coordinates": [285, 641, 339, 654], "content": " with domain ", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [339, 641, 435, 654], "content": "D_{i}=\\{0,...\\,,|D_{i}|-1\\}", "score": 0.93, "index": 36}, {"type": "text", "coordinates": [436, 641, 481, 654], "content": " is encoded", "score": 1.0, "index": 37}, {"type": "text", "coordinates": [132, 653, 146, 670], "content": "by", "score": 1.0, "index": 38}, {"type": "inline_equation", "coordinates": [147, 653, 211, 666], "content": "k_{i}\\,=\\,\\lceil l o g|D_{i}|\\rceil", "score": 0.93, "index": 39}, {"type": "text", "coordinates": [212, 653, 290, 670], "content": "Boolean variables ", "score": 1.0, "index": 40}, {"type": "inline_equation", "coordinates": [290, 654, 347, 667], "content": "x_{0}^{i},\\ldots,x_{k_{i}-1}^{i}", "score": 0.93, "index": 41}, {"type": "text", "coordinates": [347, 653, 374, 670], "content": ". Each", "score": 1.0, "index": 42}, {"type": "inline_equation", "coordinates": [375, 654, 407, 665], "content": "j~\\in~D_{i}", "score": 0.9, "index": 43}, {"type": "text", "coordinates": [408, 653, 483, 670], "content": ", corresponds to a", "score": 1.0, "index": 44}] | [{"coordinates": [236, 245, 376, 314], "index": 11, "caption": "(b)", "caption_coordinates": [353, 336, 367, 348]}] | [{"type": "inline", "coordinates": [358, 190, 365, 198], "content": "u", "caption": ""}, {"type": "inline", "coordinates": [383, 190, 389, 198], "content": "v", "caption": ""}, {"type": "inline", "coordinates": [434, 203, 442, 210], "content": "u", "caption": ""}, {"type": "inline", "coordinates": [275, 645, 284, 653], "content": "x_{i}", "caption": ""}, {"type": "inline", "coordinates": [339, 641, 435, 654], "content": "D_{i}=\\{0,...\\,,|D_{i}|-1\\}", "caption": ""}, {"type": "inline", "coordinates": [147, 653, 211, 666], "content": "k_{i}\\,=\\,\\lceil l o g|D_{i}|\\rceil", "caption": ""}, {"type": "inline", "coordinates": [290, 654, 347, 667], "content": "x_{0}^{i},\\ldots,x_{k_{i}-1}^{i}", "caption": ""}, {"type": "inline", "coordinates": [375, 654, 407, 665], "content": "j~\\in~D_{i}", "caption": ""}] | [] | [612.0, 792.0] | [{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "A BDD is reduced such that no pair of distinct nodes $u$ and $v$ are associated with the same variable and low and high successors (Fig. 3a), and no variable node $u$ has identical low and high successors (Fig. 3b). Due to these reductions, the number of nodes ", "page_idx": 3}, {"type": "image", "img_path": "images/7eba4616c041285bee63c6093edac211fe629f32875f63abcbb53f289f599bc2.jpg", "img_caption": ["(a) ", "(b) "], "img_footnote": [], "page_idx": 3}, {"type": "text", "text": "Fig. 3. (a) nodes associated to the same variable with equal low and high successors will be converted to a single node. (b) nodes causing redundant tests on a variable are eliminated. High and low edges are drawn with solid and dashed lines, respectively ", "page_idx": 3}, {"type": "text", "text": "in a BDD for many functions encountered in practice is often much smaller than the number of truth assignments of the function. Another advantage is that the reductions make BDDs canonical [11]. Large space savings can be obtained by representing a collection of BDDs in a single multi-rooted graph where the sub-graphs of the BDDs are shared. Due to the canonicity, two BDDs are identical if and only if they have the same root. Consequently, when using this representation, equivalence checking between two BDDs can be done in constant time. In addition, BDDs are easy to manipulate. Any Boolean operation on two BDDs can be carried out in time proportional to the product of their size. The size of a BDD can depend critically on the variable ordering. To find an optimal ordering is a co-NP-complete problem in itself [11], but a good heuristic for choosing an ordering is to locate dependent variables close to each other in the ordering. For a comprehensive introduction to BDDs and branching programs in general, we refer the reader to Bryant\u2019s original paper [11] and the books [12,13]. ", "page_idx": 3}, {"type": "text", "text": "3.2 Compiling the Configuration Model ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "Each of the finite domain variables $x_{i}$ with domain $D_{i}=\\{0,...\\,,|D_{i}|-1\\}$ is encoded by $k_{i}\\,=\\,\\lceil l o g|D_{i}|\\rceil$ Boolean variables $x_{0}^{i},\\ldots,x_{k_{i}-1}^{i}$ . Each $j~\\in~D_{i}$ , corresponds to a binary encoding $\\overline{{v_{0}\\ldots v_{k_{i}-1}}}$ denoted as $v_{0}\\dots\\cdot v_{k_{i}-1}=e n c(j)$ . Also, every combination of Boolean values $v_{0}\\ldots v_{k_{i}-1}$ represents some integer $j\\,\\leq\\,2^{k_{i}}\\,-\\,1$ , denoted as $j=d e c(v_{0}\\ldots v_{k_{i}-1})$ . Hence, atomic proposition $x_{i}=v$ is encoded as a Boolean expression $x_{0}^{i}\\,=\\,v_{0}\\land\\dotsc\\land x_{k_{i}-1}^{i}\\,=\\,v_{k_{i}-1}$ . 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Hence, atomic proposition", "type": "text", "cross_page": true}, {"bbox": [335, 142, 365, 151], "score": 0.89, "content": "x_{i}=v", "type": "inline_equation", "height": 9, "width": 30, "cross_page": true}, {"bbox": [366, 141, 482, 154], "score": 1.0, "content": " is encoded as a Boolean ex-", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [131, 151, 484, 169], "spans": [{"bbox": [131, 151, 170, 169], "score": 1.0, "content": "pression ", "type": "text", "cross_page": true}, {"bbox": [171, 152, 305, 166], "score": 0.89, "content": "x_{0}^{i}\\,=\\,v_{0}\\land\\dotsc\\land x_{k_{i}-1}^{i}\\,=\\,v_{k_{i}-1}", "type": "inline_equation", "height": 14, "width": 134, "cross_page": true}, {"bbox": [306, 151, 484, 169], "score": 1.0, "content": ". In addition, domain constraints are added", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [133, 164, 481, 178], "spans": [{"bbox": [133, 165, 256, 178], "score": 1.0, "content": "to forbid those assignments to", "type": "text", "cross_page": true}, {"bbox": [257, 167, 304, 176], "score": 0.88, "content": "v_{0}\\ldots v_{k_{i}-1}", "type": "inline_equation", "height": 9, "width": 47, "cross_page": true}, {"bbox": [305, 165, 450, 178], "score": 1.0, "content": " which do not translate to a value in", "type": "text", "cross_page": true}, {"bbox": [450, 164, 463, 175], "score": 0.88, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [463, 165, 481, 178], "score": 1.0, "content": ", i.e.", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [134, 176, 264, 189], "spans": [{"bbox": [134, 177, 160, 189], "score": 1.0, "content": "where", "type": "text", "cross_page": true}, {"bbox": [160, 176, 261, 189], "score": 0.91, "content": "d e c(v_{0}\\ldots v_{k_{i}-1})\\geq|D_{i}|", "type": "inline_equation", "height": 13, "width": 101, "cross_page": true}, {"bbox": [261, 177, 264, 189], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 5}], "index": 30.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [132, 641, 483, 670]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_3.png | images/0704.1394_3.png |
|
0704.1394.pdf | 4 | binary encoding $$\overline{{v_{0}\ldots v_{k_{i}-1}}}$$ denoted as $$v_{0}\dots\cdot v_{k_{i}-1}=e n c(j)$$ . Also, every combina-
tion of Boolean values $$v_{0}\ldots v_{k_{i}-1}$$ represents some integer $$j\,\leq\,2^{k_{i}}\,-\,1$$ , denoted as
$$j=d e c(v_{0}\ldots v_{k_{i}-1})$$ . Hence, atomic proposition $$x_{i}=v$$ is encoded as a Boolean ex-
pression $$x_{0}^{i}\,=\,v_{0}\land\dotsc\land x_{k_{i}-1}^{i}\,=\,v_{k_{i}-1}$$ . In addition, domain constraints are added
to forbid those assignments to $$v_{0}\ldots v_{k_{i}-1}$$ which do not translate to a value in $$D_{i}$$ , i.e.
where $$d e c(v_{0}\ldots v_{k_{i}-1})\geq|D_{i}|$$ .
Let the solution space $$S o l$$ over ordered set of variables $$x_{0}<...<x_{k-1}$$ be repre-
sented by a Binary Decision Diagram $$B(V,E,X_{b},R,v a r)$$ , where $$V$$ is the set of nodes
$$u$$ , $$E$$ is the set of edges $$e$$ and $$X_{b}=\{0,1,\ldots,|X_{b}|-1\}$$ is an ordered set of variable
indexes, labelling every non-terminal node $$u$$ with $$v a r(u)\,\leq\,|X_{b}|\,-\,1$$ and labelling
the terminal nodes $$T_{0},T_{1}$$ with index $$|X_{b}|$$ . Set of variable indexes $$X_{b}$$ is constructed
by taking the union of Boolean encoding variables $$\bigcup_{i=0}^{n-1}\{x_{0}^{i},\dotsc,x_{k_{i}-1}^{i}\}$$ and ordering
them in a natural layered way, i.e. $$x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}$$ iff $$i_{1}<i_{2}$$ or $$i_{1}=i_{2}$$ and $$j_{1}<j_{2}$$ .
Every directed edge $$\boldsymbol{e}~=~(u_{1},\bar{u}_{2})$$ has a starting vertex $$u_{1}~=~\pi_{1}(e)$$ and ending
vertex $$u_{2}=\pi_{2}(e).\,R$$ denotes the root node of the BDD.
Example 2. The BDD representing the solution space of the T-shirt example introduced
in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables: $$x_{1},x_{2}$$ and
$$x_{3}$$ , whose domain sizes are four, three and two, respectively. Each variable is repre-
sented by a vector of Boolean variables. In the figure the Boolean vector for the vari-
able $$x_{i}$$ with domain $$D_{i}$$ is $$(x_{i}^{0},x_{i}^{1},\cdot\cdot\cdot x_{i}^{l_{i}-1})$$ xlii−1), where li = ⌈lg |Di|⌉. For example, in
the figure, variable $$x_{2}$$ which corresponds to the size of the T-shirt is represented by the
Boolean vector $$(x_{2}^{0},x_{2}^{1})$$ . In the BDD any path from the root node to the terminal node
1, corresponds to one or more valid configurations. For example, the path from the root
node to the terminal node 1, with all the variables taking low values represents the valid
configuration (black, small, MIB). Another path with $$x_{1}^{0},x_{1}^{1}$$ , and $$\boldsymbol{x}_{2}^{0}$$ taking low values,
and $$x_{2}^{1}$$ taking high value represents two valid configurations: (black, medium, $$M I B$$ )
and (black, medium, $$S T W$$ ), namely. In this path the variable $$\boldsymbol{x}_{3}^{0}$$ is a don’t care variable
and hence can take both low and high value, which leads to two valid configurations.
Any path from the root node to the terminal node 0 corresponds to invalid configura-
tions. $$\diamondsuit$$
# 4 Calculating Valid Domains
Before showing the algorithms, let us first introduce the appropriate notation. If an
index $$k\ \in\ X_{b}$$ corresponds to the $$j+1$$ -st Boolean variable $$\boldsymbol{x}_{j}^{i}$$ encoding the finite
domain variable $$x_{i}$$ , we define $$v a r_{1}(k)\;=\;i$$ and $$v a r_{2}(k)\,=\,j$$ to be the appropriate
mappings. Now, given the BDD $$B(V,E,X_{b},R,v a r)$$ , $$V_{i}$$ denotes the set of all nodes
$$u\in V$$ that are labelled with a BDD variable encoding the finite domain variable $$x_{i}$$ , i.e.
$$V_{i}=\{u\in V\mid v a r_{1}(u)=i\}$$ . We think of $$V_{i}$$ as defining a layer in the BDD. We define
$$I n_{i}$$ to be the set of nodes $$u\in V_{i}$$ reachable by an edge originating from outside the $$V_{i}$$
layer, i.e. $$I n_{i}=\{u\in V_{i}|\;\exists(u^{\prime},u)\in E.\;v a r_{1}(u^{\prime})<i\}$$ . For the root node $$R$$ , labelled
with $$i_{0}=v a r_{1}(R)$$ we define $$I n_{i_{0}}=V_{i_{0}}=\{R\}$$ .
We assume that in the previous user assignment, a user fixed a value for a finite
domain variable $$x=v,x\in X$$ , extending the old partial assignment $$\rho_{o l d}$$ to the current
| <p>binary encoding $$\overline{{v_{0}\ldots v_{k_{i}-1}}}$$ denoted as $$v_{0}\dots\cdot v_{k_{i}-1}=e n c(j)$$ . Also, every combina-
tion of Boolean values $$v_{0}\ldots v_{k_{i}-1}$$ represents some integer $$j\,\leq\,2^{k_{i}}\,-\,1$$ , denoted as
$$j=d e c(v_{0}\ldots v_{k_{i}-1})$$ . Hence, atomic proposition $$x_{i}=v$$ is encoded as a Boolean ex-
pression $$x_{0}^{i}\,=\,v_{0}\land\dotsc\land x_{k_{i}-1}^{i}\,=\,v_{k_{i}-1}$$ . In addition, domain constraints are added
to forbid those assignments to $$v_{0}\ldots v_{k_{i}-1}$$ which do not translate to a value in $$D_{i}$$ , i.e.
where $$d e c(v_{0}\ldots v_{k_{i}-1})\geq|D_{i}|$$ .</p>
<p>Let the solution space $$S o l$$ over ordered set of variables $$x_{0}<...<x_{k-1}$$ be repre-
sented by a Binary Decision Diagram $$B(V,E,X_{b},R,v a r)$$ , where $$V$$ is the set of nodes
$$u$$ , $$E$$ is the set of edges $$e$$ and $$X_{b}=\{0,1,\ldots,|X_{b}|-1\}$$ is an ordered set of variable
indexes, labelling every non-terminal node $$u$$ with $$v a r(u)\,\leq\,|X_{b}|\,-\,1$$ and labelling
the terminal nodes $$T_{0},T_{1}$$ with index $$|X_{b}|$$ . Set of variable indexes $$X_{b}$$ is constructed
by taking the union of Boolean encoding variables $$\bigcup_{i=0}^{n-1}\{x_{0}^{i},\dotsc,x_{k_{i}-1}^{i}\}$$ and ordering
them in a natural layered way, i.e. $$x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}$$ iff $$i_{1}<i_{2}$$ or $$i_{1}=i_{2}$$ and $$j_{1}<j_{2}$$ .</p>
<p>Every directed edge $$\boldsymbol{e}~=~(u_{1},\bar{u}_{2})$$ has a starting vertex $$u_{1}~=~\pi_{1}(e)$$ and ending
vertex $$u_{2}=\pi_{2}(e).\,R$$ denotes the root node of the BDD.</p>
<p>Example 2. The BDD representing the solution space of the T-shirt example introduced
in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables: $$x_{1},x_{2}$$ and
$$x_{3}$$ , whose domain sizes are four, three and two, respectively. Each variable is repre-
sented by a vector of Boolean variables. In the figure the Boolean vector for the vari-
able $$x_{i}$$ with domain $$D_{i}$$ is $$(x_{i}^{0},x_{i}^{1},\cdot\cdot\cdot x_{i}^{l_{i}-1})$$ xlii−1), where li = ⌈lg |Di|⌉. For example, in
the figure, variable $$x_{2}$$ which corresponds to the size of the T-shirt is represented by the
Boolean vector $$(x_{2}^{0},x_{2}^{1})$$ . In the BDD any path from the root node to the terminal node
1, corresponds to one or more valid configurations. For example, the path from the root
node to the terminal node 1, with all the variables taking low values represents the valid
configuration (black, small, MIB). Another path with $$x_{1}^{0},x_{1}^{1}$$ , and $$\boldsymbol{x}_{2}^{0}$$ taking low values,
and $$x_{2}^{1}$$ taking high value represents two valid configurations: (black, medium, $$M I B$$ )
and (black, medium, $$S T W$$ ), namely. In this path the variable $$\boldsymbol{x}_{3}^{0}$$ is a don’t care variable
and hence can take both low and high value, which leads to two valid configurations.
Any path from the root node to the terminal node 0 corresponds to invalid configura-
tions. $$\diamondsuit$$</p>
<h1>4 Calculating Valid Domains</h1>
<p>Before showing the algorithms, let us first introduce the appropriate notation. If an
index $$k\ \in\ X_{b}$$ corresponds to the $$j+1$$ -st Boolean variable $$\boldsymbol{x}_{j}^{i}$$ encoding the finite
domain variable $$x_{i}$$ , we define $$v a r_{1}(k)\;=\;i$$ and $$v a r_{2}(k)\,=\,j$$ to be the appropriate
mappings. Now, given the BDD $$B(V,E,X_{b},R,v a r)$$ , $$V_{i}$$ denotes the set of all nodes
$$u\in V$$ that are labelled with a BDD variable encoding the finite domain variable $$x_{i}$$ , i.e.
$$V_{i}=\{u\in V\mid v a r_{1}(u)=i\}$$ . We think of $$V_{i}$$ as defining a layer in the BDD. We define
$$I n_{i}$$ to be the set of nodes $$u\in V_{i}$$ reachable by an edge originating from outside the $$V_{i}$$
layer, i.e. $$I n_{i}=\{u\in V_{i}|\;\exists(u^{\prime},u)\in E.\;v a r_{1}(u^{\prime})<i\}$$ . For the root node $$R$$ , labelled
with $$i_{0}=v a r_{1}(R)$$ we define $$I n_{i_{0}}=V_{i_{0}}=\{R\}$$ .</p>
<p>We assume that in the previous user assignment, a user fixed a value for a finite
domain variable $$x=v,x\in X$$ , extending the old partial assignment $$\rho_{o l d}$$ to the current</p>
| [{"type": "text", "coordinates": [133, 116, 482, 188], "content": "binary encoding $$\\overline{{v_{0}\\ldots v_{k_{i}-1}}}$$ denoted as $$v_{0}\\dots\\cdot v_{k_{i}-1}=e n c(j)$$ . Also, every combina-\ntion of Boolean values $$v_{0}\\ldots v_{k_{i}-1}$$ represents some integer $$j\\,\\leq\\,2^{k_{i}}\\,-\\,1$$ , denoted as\n$$j=d e c(v_{0}\\ldots v_{k_{i}-1})$$ . Hence, atomic proposition $$x_{i}=v$$ is encoded as a Boolean ex-\npression $$x_{0}^{i}\\,=\\,v_{0}\\land\\dotsc\\land x_{k_{i}-1}^{i}\\,=\\,v_{k_{i}-1}$$ . In addition, domain constraints are added\nto forbid those assignments to $$v_{0}\\ldots v_{k_{i}-1}$$ which do not translate to a value in $$D_{i}$$ , i.e.\nwhere $$d e c(v_{0}\\ldots v_{k_{i}-1})\\geq|D_{i}|$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [133, 189, 482, 277], "content": "Let the solution space $$S o l$$ over ordered set of variables $$x_{0}<...<x_{k-1}$$ be repre-\nsented by a Binary Decision Diagram $$B(V,E,X_{b},R,v a r)$$ , where $$V$$ is the set of nodes\n$$u$$ , $$E$$ is the set of edges $$e$$ and $$X_{b}=\\{0,1,\\ldots,|X_{b}|-1\\}$$ is an ordered set of variable\nindexes, labelling every non-terminal node $$u$$ with $$v a r(u)\\,\\leq\\,|X_{b}|\\,-\\,1$$ and labelling\nthe terminal nodes $$T_{0},T_{1}$$ with index $$|X_{b}|$$ . Set of variable indexes $$X_{b}$$ is constructed\nby taking the union of Boolean encoding variables $$\\bigcup_{i=0}^{n-1}\\{x_{0}^{i},\\dotsc,x_{k_{i}-1}^{i}\\}$$ and ordering\nthem in a natural layered way, i.e. $$x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}$$ iff $$i_{1}<i_{2}$$ or $$i_{1}=i_{2}$$ and $$j_{1}<j_{2}$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [134, 276, 482, 299], "content": "Every directed edge $$\\boldsymbol{e}~=~(u_{1},\\bar{u}_{2})$$ has a starting vertex $$u_{1}~=~\\pi_{1}(e)$$ and ending\nvertex $$u_{2}=\\pi_{2}(e).\\,R$$ denotes the root node of the BDD.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [134, 306, 482, 484], "content": "Example 2. The BDD representing the solution space of the T-shirt example introduced\nin Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables: $$x_{1},x_{2}$$ and\n$$x_{3}$$ , whose domain sizes are four, three and two, respectively. Each variable is repre-\nsented by a vector of Boolean variables. In the figure the Boolean vector for the vari-\nable $$x_{i}$$ with domain $$D_{i}$$ is $$(x_{i}^{0},x_{i}^{1},\\cdot\\cdot\\cdot x_{i}^{l_{i}-1})$$ xlii\u22121), where li = \u2308lg |Di|\u2309. For example, in\nthe figure, variable $$x_{2}$$ which corresponds to the size of the T-shirt is represented by the\nBoolean vector $$(x_{2}^{0},x_{2}^{1})$$ . In the BDD any path from the root node to the terminal node\n1, corresponds to one or more valid configurations. For example, the path from the root\nnode to the terminal node 1, with all the variables taking low values represents the valid\nconfiguration (black, small, MIB). Another path with $$x_{1}^{0},x_{1}^{1}$$ , and $$\\boldsymbol{x}_{2}^{0}$$ taking low values,\nand $$x_{2}^{1}$$ taking high value represents two valid configurations: (black, medium, $$M I B$$ )\nand (black, medium, $$S T W$$ ), namely. In this path the variable $$\\boldsymbol{x}_{3}^{0}$$ is a don\u2019t care variable\nand hence can take both low and high value, which leads to two valid configurations.\nAny path from the root node to the terminal node 0 corresponds to invalid configura-\ntions. $$\\diamondsuit$$", "block_type": "text", "index": 4}, {"type": "title", "coordinates": [134, 510, 290, 524], "content": "4 Calculating Valid Domains", "block_type": "title", "index": 5}, {"type": "text", "coordinates": [133, 533, 482, 642], "content": "Before showing the algorithms, let us first introduce the appropriate notation. If an\nindex $$k\\ \\in\\ X_{b}$$ corresponds to the $$j+1$$ -st Boolean variable $$\\boldsymbol{x}_{j}^{i}$$ encoding the finite\ndomain variable $$x_{i}$$ , we define $$v a r_{1}(k)\\;=\\;i$$ and $$v a r_{2}(k)\\,=\\,j$$ to be the appropriate\nmappings. Now, given the BDD $$B(V,E,X_{b},R,v a r)$$ , $$V_{i}$$ denotes the set of all nodes\n$$u\\in V$$ that are labelled with a BDD variable encoding the finite domain variable $$x_{i}$$ , i.e.\n$$V_{i}=\\{u\\in V\\mid v a r_{1}(u)=i\\}$$ . We think of $$V_{i}$$ as defining a layer in the BDD. We define\n$$I n_{i}$$ to be the set of nodes $$u\\in V_{i}$$ reachable by an edge originating from outside the $$V_{i}$$\nlayer, i.e. $$I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}$$ . For the root node $$R$$ , labelled\nwith $$i_{0}=v a r_{1}(R)$$ we define $$I n_{i_{0}}=V_{i_{0}}=\\{R\\}$$ .", "block_type": "text", "index": 6}, {"type": "text", "coordinates": [133, 642, 482, 665], "content": "We assume that in the previous user assignment, a user fixed a value for a finite\ndomain variable $$x=v,x\\in X$$ , extending the old partial assignment $$\\rho_{o l d}$$ to the current", "block_type": "text", "index": 7}] | [{"type": "text", "coordinates": [133, 117, 201, 130], "content": "binary encoding", "score": 1.0, "index": 1}, {"type": "inline_equation", "coordinates": [201, 118, 250, 128], "content": "\\overline{{v_{0}\\ldots v_{k_{i}-1}}}", "score": 0.91, "index": 2}, {"type": "text", "coordinates": [251, 117, 297, 130], "content": " denoted as", "score": 1.0, "index": 3}, {"type": "inline_equation", "coordinates": [298, 117, 388, 128], "content": "v_{0}\\dots\\cdot v_{k_{i}-1}=e n c(j)", "score": 0.91, "index": 4}, {"type": "text", "coordinates": [389, 117, 480, 130], "content": ". Also, every combina-", "score": 1.0, "index": 5}, {"type": "text", "coordinates": [132, 128, 228, 142], "content": "tion of Boolean values", "score": 1.0, "index": 6}, {"type": "inline_equation", "coordinates": [229, 131, 277, 140], "content": "v_{0}\\ldots v_{k_{i}-1}", "score": 0.89, "index": 7}, {"type": "text", "coordinates": [277, 128, 378, 142], "content": " represents some integer ", "score": 1.0, "index": 8}, {"type": "inline_equation", "coordinates": [379, 128, 432, 140], "content": "j\\,\\leq\\,2^{k_{i}}\\,-\\,1", "score": 0.92, "index": 9}, {"type": "text", "coordinates": [432, 128, 482, 142], "content": ", denoted as", "score": 1.0, "index": 10}, {"type": "inline_equation", "coordinates": [133, 140, 222, 152], "content": "j=d e c(v_{0}\\ldots v_{k_{i}-1})", "score": 0.91, "index": 11}, {"type": "text", "coordinates": [223, 141, 335, 154], "content": ". Hence, atomic proposition", "score": 1.0, "index": 12}, {"type": "inline_equation", "coordinates": [335, 142, 365, 151], "content": "x_{i}=v", "score": 0.89, "index": 13}, {"type": "text", "coordinates": [366, 141, 482, 154], "content": " is encoded as a Boolean ex-", "score": 1.0, "index": 14}, {"type": "text", "coordinates": [131, 151, 170, 169], "content": "pression ", "score": 1.0, "index": 15}, {"type": "inline_equation", "coordinates": [171, 152, 305, 166], "content": "x_{0}^{i}\\,=\\,v_{0}\\land\\dotsc\\land x_{k_{i}-1}^{i}\\,=\\,v_{k_{i}-1}", "score": 0.89, "index": 16}, {"type": "text", "coordinates": [306, 151, 484, 169], "content": ". In addition, domain constraints are added", "score": 1.0, "index": 17}, {"type": "text", "coordinates": [133, 165, 256, 178], "content": "to forbid those assignments to", "score": 1.0, "index": 18}, {"type": "inline_equation", "coordinates": [257, 167, 304, 176], "content": "v_{0}\\ldots v_{k_{i}-1}", "score": 0.88, "index": 19}, {"type": "text", "coordinates": [305, 165, 450, 178], "content": " which do not translate to a value in", "score": 1.0, "index": 20}, {"type": "inline_equation", "coordinates": [450, 164, 463, 175], "content": "D_{i}", "score": 0.88, "index": 21}, {"type": "text", "coordinates": [463, 165, 481, 178], "content": ", i.e.", "score": 1.0, "index": 22}, {"type": "text", "coordinates": [134, 177, 160, 189], "content": "where", "score": 1.0, "index": 23}, {"type": "inline_equation", "coordinates": [160, 176, 261, 189], "content": "d e c(v_{0}\\ldots v_{k_{i}-1})\\geq|D_{i}|", "score": 0.91, "index": 24}, {"type": "text", "coordinates": [261, 177, 264, 189], "content": ".", "score": 1.0, "index": 25}, {"type": "text", "coordinates": [148, 189, 238, 201], "content": "Let the solution space", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [238, 189, 254, 199], "content": "S o l", "score": 0.78, "index": 27}, {"type": "text", "coordinates": [254, 189, 371, 201], "content": " over ordered set of variables", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [371, 190, 442, 200], "content": "x_{0}<...<x_{k-1}", "score": 0.9, "index": 29}, {"type": "text", "coordinates": [442, 189, 480, 201], "content": " be repre-", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [133, 201, 284, 212], "content": "sented by a Binary Decision Diagram", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [285, 200, 368, 212], "content": "B(V,E,X_{b},R,v a r)", "score": 0.91, "index": 32}, {"type": "text", "coordinates": [368, 201, 398, 212], "content": ", where", "score": 1.0, "index": 33}, {"type": "inline_equation", "coordinates": [398, 201, 408, 210], "content": "V", "score": 0.78, "index": 34}, {"type": "text", "coordinates": [408, 201, 481, 212], "content": "is the set of nodes", "score": 1.0, "index": 35}, {"type": "inline_equation", "coordinates": [133, 214, 140, 222], "content": "u", "score": 0.44, "index": 36}, {"type": "text", "coordinates": [141, 213, 144, 225], "content": ",", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [144, 212, 154, 222], "content": "E", "score": 0.63, "index": 38}, {"type": "text", "coordinates": [154, 213, 230, 225], "content": " is the set of edges", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [230, 214, 237, 222], "content": "e", "score": 0.73, "index": 40}, {"type": "text", "coordinates": [237, 213, 255, 225], "content": " and", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [255, 212, 365, 224], "content": "X_{b}=\\{0,1,\\ldots,|X_{b}|-1\\}", "score": 0.92, "index": 42}, {"type": "text", "coordinates": [366, 213, 482, 225], "content": " is an ordered set of variable", "score": 1.0, "index": 43}, {"type": "text", "coordinates": [133, 225, 310, 237], "content": "indexes, labelling every non-terminal node", "score": 1.0, "index": 44}, {"type": "inline_equation", "coordinates": [311, 226, 318, 234], "content": "u", "score": 0.75, "index": 45}, {"type": "text", "coordinates": [318, 225, 341, 237], "content": " with ", "score": 1.0, "index": 46}, {"type": "inline_equation", "coordinates": [342, 224, 424, 236], "content": "v a r(u)\\,\\leq\\,|X_{b}|\\,-\\,1", "score": 0.92, "index": 47}, {"type": "text", "coordinates": [424, 225, 481, 237], "content": " and labelling", "score": 1.0, "index": 48}, {"type": "text", "coordinates": [133, 237, 212, 249], "content": "the terminal nodes ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [212, 236, 239, 248], "content": "T_{0},T_{1}", "score": 0.92, "index": 50}, {"type": "text", "coordinates": [239, 237, 287, 249], "content": " with index ", "score": 1.0, "index": 51}, {"type": "inline_equation", "coordinates": [288, 236, 306, 248], "content": "|X_{b}|", "score": 0.92, "index": 52}, {"type": "text", "coordinates": [306, 237, 407, 249], "content": ". Set of variable indexes", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [408, 237, 421, 247], "content": "X_{b}", "score": 0.87, "index": 54}, {"type": "text", "coordinates": [421, 237, 482, 249], "content": " is constructed", "score": 1.0, "index": 55}, {"type": "text", "coordinates": [133, 249, 335, 264], "content": "by taking the union of Boolean encoding variables", "score": 1.0, "index": 56}, {"type": "inline_equation", "coordinates": [336, 248, 427, 262], "content": "\\bigcup_{i=0}^{n-1}\\{x_{0}^{i},\\dotsc,x_{k_{i}-1}^{i}\\}", "score": 0.94, "index": 57}, {"type": "text", "coordinates": [428, 249, 482, 264], "content": " and ordering", "score": 1.0, "index": 58}, {"type": "text", "coordinates": [133, 264, 270, 278], "content": "them in a natural layered way, i.e.", "score": 1.0, "index": 59}, {"type": "inline_equation", "coordinates": [270, 261, 311, 276], "content": "x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}", "score": 0.93, "index": 60}, {"type": "text", "coordinates": [311, 261, 324, 279], "content": " iff", "score": 1.0, "index": 61}, {"type": "inline_equation", "coordinates": [325, 263, 355, 275], "content": "i_{1}<i_{2}", "score": 0.9, "index": 62}, {"type": "text", "coordinates": [355, 261, 367, 279], "content": " or", "score": 1.0, "index": 63}, {"type": "inline_equation", "coordinates": [367, 263, 397, 274], "content": "i_{1}=i_{2}", "score": 0.9, "index": 64}, {"type": "text", "coordinates": [397, 261, 415, 279], "content": " and", "score": 1.0, "index": 65}, {"type": "inline_equation", "coordinates": [415, 263, 447, 275], "content": "j_{1}<j_{2}", "score": 0.91, "index": 66}, {"type": "text", "coordinates": [447, 261, 451, 279], "content": ".", "score": 1.0, "index": 67}, {"type": "text", "coordinates": [148, 275, 234, 288], "content": "Every directed edge ", "score": 1.0, "index": 68}, {"type": "inline_equation", "coordinates": [234, 275, 290, 287], "content": "\\boldsymbol{e}~=~(u_{1},\\bar{u}_{2})", "score": 0.92, "index": 69}, {"type": "text", "coordinates": [290, 275, 379, 288], "content": " has a starting vertex", "score": 1.0, "index": 70}, {"type": "inline_equation", "coordinates": [380, 275, 431, 286], "content": "u_{1}~=~\\pi_{1}(e)", "score": 0.93, "index": 71}, {"type": "text", "coordinates": [432, 275, 480, 288], "content": " and ending", "score": 1.0, "index": 72}, {"type": "text", "coordinates": [134, 288, 160, 299], "content": "vertex", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [160, 286, 220, 299], "content": "u_{2}=\\pi_{2}(e).\\,R", "score": 0.78, "index": 74}, {"type": "text", "coordinates": [221, 288, 360, 299], "content": " denotes the root node of the BDD.", "score": 1.0, "index": 75}, {"type": "text", "coordinates": [134, 307, 482, 318], "content": "Example 2. The BDD representing the solution space of the T-shirt example introduced", "score": 1.0, "index": 76}, {"type": "text", "coordinates": [133, 318, 437, 330], "content": "in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables:", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [438, 320, 464, 329], "content": "x_{1},x_{2}", "score": 0.89, "index": 78}, {"type": "text", "coordinates": [464, 318, 481, 330], "content": " and", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [133, 332, 145, 342], "content": "x_{3}", "score": 0.82, "index": 80}, {"type": "text", "coordinates": [145, 331, 482, 343], "content": ", whose domain sizes are four, three and two, respectively. Each variable is repre-", "score": 1.0, "index": 81}, {"type": "text", "coordinates": [134, 343, 481, 354], "content": "sented by a vector of Boolean variables. In the figure the Boolean vector for the vari-", "score": 1.0, "index": 82}, {"type": "text", "coordinates": [132, 351, 153, 370], "content": "able", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [153, 355, 164, 365], "content": "x_{i}", "score": 0.85, "index": 84}, {"type": "text", "coordinates": [164, 351, 219, 370], "content": " with domain", "score": 1.0, "index": 85}, {"type": "inline_equation", "coordinates": [220, 354, 233, 365], "content": "D_{i}", "score": 0.88, "index": 86}, {"type": "text", "coordinates": [233, 351, 245, 370], "content": " is ", "score": 1.0, "index": 87}, {"type": "inline_equation", "coordinates": [246, 353, 318, 366], "content": "(x_{i}^{0},x_{i}^{1},\\cdot\\cdot\\cdot x_{i}^{l_{i}-1})", "score": 0.94, "index": 88}, {"type": "text", "coordinates": [290, 352, 483, 369], "content": " xlii\u22121), where li = \u2308lg |Di|\u2309. For example, in", "score": 1.0, "index": 89}, {"type": "text", "coordinates": [133, 366, 210, 379], "content": "the figure, variable", "score": 1.0, "index": 90}, {"type": "inline_equation", "coordinates": [210, 367, 222, 377], "content": "x_{2}", "score": 0.86, "index": 91}, {"type": "text", "coordinates": [222, 366, 481, 379], "content": " which corresponds to the size of the T-shirt is represented by the", "score": 1.0, "index": 92}, {"type": "text", "coordinates": [132, 377, 197, 393], "content": "Boolean vector ", "score": 1.0, "index": 93}, {"type": "inline_equation", "coordinates": [198, 377, 230, 390], "content": "(x_{2}^{0},x_{2}^{1})", "score": 0.93, "index": 94}, {"type": "text", "coordinates": [230, 377, 483, 393], "content": ". In the BDD any path from the root node to the terminal node", "score": 1.0, "index": 95}, {"type": "text", "coordinates": [133, 391, 482, 403], "content": "1, corresponds to one or more valid configurations. For example, the path from the root", "score": 1.0, "index": 96}, {"type": "text", "coordinates": [134, 403, 481, 414], "content": "node to the terminal node 1, with all the variables taking low values represents the valid", "score": 1.0, "index": 97}, {"type": "text", "coordinates": [133, 413, 350, 429], "content": "configuration (black, small, MIB). Another path with", "score": 1.0, "index": 98}, {"type": "inline_equation", "coordinates": [350, 413, 376, 426], "content": "x_{1}^{0},x_{1}^{1}", "score": 0.79, "index": 99}, {"type": "text", "coordinates": [376, 413, 396, 429], "content": ", and", "score": 1.0, "index": 100}, {"type": "inline_equation", "coordinates": [396, 413, 407, 426], "content": "\\boldsymbol{x}_{2}^{0}", "score": 0.9, "index": 101}, {"type": "text", "coordinates": [408, 413, 483, 429], "content": " taking low values,", "score": 1.0, "index": 102}, {"type": "text", "coordinates": [133, 426, 150, 438], "content": "and", "score": 1.0, "index": 103}, {"type": "inline_equation", "coordinates": [151, 425, 162, 438], "content": "x_{2}^{1}", "score": 0.9, "index": 104}, {"type": "text", "coordinates": [163, 426, 455, 438], "content": " taking high value represents two valid configurations: (black, medium,", "score": 1.0, "index": 105}, {"type": "inline_equation", "coordinates": [455, 425, 477, 436], "content": "M I B", "score": 0.29, "index": 106}, {"type": "text", "coordinates": [477, 426, 481, 438], "content": ")", "score": 1.0, "index": 107}, {"type": "text", "coordinates": [133, 438, 219, 452], "content": "and (black, medium,", "score": 1.0, "index": 108}, {"type": "inline_equation", "coordinates": [220, 437, 246, 448], "content": "S T W", "score": 0.3, "index": 109}, {"type": "text", "coordinates": [246, 438, 379, 452], "content": "), namely. In this path the variable", "score": 1.0, "index": 110}, {"type": "inline_equation", "coordinates": [379, 437, 390, 450], "content": "\\boldsymbol{x}_{3}^{0}", "score": 0.9, "index": 111}, {"type": "text", "coordinates": [391, 438, 483, 452], "content": " is a don\u2019t care variable", "score": 1.0, "index": 112}, {"type": "text", "coordinates": [133, 450, 480, 462], "content": "and hence can take both low and high value, which leads to two valid configurations.", "score": 1.0, "index": 113}, {"type": "text", "coordinates": [134, 462, 480, 474], "content": "Any path from the root node to the terminal node 0 corresponds to invalid configura-", "score": 1.0, "index": 114}, {"type": "text", "coordinates": [133, 474, 157, 486], "content": "tions.", "score": 1.0, "index": 115}, {"type": "inline_equation", "coordinates": [472, 473, 481, 485], "content": "\\diamondsuit", "score": 0.47, "index": 116}, {"type": "text", "coordinates": [133, 511, 289, 524], "content": "4 Calculating Valid Domains", "score": 1.0, "index": 117}, {"type": "text", "coordinates": [133, 535, 481, 547], "content": "Before showing the algorithms, let us first introduce the appropriate notation. If an", "score": 1.0, "index": 118}, {"type": "text", "coordinates": [133, 547, 159, 559], "content": "index ", "score": 1.0, "index": 119}, {"type": "inline_equation", "coordinates": [159, 546, 195, 557], "content": "k\\ \\in\\ X_{b}", "score": 0.91, "index": 120}, {"type": "text", "coordinates": [196, 547, 277, 559], "content": " corresponds to the", "score": 1.0, "index": 121}, {"type": "inline_equation", "coordinates": [278, 546, 303, 558], "content": "j+1", "score": 0.88, "index": 122}, {"type": "text", "coordinates": [303, 547, 388, 559], "content": "-st Boolean variable", "score": 1.0, "index": 123}, {"type": "inline_equation", "coordinates": [388, 546, 399, 559], "content": "\\boldsymbol{x}_{j}^{i}", "score": 0.89, "index": 124}, {"type": "text", "coordinates": [400, 547, 481, 559], "content": " encoding the finite", "score": 1.0, "index": 125}, {"type": "text", "coordinates": [133, 559, 201, 571], "content": "domain variable", "score": 1.0, "index": 126}, {"type": "inline_equation", "coordinates": [202, 560, 212, 569], "content": "x_{i}", "score": 0.83, "index": 127}, {"type": "text", "coordinates": [213, 559, 261, 571], "content": ", we define ", "score": 1.0, "index": 128}, {"type": "inline_equation", "coordinates": [261, 558, 315, 570], "content": "v a r_{1}(k)\\;=\\;i", "score": 0.9, "index": 129}, {"type": "text", "coordinates": [316, 559, 336, 571], "content": " and ", "score": 1.0, "index": 130}, {"type": "inline_equation", "coordinates": [336, 558, 392, 570], "content": "v a r_{2}(k)\\,=\\,j", "score": 0.92, "index": 131}, {"type": "text", "coordinates": [392, 559, 481, 571], "content": " to be the appropriate", "score": 1.0, "index": 132}, {"type": "text", "coordinates": [133, 571, 266, 583], "content": "mappings. Now, given the BDD", "score": 1.0, "index": 133}, {"type": "inline_equation", "coordinates": [267, 570, 351, 582], "content": "B(V,E,X_{b},R,v a r)", "score": 0.9, "index": 134}, {"type": "text", "coordinates": [351, 571, 355, 583], "content": ",", "score": 1.0, "index": 135}, {"type": "inline_equation", "coordinates": [355, 570, 366, 581], "content": "V_{i}", "score": 0.81, "index": 136}, {"type": "text", "coordinates": [366, 571, 482, 583], "content": " denotes the set of all nodes", "score": 1.0, "index": 137}, {"type": "inline_equation", "coordinates": [133, 582, 160, 592], "content": "u\\in V", "score": 0.9, "index": 138}, {"type": "text", "coordinates": [161, 583, 453, 595], "content": "that are labelled with a BDD variable encoding the finite domain variable", "score": 1.0, "index": 139}, {"type": "inline_equation", "coordinates": [453, 583, 463, 593], "content": "x_{i}", "score": 0.85, "index": 140}, {"type": "text", "coordinates": [464, 583, 481, 595], "content": ", i.e.", "score": 1.0, "index": 141}, {"type": "inline_equation", "coordinates": [133, 593, 251, 606], "content": "V_{i}=\\{u\\in V\\mid v a r_{1}(u)=i\\}", "score": 0.93, "index": 142}, {"type": "text", "coordinates": [252, 594, 303, 607], "content": ". We think of", "score": 1.0, "index": 143}, {"type": "inline_equation", "coordinates": [303, 594, 313, 605], "content": "V_{i}", "score": 0.88, "index": 144}, {"type": "text", "coordinates": [314, 594, 482, 607], "content": " as defining a layer in the BDD. We define", "score": 1.0, "index": 145}, {"type": "inline_equation", "coordinates": [133, 606, 149, 617], "content": "I n_{i}", "score": 0.88, "index": 146}, {"type": "text", "coordinates": [149, 606, 238, 619], "content": " to be the set of nodes", "score": 1.0, "index": 147}, {"type": "inline_equation", "coordinates": [238, 606, 267, 617], "content": "u\\in V_{i}", "score": 0.9, "index": 148}, {"type": "text", "coordinates": [267, 606, 470, 619], "content": " reachable by an edge originating from outside the", "score": 1.0, "index": 149}, {"type": "inline_equation", "coordinates": [470, 606, 480, 617], "content": "V_{i}", "score": 0.85, "index": 150}, {"type": "text", "coordinates": [134, 619, 172, 630], "content": "layer, i.e.", "score": 1.0, "index": 151}, {"type": "inline_equation", "coordinates": [172, 617, 359, 630], "content": "I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}", "score": 0.81, "index": 152}, {"type": "text", "coordinates": [359, 619, 435, 630], "content": ". For the root node ", "score": 1.0, "index": 153}, {"type": "inline_equation", "coordinates": [435, 618, 444, 628], "content": "R", "score": 0.76, "index": 154}, {"type": "text", "coordinates": [444, 619, 481, 630], "content": ", labelled", "score": 1.0, "index": 155}, {"type": "text", "coordinates": [133, 630, 154, 642], "content": "with ", "score": 1.0, "index": 156}, {"type": "inline_equation", "coordinates": [154, 630, 210, 641], "content": "i_{0}=v a r_{1}(R)", "score": 0.93, "index": 157}, {"type": "text", "coordinates": [211, 630, 253, 642], "content": " we define", "score": 1.0, "index": 158}, {"type": "inline_equation", "coordinates": [253, 630, 330, 642], "content": "I n_{i_{0}}=V_{i_{0}}=\\{R\\}", "score": 0.92, "index": 159}, {"type": "text", "coordinates": [330, 630, 333, 642], "content": ".", "score": 1.0, "index": 160}, {"type": "text", "coordinates": [148, 642, 481, 654], "content": "We assume that in the previous user assignment, a user fixed a value for a finite", "score": 1.0, "index": 161}, {"type": "text", "coordinates": [133, 654, 200, 667], "content": "domain variable", "score": 1.0, "index": 162}, {"type": "inline_equation", "coordinates": [200, 654, 257, 665], "content": "x=v,x\\in X", "score": 0.92, "index": 163}, {"type": "text", "coordinates": [257, 654, 407, 667], "content": ", extending the old partial assignment", "score": 1.0, "index": 164}, {"type": "inline_equation", "coordinates": [407, 655, 425, 665], "content": "\\rho_{o l d}", "score": 0.88, "index": 165}, {"type": "text", "coordinates": [425, 654, 482, 667], "content": " to the current", "score": 1.0, "index": 166}] | [] | [{"type": "inline", "coordinates": [201, 118, 250, 128], "content": "\\overline{{v_{0}\\ldots v_{k_{i}-1}}}", "caption": ""}, {"type": "inline", "coordinates": [298, 117, 388, 128], "content": "v_{0}\\dots\\cdot v_{k_{i}-1}=e n c(j)", "caption": ""}, {"type": "inline", "coordinates": [229, 131, 277, 140], "content": "v_{0}\\ldots v_{k_{i}-1}", "caption": 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"content": "I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}", "caption": ""}, {"type": "inline", "coordinates": [435, 618, 444, 628], "content": "R", "caption": ""}, {"type": "inline", "coordinates": [154, 630, 210, 641], "content": "i_{0}=v a r_{1}(R)", "caption": ""}, {"type": "inline", "coordinates": [253, 630, 330, 642], "content": "I n_{i_{0}}=V_{i_{0}}=\\{R\\}", "caption": ""}, {"type": "inline", "coordinates": [200, 654, 257, 665], "content": "x=v,x\\in X", "caption": ""}, {"type": "inline", "coordinates": [407, 655, 425, 665], "content": "\\rho_{o l d}", "caption": ""}] | [] | [612.0, 792.0] | [{"type": "text", "text": "", "page_idx": 4}, {"type": "text", "text": "Let the solution space $S o l$ over ordered set of variables $x_{0}<...<x_{k-1}$ be represented by a Binary Decision Diagram $B(V,E,X_{b},R,v a r)$ , where $V$ is the set of nodes $u$ , $E$ is the set of edges $e$ and $X_{b}=\\{0,1,\\ldots,|X_{b}|-1\\}$ is an ordered set of variable indexes, labelling every non-terminal node $u$ with $v a r(u)\\,\\leq\\,|X_{b}|\\,-\\,1$ and labelling the terminal nodes $T_{0},T_{1}$ with index $|X_{b}|$ . Set of variable indexes $X_{b}$ is constructed by taking the union of Boolean encoding variables $\\bigcup_{i=0}^{n-1}\\{x_{0}^{i},\\dotsc,x_{k_{i}-1}^{i}\\}$ and ordering them in a natural layered way, i.e. $x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}$ iff $i_{1}<i_{2}$ or $i_{1}=i_{2}$ and $j_{1}<j_{2}$ . ", "page_idx": 4}, {"type": "text", "text": "Every directed edge $\\boldsymbol{e}~=~(u_{1},\\bar{u}_{2})$ has a starting vertex $u_{1}~=~\\pi_{1}(e)$ and ending vertex $u_{2}=\\pi_{2}(e).\\,R$ denotes the root node of the BDD. ", "page_idx": 4}, {"type": "text", "text": "Example 2. The BDD representing the solution space of the T-shirt example introduced in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables: $x_{1},x_{2}$ and $x_{3}$ , whose domain sizes are four, three and two, respectively. Each variable is represented by a vector of Boolean variables. In the figure the Boolean vector for the variable $x_{i}$ with domain $D_{i}$ is $(x_{i}^{0},x_{i}^{1},\\cdot\\cdot\\cdot x_{i}^{l_{i}-1})$ xlii\u22121), where li = \u2308lg |Di|\u2309. For example, in the figure, variable $x_{2}$ which corresponds to the size of the T-shirt is represented by the Boolean vector $(x_{2}^{0},x_{2}^{1})$ . In the BDD any path from the root node to the terminal node 1, corresponds to one or more valid configurations. For example, the path from the root node to the terminal node 1, with all the variables taking low values represents the valid configuration (black, small, MIB). Another path with $x_{1}^{0},x_{1}^{1}$ , and $\\boldsymbol{x}_{2}^{0}$ taking low values, and $x_{2}^{1}$ taking high value represents two valid configurations: (black, medium, $M I B$ ) and (black, medium, $S T W$ ), namely. In this path the variable $\\boldsymbol{x}_{3}^{0}$ is a don\u2019t care variable and hence can take both low and high value, which leads to two valid configurations. Any path from the root node to the terminal node 0 corresponds to invalid configurations. $\\diamondsuit$ ", "page_idx": 4}, {"type": "text", "text": "4 Calculating Valid Domains ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Before showing the algorithms, let us first introduce the appropriate notation. If an index $k\\ \\in\\ X_{b}$ corresponds to the $j+1$ -st Boolean variable $\\boldsymbol{x}_{j}^{i}$ encoding the finite domain variable $x_{i}$ , we define $v a r_{1}(k)\\;=\\;i$ and $v a r_{2}(k)\\,=\\,j$ to be the appropriate mappings. Now, given the BDD $B(V,E,X_{b},R,v a r)$ , $V_{i}$ denotes the set of all nodes $u\\in V$ that are labelled with a BDD variable encoding the finite domain variable $x_{i}$ , i.e. $V_{i}=\\{u\\in V\\mid v a r_{1}(u)=i\\}$ . We think of $V_{i}$ as defining a layer in the BDD. We define $I n_{i}$ to be the set of nodes $u\\in V_{i}$ reachable by an edge originating from outside the $V_{i}$ layer, i.e. $I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}$ . For the root node $R$ , labelled with $i_{0}=v a r_{1}(R)$ we define $I n_{i_{0}}=V_{i_{0}}=\\{R\\}$ . ", "page_idx": 4}, {"type": "text", "text": "We assume that in the previous user assignment, a user fixed a value for a finite domain variable $x=v,x\\in X$ , extending the old partial assignment $\\rho_{o l d}$ to the current assignment $\\rho\\,=\\,\\rho_{o l d}\\cup\\{(x,v)\\}$ . For every variable $x_{i}\\ \\in\\ X$ , old valid domains are denoted as $D_{i}^{\\rho_{o l d}},i=0,\\dots,n-1$ . and the old BDD $B^{\\rho_{o l d}}$ is reduced to the restricted BDD, $B^{\\rho}(V,E,X_{b},v a r)$ . The $C V D$ functionality is to calculate valid domains $D_{i}^{\\rho}$ for remaining unassigned variables $x_{i}\\not\\in d o m(\\rho)$ by extracting values from the newly restricted BDD $B^{\\rho}(V,E,X_{b},v a r)$ . 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Set of variable indexes", "type": "text"}, {"bbox": [408, 237, 421, 247], "score": 0.87, "content": "X_{b}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [421, 237, 482, 249], "score": 1.0, "content": " is constructed", "type": "text"}], "index": 10}, {"bbox": [133, 248, 482, 264], "spans": [{"bbox": [133, 249, 335, 264], "score": 1.0, "content": "by taking the union of Boolean encoding variables", "type": "text"}, {"bbox": [336, 248, 427, 262], "score": 0.94, "content": "\\bigcup_{i=0}^{n-1}\\{x_{0}^{i},\\dotsc,x_{k_{i}-1}^{i}\\}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [428, 249, 482, 264], "score": 1.0, "content": " and ordering", "type": "text"}], "index": 11}, {"bbox": [133, 261, 451, 279], "spans": [{"bbox": [133, 264, 270, 278], "score": 1.0, "content": "them in a natural layered way, i.e.", "type": "text"}, {"bbox": [270, 261, 311, 276], "score": 0.93, "content": "x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}", "type": "inline_equation", "height": 15, "width": 41}, {"bbox": [311, 261, 324, 279], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 263, 355, 275], "score": 0.9, "content": "i_{1}<i_{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [355, 261, 367, 279], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [367, 263, 397, 274], "score": 0.9, "content": "i_{1}=i_{2}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [397, 261, 415, 279], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [415, 263, 447, 275], "score": 0.91, "content": "j_{1}<j_{2}", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [447, 261, 451, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9}, {"type": "text", "bbox": [134, 276, 482, 299], "lines": [{"bbox": [148, 275, 480, 288], "spans": [{"bbox": [148, 275, 234, 288], "score": 1.0, "content": "Every directed edge ", "type": "text"}, {"bbox": [234, 275, 290, 287], "score": 0.92, "content": "\\boldsymbol{e}~=~(u_{1},\\bar{u}_{2})", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [290, 275, 379, 288], "score": 1.0, "content": " has a starting vertex", "type": "text"}, {"bbox": [380, 275, 431, 286], "score": 0.93, "content": "u_{1}~=~\\pi_{1}(e)", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [432, 275, 480, 288], "score": 1.0, "content": " and ending", "type": "text"}], "index": 13}, {"bbox": [134, 286, 360, 299], "spans": [{"bbox": [134, 288, 160, 299], "score": 1.0, "content": "vertex", "type": "text"}, {"bbox": [160, 286, 220, 299], "score": 0.78, "content": "u_{2}=\\pi_{2}(e).\\,R", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [221, 288, 360, 299], "score": 1.0, "content": " denotes the root node of the BDD.", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [134, 306, 482, 484], "lines": [{"bbox": [134, 307, 482, 318], "spans": [{"bbox": [134, 307, 482, 318], "score": 1.0, "content": "Example 2. The BDD representing the solution space of the T-shirt example introduced", "type": "text"}], "index": 15}, {"bbox": [133, 318, 481, 330], "spans": [{"bbox": [133, 318, 437, 330], "score": 1.0, "content": "in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables:", "type": "text"}, {"bbox": [438, 320, 464, 329], "score": 0.89, "content": "x_{1},x_{2}", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [464, 318, 481, 330], "score": 1.0, "content": " and", "type": "text"}], "index": 16}, {"bbox": [133, 331, 482, 343], "spans": [{"bbox": [133, 332, 145, 342], "score": 0.82, "content": "x_{3}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [145, 331, 482, 343], "score": 1.0, "content": ", whose domain sizes are four, three and two, respectively. Each variable is repre-", "type": "text"}], "index": 17}, {"bbox": [134, 343, 481, 354], "spans": [{"bbox": [134, 343, 481, 354], "score": 1.0, "content": "sented by a vector of Boolean variables. In the figure the Boolean vector for the vari-", "type": "text"}], "index": 18}, {"bbox": [132, 351, 483, 370], "spans": [{"bbox": [132, 351, 153, 370], "score": 1.0, "content": "able", "type": "text"}, {"bbox": [153, 355, 164, 365], "score": 0.85, "content": "x_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [164, 351, 219, 370], "score": 1.0, "content": " with domain", "type": "text"}, {"bbox": [220, 354, 233, 365], "score": 0.88, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [233, 351, 245, 370], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [246, 353, 318, 366], "score": 0.94, "content": "(x_{i}^{0},x_{i}^{1},\\cdot\\cdot\\cdot x_{i}^{l_{i}-1})", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [290, 352, 483, 369], "score": 1.0, "content": " xlii\u22121), where li = \u2308lg |Di|\u2309. For example, in", "type": "text"}], "index": 19}, {"bbox": [133, 366, 481, 379], "spans": [{"bbox": [133, 366, 210, 379], "score": 1.0, "content": "the figure, variable", "type": "text"}, {"bbox": [210, 367, 222, 377], "score": 0.86, "content": "x_{2}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [222, 366, 481, 379], "score": 1.0, "content": " which corresponds to the size of the T-shirt is represented by the", "type": "text"}], "index": 20}, {"bbox": [132, 377, 483, 393], "spans": [{"bbox": [132, 377, 197, 393], "score": 1.0, "content": "Boolean vector ", "type": "text"}, {"bbox": [198, 377, 230, 390], "score": 0.93, "content": "(x_{2}^{0},x_{2}^{1})", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [230, 377, 483, 393], "score": 1.0, "content": ". In the BDD any path from the root node to the terminal node", "type": "text"}], "index": 21}, {"bbox": [133, 391, 482, 403], "spans": [{"bbox": [133, 391, 482, 403], "score": 1.0, "content": "1, corresponds to one or more valid configurations. For example, the path from the root", "type": "text"}], "index": 22}, {"bbox": [134, 403, 481, 414], "spans": [{"bbox": [134, 403, 481, 414], "score": 1.0, "content": "node to the terminal node 1, with all the variables taking low values represents the valid", "type": "text"}], "index": 23}, {"bbox": [133, 413, 483, 429], "spans": [{"bbox": [133, 413, 350, 429], "score": 1.0, "content": "configuration (black, small, MIB). Another path with", "type": "text"}, {"bbox": [350, 413, 376, 426], "score": 0.79, "content": "x_{1}^{0},x_{1}^{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [376, 413, 396, 429], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [396, 413, 407, 426], "score": 0.9, "content": "\\boldsymbol{x}_{2}^{0}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [408, 413, 483, 429], "score": 1.0, "content": " taking low values,", "type": "text"}], "index": 24}, {"bbox": [133, 425, 481, 438], "spans": [{"bbox": [133, 426, 150, 438], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [151, 425, 162, 438], "score": 0.9, "content": "x_{2}^{1}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [163, 426, 455, 438], "score": 1.0, "content": " taking high value represents two valid configurations: (black, medium,", "type": "text"}, {"bbox": [455, 425, 477, 436], "score": 0.29, "content": "M I B", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [477, 426, 481, 438], "score": 1.0, "content": ")", "type": "text"}], "index": 25}, {"bbox": [133, 437, 483, 452], "spans": [{"bbox": [133, 438, 219, 452], "score": 1.0, "content": "and (black, medium,", "type": "text"}, {"bbox": [220, 437, 246, 448], "score": 0.3, "content": "S T W", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [246, 438, 379, 452], "score": 1.0, "content": "), namely. In this path the variable", "type": "text"}, {"bbox": [379, 437, 390, 450], "score": 0.9, "content": "\\boldsymbol{x}_{3}^{0}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [391, 438, 483, 452], "score": 1.0, "content": " is a don\u2019t care variable", "type": "text"}], "index": 26}, {"bbox": [133, 450, 480, 462], "spans": [{"bbox": [133, 450, 480, 462], "score": 1.0, "content": "and hence can take both low and high value, which leads to two valid configurations.", "type": "text"}], "index": 27}, {"bbox": [134, 462, 480, 474], "spans": [{"bbox": [134, 462, 480, 474], "score": 1.0, "content": "Any path from the root node to the terminal node 0 corresponds to invalid configura-", "type": "text"}], "index": 28}, {"bbox": [133, 473, 481, 486], "spans": [{"bbox": [133, 474, 157, 486], "score": 1.0, "content": "tions.", "type": "text"}, {"bbox": [472, 473, 481, 485], "score": 0.47, "content": "\\diamondsuit", "type": "inline_equation", "height": 12, "width": 9}], "index": 29}], "index": 22}, {"type": "title", "bbox": [134, 510, 290, 524], "lines": [{"bbox": [133, 511, 289, 524], "spans": [{"bbox": [133, 511, 289, 524], "score": 1.0, "content": "4 Calculating Valid Domains", "type": "text"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [133, 533, 482, 642], "lines": [{"bbox": [133, 535, 481, 547], "spans": [{"bbox": [133, 535, 481, 547], "score": 1.0, "content": "Before showing the algorithms, let us first introduce the appropriate notation. If an", "type": "text"}], "index": 31}, {"bbox": [133, 546, 481, 559], "spans": [{"bbox": [133, 547, 159, 559], "score": 1.0, "content": "index ", "type": "text"}, {"bbox": [159, 546, 195, 557], "score": 0.91, "content": "k\\ \\in\\ X_{b}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [196, 547, 277, 559], "score": 1.0, "content": " corresponds to the", "type": "text"}, {"bbox": [278, 546, 303, 558], "score": 0.88, "content": "j+1", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [303, 547, 388, 559], "score": 1.0, "content": "-st Boolean variable", "type": "text"}, {"bbox": [388, 546, 399, 559], "score": 0.89, "content": "\\boldsymbol{x}_{j}^{i}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [400, 547, 481, 559], "score": 1.0, "content": " encoding the finite", "type": "text"}], "index": 32}, {"bbox": [133, 558, 481, 571], "spans": [{"bbox": [133, 559, 201, 571], "score": 1.0, "content": "domain variable", "type": "text"}, {"bbox": [202, 560, 212, 569], "score": 0.83, "content": "x_{i}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [213, 559, 261, 571], "score": 1.0, "content": ", we define ", "type": "text"}, {"bbox": [261, 558, 315, 570], "score": 0.9, "content": "v a r_{1}(k)\\;=\\;i", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [316, 559, 336, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 558, 392, 570], "score": 0.92, "content": "v a r_{2}(k)\\,=\\,j", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [392, 559, 481, 571], "score": 1.0, "content": " to be the appropriate", "type": "text"}], "index": 33}, {"bbox": [133, 570, 482, 583], "spans": [{"bbox": [133, 571, 266, 583], "score": 1.0, "content": "mappings. Now, given the BDD", "type": "text"}, {"bbox": [267, 570, 351, 582], "score": 0.9, "content": "B(V,E,X_{b},R,v a r)", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [351, 571, 355, 583], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [355, 570, 366, 581], "score": 0.81, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [366, 571, 482, 583], "score": 1.0, "content": " denotes the set of all nodes", "type": "text"}], "index": 34}, {"bbox": [133, 582, 481, 595], "spans": [{"bbox": [133, 582, 160, 592], "score": 0.9, "content": "u\\in V", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [161, 583, 453, 595], "score": 1.0, "content": "that are labelled with a BDD variable encoding the finite domain variable", "type": "text"}, {"bbox": [453, 583, 463, 593], "score": 0.85, "content": "x_{i}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [464, 583, 481, 595], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 35}, {"bbox": [133, 593, 482, 607], "spans": [{"bbox": [133, 593, 251, 606], "score": 0.93, "content": "V_{i}=\\{u\\in V\\mid v a r_{1}(u)=i\\}", "type": "inline_equation", "height": 13, "width": 118}, {"bbox": [252, 594, 303, 607], "score": 1.0, "content": ". We think of", "type": "text"}, {"bbox": [303, 594, 313, 605], "score": 0.88, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [314, 594, 482, 607], "score": 1.0, "content": " as defining a layer in the BDD. We define", "type": "text"}], "index": 36}, {"bbox": [133, 606, 480, 619], "spans": [{"bbox": [133, 606, 149, 617], "score": 0.88, "content": "I n_{i}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [149, 606, 238, 619], "score": 1.0, "content": " to be the set of nodes", "type": "text"}, {"bbox": [238, 606, 267, 617], "score": 0.9, "content": "u\\in V_{i}", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [267, 606, 470, 619], "score": 1.0, "content": " reachable by an edge originating from outside the", "type": "text"}, {"bbox": [470, 606, 480, 617], "score": 0.85, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 10}], "index": 37}, {"bbox": [134, 617, 481, 630], "spans": [{"bbox": [134, 619, 172, 630], "score": 1.0, "content": "layer, i.e.", "type": "text"}, {"bbox": [172, 617, 359, 630], "score": 0.81, "content": "I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [359, 619, 435, 630], "score": 1.0, "content": ". For the root node ", "type": "text"}, {"bbox": [435, 618, 444, 628], "score": 0.76, "content": "R", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [444, 619, 481, 630], "score": 1.0, "content": ", labelled", "type": "text"}], "index": 38}, {"bbox": [133, 630, 333, 642], "spans": [{"bbox": [133, 630, 154, 642], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [154, 630, 210, 641], "score": 0.93, "content": "i_{0}=v a r_{1}(R)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [211, 630, 253, 642], "score": 1.0, "content": " we define", "type": "text"}, {"bbox": [253, 630, 330, 642], "score": 0.92, "content": "I n_{i_{0}}=V_{i_{0}}=\\{R\\}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [330, 630, 333, 642], "score": 1.0, "content": ".", "type": "text"}], "index": 39}], "index": 35}, {"type": "text", "bbox": [133, 642, 482, 665], "lines": [{"bbox": [148, 642, 481, 654], "spans": [{"bbox": [148, 642, 481, 654], "score": 1.0, "content": "We assume that in the previous user assignment, a user fixed a value for a finite", "type": "text"}], "index": 40}, {"bbox": [133, 654, 482, 667], "spans": [{"bbox": [133, 654, 200, 667], "score": 1.0, "content": "domain variable", "type": "text"}, {"bbox": [200, 654, 257, 665], "score": 0.92, "content": "x=v,x\\in X", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [257, 654, 407, 667], "score": 1.0, "content": ", extending the old partial assignment", "type": "text"}, {"bbox": [407, 655, 425, 665], "score": 0.88, "content": "\\rho_{o l d}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [425, 654, 482, 667], "score": 1.0, "content": " to the current", "type": "text"}], "index": 41}], "index": 40.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [304, 685, 310, 693], "lines": [{"bbox": [304, 685, 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"content": " is the set of edges", "type": "text"}, {"bbox": [230, 214, 237, 222], "score": 0.73, "content": "e", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [237, 213, 255, 225], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [255, 212, 365, 224], "score": 0.92, "content": "X_{b}=\\{0,1,\\ldots,|X_{b}|-1\\}", "type": "inline_equation", "height": 12, "width": 110}, {"bbox": [366, 213, 482, 225], "score": 1.0, "content": " is an ordered set of variable", "type": "text"}], "index": 8}, {"bbox": [133, 224, 481, 237], "spans": [{"bbox": [133, 225, 310, 237], "score": 1.0, "content": "indexes, labelling every non-terminal node", "type": "text"}, {"bbox": [311, 226, 318, 234], "score": 0.75, "content": "u", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [318, 225, 341, 237], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [342, 224, 424, 236], "score": 0.92, "content": "v a r(u)\\,\\leq\\,|X_{b}|\\,-\\,1", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [424, 225, 481, 237], "score": 1.0, "content": " and labelling", "type": "text"}], "index": 9}, {"bbox": [133, 236, 482, 249], "spans": [{"bbox": [133, 237, 212, 249], "score": 1.0, "content": "the terminal nodes ", "type": "text"}, {"bbox": [212, 236, 239, 248], "score": 0.92, "content": "T_{0},T_{1}", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [239, 237, 287, 249], "score": 1.0, "content": " with index ", "type": "text"}, {"bbox": [288, 236, 306, 248], "score": 0.92, "content": "|X_{b}|", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [306, 237, 407, 249], "score": 1.0, "content": ". Set of variable indexes", "type": "text"}, {"bbox": [408, 237, 421, 247], "score": 0.87, "content": "X_{b}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [421, 237, 482, 249], "score": 1.0, "content": " is constructed", "type": "text"}], "index": 10}, {"bbox": [133, 248, 482, 264], "spans": [{"bbox": [133, 249, 335, 264], "score": 1.0, "content": "by taking the union of Boolean encoding variables", "type": "text"}, {"bbox": [336, 248, 427, 262], "score": 0.94, "content": "\\bigcup_{i=0}^{n-1}\\{x_{0}^{i},\\dotsc,x_{k_{i}-1}^{i}\\}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [428, 249, 482, 264], "score": 1.0, "content": " and ordering", "type": "text"}], "index": 11}, {"bbox": [133, 261, 451, 279], "spans": [{"bbox": [133, 264, 270, 278], "score": 1.0, "content": "them in a natural layered way, i.e.", "type": "text"}, {"bbox": [270, 261, 311, 276], "score": 0.93, "content": "x_{j_{1}}^{i_{1}}<x_{j_{2}}^{i_{2}}", "type": "inline_equation", "height": 15, "width": 41}, {"bbox": [311, 261, 324, 279], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 263, 355, 275], "score": 0.9, "content": "i_{1}<i_{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [355, 261, 367, 279], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [367, 263, 397, 274], "score": 0.9, "content": "i_{1}=i_{2}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [397, 261, 415, 279], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [415, 263, 447, 275], "score": 0.91, "content": "j_{1}<j_{2}", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [447, 261, 451, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [133, 189, 482, 279]}, {"type": "text", "bbox": [134, 276, 482, 299], "lines": [{"bbox": [148, 275, 480, 288], "spans": [{"bbox": [148, 275, 234, 288], "score": 1.0, "content": "Every directed edge ", "type": "text"}, {"bbox": [234, 275, 290, 287], "score": 0.92, "content": "\\boldsymbol{e}~=~(u_{1},\\bar{u}_{2})", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [290, 275, 379, 288], "score": 1.0, "content": " has a starting vertex", "type": "text"}, {"bbox": [380, 275, 431, 286], "score": 0.93, "content": "u_{1}~=~\\pi_{1}(e)", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [432, 275, 480, 288], "score": 1.0, "content": " and ending", "type": "text"}], "index": 13}, {"bbox": [134, 286, 360, 299], "spans": [{"bbox": [134, 288, 160, 299], "score": 1.0, "content": "vertex", "type": "text"}, {"bbox": [160, 286, 220, 299], "score": 0.78, "content": "u_{2}=\\pi_{2}(e).\\,R", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [221, 288, 360, 299], "score": 1.0, "content": " denotes the root node of the BDD.", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [134, 275, 480, 299]}, {"type": "text", "bbox": [134, 306, 482, 484], "lines": [{"bbox": [134, 307, 482, 318], "spans": [{"bbox": [134, 307, 482, 318], "score": 1.0, "content": "Example 2. The BDD representing the solution space of the T-shirt example introduced", "type": "text"}], "index": 15}, {"bbox": [133, 318, 481, 330], "spans": [{"bbox": [133, 318, 437, 330], "score": 1.0, "content": "in Sect. 2 is shown in Fig. 4. In the T-shirt example there are three variables:", "type": "text"}, {"bbox": [438, 320, 464, 329], "score": 0.89, "content": "x_{1},x_{2}", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [464, 318, 481, 330], "score": 1.0, "content": " and", "type": "text"}], "index": 16}, {"bbox": [133, 331, 482, 343], "spans": [{"bbox": [133, 332, 145, 342], "score": 0.82, "content": "x_{3}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [145, 331, 482, 343], "score": 1.0, "content": ", whose domain sizes are four, three and two, respectively. Each variable is repre-", "type": "text"}], "index": 17}, {"bbox": [134, 343, 481, 354], "spans": [{"bbox": [134, 343, 481, 354], "score": 1.0, "content": "sented by a vector of Boolean variables. In the figure the Boolean vector for the vari-", "type": "text"}], "index": 18}, {"bbox": [132, 351, 483, 370], "spans": [{"bbox": [132, 351, 153, 370], "score": 1.0, "content": "able", "type": "text"}, {"bbox": [153, 355, 164, 365], "score": 0.85, "content": "x_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [164, 351, 219, 370], "score": 1.0, "content": " with domain", "type": "text"}, {"bbox": [220, 354, 233, 365], "score": 0.88, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [233, 351, 245, 370], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [246, 353, 318, 366], "score": 0.94, "content": "(x_{i}^{0},x_{i}^{1},\\cdot\\cdot\\cdot x_{i}^{l_{i}-1})", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [290, 352, 483, 369], "score": 1.0, "content": " xlii\u22121), where li = \u2308lg |Di|\u2309. For example, in", "type": "text"}], "index": 19}, {"bbox": [133, 366, 481, 379], "spans": [{"bbox": [133, 366, 210, 379], "score": 1.0, "content": "the figure, variable", "type": "text"}, {"bbox": [210, 367, 222, 377], "score": 0.86, "content": "x_{2}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [222, 366, 481, 379], "score": 1.0, "content": " which corresponds to the size of the T-shirt is represented by the", "type": "text"}], "index": 20}, {"bbox": [132, 377, 483, 393], "spans": [{"bbox": [132, 377, 197, 393], "score": 1.0, "content": "Boolean vector ", "type": "text"}, {"bbox": [198, 377, 230, 390], "score": 0.93, "content": "(x_{2}^{0},x_{2}^{1})", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [230, 377, 483, 393], "score": 1.0, "content": ". In the BDD any path from the root node to the terminal node", "type": "text"}], "index": 21}, {"bbox": [133, 391, 482, 403], "spans": [{"bbox": [133, 391, 482, 403], "score": 1.0, "content": "1, corresponds to one or more valid configurations. For example, the path from the root", "type": "text"}], "index": 22}, {"bbox": [134, 403, 481, 414], "spans": [{"bbox": [134, 403, 481, 414], "score": 1.0, "content": "node to the terminal node 1, with all the variables taking low values represents the valid", "type": "text"}], "index": 23}, {"bbox": [133, 413, 483, 429], "spans": [{"bbox": [133, 413, 350, 429], "score": 1.0, "content": "configuration (black, small, MIB). Another path with", "type": "text"}, {"bbox": [350, 413, 376, 426], "score": 0.79, "content": "x_{1}^{0},x_{1}^{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [376, 413, 396, 429], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [396, 413, 407, 426], "score": 0.9, "content": "\\boldsymbol{x}_{2}^{0}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [408, 413, 483, 429], "score": 1.0, "content": " taking low values,", "type": "text"}], "index": 24}, {"bbox": [133, 425, 481, 438], "spans": [{"bbox": [133, 426, 150, 438], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [151, 425, 162, 438], "score": 0.9, "content": "x_{2}^{1}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [163, 426, 455, 438], "score": 1.0, "content": " taking high value represents two valid configurations: (black, medium,", "type": "text"}, {"bbox": [455, 425, 477, 436], "score": 0.29, "content": "M I B", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [477, 426, 481, 438], "score": 1.0, "content": ")", "type": "text"}], "index": 25}, {"bbox": [133, 437, 483, 452], "spans": [{"bbox": [133, 438, 219, 452], "score": 1.0, "content": "and (black, medium,", "type": "text"}, {"bbox": [220, 437, 246, 448], "score": 0.3, "content": "S T W", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [246, 438, 379, 452], "score": 1.0, "content": "), namely. In this path the variable", "type": "text"}, {"bbox": [379, 437, 390, 450], "score": 0.9, "content": "\\boldsymbol{x}_{3}^{0}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [391, 438, 483, 452], "score": 1.0, "content": " is a don\u2019t care variable", "type": "text"}], "index": 26}, {"bbox": [133, 450, 480, 462], "spans": [{"bbox": [133, 450, 480, 462], "score": 1.0, "content": "and hence can take both low and high value, which leads to two valid configurations.", "type": "text"}], "index": 27}, {"bbox": [134, 462, 480, 474], "spans": [{"bbox": [134, 462, 480, 474], "score": 1.0, "content": "Any path from the root node to the terminal node 0 corresponds to invalid configura-", "type": "text"}], "index": 28}, {"bbox": [133, 473, 481, 486], "spans": [{"bbox": [133, 474, 157, 486], "score": 1.0, "content": "tions.", "type": "text"}, {"bbox": [472, 473, 481, 485], "score": 0.47, "content": "\\diamondsuit", "type": "inline_equation", "height": 12, "width": 9}], "index": 29}], "index": 22, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [132, 307, 483, 486]}, {"type": "title", "bbox": [134, 510, 290, 524], "lines": [{"bbox": [133, 511, 289, 524], "spans": [{"bbox": [133, 511, 289, 524], "score": 1.0, "content": "4 Calculating Valid Domains", "type": "text"}], "index": 30}], "index": 30, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [133, 533, 482, 642], "lines": [{"bbox": [133, 535, 481, 547], "spans": [{"bbox": [133, 535, 481, 547], "score": 1.0, "content": "Before showing the algorithms, let us first introduce the appropriate notation. If an", "type": "text"}], "index": 31}, {"bbox": [133, 546, 481, 559], "spans": [{"bbox": [133, 547, 159, 559], "score": 1.0, "content": "index ", "type": "text"}, {"bbox": [159, 546, 195, 557], "score": 0.91, "content": "k\\ \\in\\ X_{b}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [196, 547, 277, 559], "score": 1.0, "content": " corresponds to the", "type": "text"}, {"bbox": [278, 546, 303, 558], "score": 0.88, "content": "j+1", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [303, 547, 388, 559], "score": 1.0, "content": "-st Boolean variable", "type": "text"}, {"bbox": [388, 546, 399, 559], "score": 0.89, "content": "\\boldsymbol{x}_{j}^{i}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [400, 547, 481, 559], "score": 1.0, "content": " encoding the finite", "type": "text"}], "index": 32}, {"bbox": [133, 558, 481, 571], "spans": [{"bbox": [133, 559, 201, 571], "score": 1.0, "content": "domain variable", "type": "text"}, {"bbox": [202, 560, 212, 569], "score": 0.83, "content": "x_{i}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [213, 559, 261, 571], "score": 1.0, "content": ", we define ", "type": "text"}, {"bbox": [261, 558, 315, 570], "score": 0.9, "content": "v a r_{1}(k)\\;=\\;i", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [316, 559, 336, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 558, 392, 570], "score": 0.92, "content": "v a r_{2}(k)\\,=\\,j", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [392, 559, 481, 571], "score": 1.0, "content": " to be the appropriate", "type": "text"}], "index": 33}, {"bbox": [133, 570, 482, 583], "spans": [{"bbox": [133, 571, 266, 583], "score": 1.0, "content": "mappings. Now, given the BDD", "type": "text"}, {"bbox": [267, 570, 351, 582], "score": 0.9, "content": "B(V,E,X_{b},R,v a r)", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [351, 571, 355, 583], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [355, 570, 366, 581], "score": 0.81, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [366, 571, 482, 583], "score": 1.0, "content": " denotes the set of all nodes", "type": "text"}], "index": 34}, {"bbox": [133, 582, 481, 595], "spans": [{"bbox": [133, 582, 160, 592], "score": 0.9, "content": "u\\in V", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [161, 583, 453, 595], "score": 1.0, "content": "that are labelled with a BDD variable encoding the finite domain variable", "type": "text"}, {"bbox": [453, 583, 463, 593], "score": 0.85, "content": "x_{i}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [464, 583, 481, 595], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 35}, {"bbox": [133, 593, 482, 607], "spans": [{"bbox": [133, 593, 251, 606], "score": 0.93, "content": "V_{i}=\\{u\\in V\\mid v a r_{1}(u)=i\\}", "type": "inline_equation", "height": 13, "width": 118}, {"bbox": [252, 594, 303, 607], "score": 1.0, "content": ". We think of", "type": "text"}, {"bbox": [303, 594, 313, 605], "score": 0.88, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [314, 594, 482, 607], "score": 1.0, "content": " as defining a layer in the BDD. We define", "type": "text"}], "index": 36}, {"bbox": [133, 606, 480, 619], "spans": [{"bbox": [133, 606, 149, 617], "score": 0.88, "content": "I n_{i}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [149, 606, 238, 619], "score": 1.0, "content": " to be the set of nodes", "type": "text"}, {"bbox": [238, 606, 267, 617], "score": 0.9, "content": "u\\in V_{i}", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [267, 606, 470, 619], "score": 1.0, "content": " reachable by an edge originating from outside the", "type": "text"}, {"bbox": [470, 606, 480, 617], "score": 0.85, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 10}], "index": 37}, {"bbox": [134, 617, 481, 630], "spans": [{"bbox": [134, 619, 172, 630], "score": 1.0, "content": "layer, i.e.", "type": "text"}, {"bbox": [172, 617, 359, 630], "score": 0.81, "content": "I n_{i}=\\{u\\in V_{i}|\\;\\exists(u^{\\prime},u)\\in E.\\;v a r_{1}(u^{\\prime})<i\\}", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [359, 619, 435, 630], "score": 1.0, "content": ". For the root node ", "type": "text"}, {"bbox": [435, 618, 444, 628], "score": 0.76, "content": "R", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [444, 619, 481, 630], "score": 1.0, "content": ", labelled", "type": "text"}], "index": 38}, {"bbox": [133, 630, 333, 642], "spans": [{"bbox": [133, 630, 154, 642], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [154, 630, 210, 641], "score": 0.93, "content": "i_{0}=v a r_{1}(R)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [211, 630, 253, 642], "score": 1.0, "content": " we define", "type": "text"}, {"bbox": [253, 630, 330, 642], "score": 0.92, "content": "I n_{i_{0}}=V_{i_{0}}=\\{R\\}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [330, 630, 333, 642], "score": 1.0, "content": ".", "type": "text"}], "index": 39}], "index": 35, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [133, 535, 482, 642]}, {"type": "text", "bbox": [133, 642, 482, 665], "lines": [{"bbox": [148, 642, 481, 654], "spans": [{"bbox": [148, 642, 481, 654], "score": 1.0, "content": "We assume that in the previous user assignment, a user fixed a value for a finite", "type": "text"}], "index": 40}, {"bbox": [133, 654, 482, 667], "spans": [{"bbox": [133, 654, 200, 667], "score": 1.0, "content": "domain variable", "type": "text"}, {"bbox": [200, 654, 257, 665], "score": 0.92, "content": "x=v,x\\in X", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [257, 654, 407, 667], "score": 1.0, "content": ", extending the old partial assignment", "type": "text"}, {"bbox": [407, 655, 425, 665], "score": 0.88, "content": "\\rho_{o l d}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [425, 654, 482, 667], "score": 1.0, "content": " to the current", "type": "text"}], "index": 41}, {"bbox": [134, 405, 481, 419], "spans": [{"bbox": [134, 406, 181, 419], "score": 1.0, "content": "assignment", "type": "text", "cross_page": true}, {"bbox": [182, 405, 267, 417], "score": 0.9, "content": "\\rho\\,=\\,\\rho_{o l d}\\cup\\{(x,v)\\}", "type": "inline_equation", "height": 12, "width": 85, "cross_page": true}, {"bbox": [267, 406, 349, 419], "score": 1.0, "content": ". For every variable", "type": "text", "cross_page": true}, {"bbox": [350, 405, 385, 416], "score": 0.92, "content": "x_{i}\\ \\in\\ X", "type": "inline_equation", "height": 11, "width": 35, "cross_page": true}, {"bbox": [386, 406, 481, 419], "score": 1.0, "content": ", old valid domains are", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [133, 415, 483, 432], "spans": [{"bbox": [133, 415, 178, 432], "score": 1.0, "content": "denoted as", "type": "text", "cross_page": true}, {"bbox": [178, 417, 275, 429], "score": 0.86, "content": "D_{i}^{\\rho_{o l d}},i=0,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 97, "cross_page": true}, {"bbox": [275, 415, 348, 432], "score": 1.0, "content": ". and the old BDD", "type": "text", "cross_page": true}, {"bbox": [349, 417, 372, 427], "score": 0.86, "content": "B^{\\rho_{o l d}}", "type": "inline_equation", "height": 10, "width": 23, "cross_page": true}, {"bbox": [372, 415, 483, 432], "score": 1.0, "content": " is reduced to the restricted", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [134, 428, 480, 442], "spans": [{"bbox": [134, 430, 160, 442], "score": 1.0, "content": "BDD,", "type": "text", "cross_page": true}, {"bbox": [161, 429, 237, 441], "score": 0.91, "content": "B^{\\rho}(V,E,X_{b},v a r)", "type": "inline_equation", "height": 12, "width": 76, "cross_page": true}, {"bbox": [237, 430, 261, 442], "score": 1.0, "content": ". The", "type": "text", "cross_page": true}, {"bbox": [261, 429, 287, 439], "score": 0.8, "content": "C V D", "type": "inline_equation", "height": 10, "width": 26, "cross_page": true}, {"bbox": [288, 430, 466, 442], "score": 1.0, "content": " functionality is to calculate valid domains ", "type": "text", "cross_page": true}, {"bbox": [466, 428, 480, 441], "score": 0.89, "content": "D_{i}^{\\rho}", "type": "inline_equation", "height": 13, "width": 14, "cross_page": true}], "index": 5}, {"bbox": [134, 441, 480, 453], "spans": [{"bbox": [134, 442, 277, 453], "score": 1.0, "content": "for remaining unassigned variables", "type": "text", "cross_page": true}, {"bbox": [277, 441, 332, 452], "score": 0.94, "content": "x_{i}\\not\\in d o m(\\rho)", "type": "inline_equation", "height": 11, "width": 55, "cross_page": true}, {"bbox": [333, 442, 480, 453], "score": 1.0, "content": " by extracting values from the newly", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [133, 452, 276, 466], "spans": [{"bbox": [133, 453, 196, 466], "score": 1.0, "content": "restricted BDD", "type": "text", "cross_page": true}, {"bbox": [197, 452, 273, 465], "score": 0.92, "content": "B^{\\rho}(V,E,X_{b},v a r)", "type": "inline_equation", "height": 13, "width": 76, "cross_page": true}, {"bbox": [273, 453, 276, 466], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 7}], "index": 40.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [133, 642, 482, 667]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_4.png | images/0704.1394_4.png |
|
0704.1394.pdf | 5 | assignment $$\rho\,=\,\rho_{o l d}\cup\{(x,v)\}$$ . For every variable $$x_{i}\ \in\ X$$ , old valid domains are
denoted as $$D_{i}^{\rho_{o l d}},i=0,\dots,n-1$$ . and the old BDD $$B^{\rho_{o l d}}$$ is reduced to the restricted
BDD, $$B^{\rho}(V,E,X_{b},v a r)$$ . The $$C V D$$ functionality is to calculate valid domains $$D_{i}^{\rho}$$
for remaining unassigned variables $$x_{i}\not\in d o m(\rho)$$ by extracting values from the newly
restricted BDD $$B^{\rho}(V,E,X_{b},v a r)$$ .
To simplify the following discussion, we will analyze the isolated execution of the
$$C V D$$ algorithms over a given BDD $$B(V,E,X_{b},v a r)$$ . The task is to calculate valid
domains $$V D_{i}$$ from the starting domains $$D_{i}$$ . The user-configurator interaction can be
modelled as a sequence of these executions over restricted BDDs $$B^{\rho}$$ , where the valid
domains are $$D_{i}^{\rho}$$ and the starting domains are $$D_{i}^{\rho_{o l d}}$$ .
The $$C V D$$ functionality is delivered by executing two algorithms presented in Fig.
5 and Fig. 6. The first algorithm is based on the key idea that if there is an edge $$e=$$
$$(u_{1},u_{2})$$ crossing over $$V_{j}$$ , i.e. $$v a r_{1}(u_{1})\,<\,j\,<\,v a r_{1}(u_{2})$$ then we can include all the
values from $$D_{j}$$ into a valid domain $$V D_{j}\leftarrow D_{j}$$ .
We refer to $$e$$ as a long edge of length $$v a r_{1}(u_{2})\,-\,v a r_{1}(u_{1})$$ . Note that it skips
$$v a r(u_{2})-v a r(u_{1})$$ Boolean variables, and therefore compactly represents the part of a
solution space of size $$2^{v a r\left(u_{2}\right)-v a r\left(u_{1}\right)}$$ .
For the remaining variables $$x_{i}$$ , whose valid domain was not copied by $$C V D\mathrm{~-~}$$
Skipped, we execute $$C V D(B,x_{i})$$ from Fig. 6. There, for each value $$j$$ in a domain $$D_{i}^{\prime}$$
we check whether it can be part of the domain $$D_{i}$$ . The key idea is that if $$j\in D_{i}$$ then
there must be $$u\in V_{i}$$ such that traversing the BDD from $$u$$ with binary encoding of $$j$$
| <p>assignment $$\rho\,=\,\rho_{o l d}\cup\{(x,v)\}$$ . For every variable $$x_{i}\ \in\ X$$ , old valid domains are
denoted as $$D_{i}^{\rho_{o l d}},i=0,\dots,n-1$$ . and the old BDD $$B^{\rho_{o l d}}$$ is reduced to the restricted
BDD, $$B^{\rho}(V,E,X_{b},v a r)$$ . The $$C V D$$ functionality is to calculate valid domains $$D_{i}^{\rho}$$
for remaining unassigned variables $$x_{i}\not\in d o m(\rho)$$ by extracting values from the newly
restricted BDD $$B^{\rho}(V,E,X_{b},v a r)$$ .</p>
<p>To simplify the following discussion, we will analyze the isolated execution of the
$$C V D$$ algorithms over a given BDD $$B(V,E,X_{b},v a r)$$ . The task is to calculate valid
domains $$V D_{i}$$ from the starting domains $$D_{i}$$ . The user-configurator interaction can be
modelled as a sequence of these executions over restricted BDDs $$B^{\rho}$$ , where the valid
domains are $$D_{i}^{\rho}$$ and the starting domains are $$D_{i}^{\rho_{o l d}}$$ .</p>
<p>The $$C V D$$ functionality is delivered by executing two algorithms presented in Fig.
5 and Fig. 6. The first algorithm is based on the key idea that if there is an edge $$e=$$
$$(u_{1},u_{2})$$ crossing over $$V_{j}$$ , i.e. $$v a r_{1}(u_{1})\,<\,j\,<\,v a r_{1}(u_{2})$$ then we can include all the
values from $$D_{j}$$ into a valid domain $$V D_{j}\leftarrow D_{j}$$ .</p>
<p>We refer to $$e$$ as a long edge of length $$v a r_{1}(u_{2})\,-\,v a r_{1}(u_{1})$$ . Note that it skips
$$v a r(u_{2})-v a r(u_{1})$$ Boolean variables, and therefore compactly represents the part of a
solution space of size $$2^{v a r\left(u_{2}\right)-v a r\left(u_{1}\right)}$$ .</p>
<p>For the remaining variables $$x_{i}$$ , whose valid domain was not copied by $$C V D\mathrm{~-~}$$
Skipped, we execute $$C V D(B,x_{i})$$ from Fig. 6. There, for each value $$j$$ in a domain $$D_{i}^{\prime}$$
we check whether it can be part of the domain $$D_{i}$$ . The key idea is that if $$j\in D_{i}$$ then
there must be $$u\in V_{i}$$ such that traversing the BDD from $$u$$ with binary encoding of $$j$$</p>
| [{"type": "image", "coordinates": [222, 114, 374, 317], "content": "", "block_type": "image", "index": 1}, {"type": "text", "coordinates": [134, 405, 481, 464], "content": "assignment $$\\rho\\,=\\,\\rho_{o l d}\\cup\\{(x,v)\\}$$ . For every variable $$x_{i}\\ \\in\\ X$$ , old valid domains are\ndenoted as $$D_{i}^{\\rho_{o l d}},i=0,\\dots,n-1$$ . and the old BDD $$B^{\\rho_{o l d}}$$ is reduced to the restricted\nBDD, $$B^{\\rho}(V,E,X_{b},v a r)$$ . The $$C V D$$ functionality is to calculate valid domains $$D_{i}^{\\rho}$$\nfor remaining unassigned variables $$x_{i}\\not\\in d o m(\\rho)$$ by extracting values from the newly\nrestricted BDD $$B^{\\rho}(V,E,X_{b},v a r)$$ .", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [134, 467, 481, 527], "content": "To simplify the following discussion, we will analyze the isolated execution of the\n$$C V D$$ algorithms over a given BDD $$B(V,E,X_{b},v a r)$$ . The task is to calculate valid\ndomains $$V D_{i}$$ from the starting domains $$D_{i}$$ . The user-configurator interaction can be\nmodelled as a sequence of these executions over restricted BDDs $$B^{\\rho}$$ , where the valid\ndomains are $$D_{i}^{\\rho}$$ and the starting domains are $$D_{i}^{\\rho_{o l d}}$$ .", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [134, 529, 481, 577], "content": "The $$C V D$$ functionality is delivered by executing two algorithms presented in Fig.\n5 and Fig. 6. The first algorithm is based on the key idea that if there is an edge $$e=$$\n$$(u_{1},u_{2})$$ crossing over $$V_{j}$$ , i.e. $$v a r_{1}(u_{1})\\,<\\,j\\,<\\,v a r_{1}(u_{2})$$ then we can include all the\nvalues from $$D_{j}$$ into a valid domain $$V D_{j}\\leftarrow D_{j}$$ .", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [134, 580, 481, 615], "content": "We refer to $$e$$ as a long edge of length $$v a r_{1}(u_{2})\\,-\\,v a r_{1}(u_{1})$$ . Note that it skips\n$$v a r(u_{2})-v a r(u_{1})$$ Boolean variables, and therefore compactly represents the part of a\nsolution space of size $$2^{v a r\\left(u_{2}\\right)-v a r\\left(u_{1}\\right)}$$ .", "block_type": "text", "index": 5}, {"type": "text", "coordinates": [134, 618, 481, 665], "content": "For the remaining variables $$x_{i}$$ , whose valid domain was not copied by $$C V D\\mathrm{~-~}$$\nSkipped, we execute $$C V D(B,x_{i})$$ from Fig. 6. There, for each value $$j$$ in a domain $$D_{i}^{\\prime}$$\nwe check whether it can be part of the domain $$D_{i}$$ . 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The first algorithm is based on the key idea that if there is an edge", "type": "text"}, {"bbox": [463, 542, 482, 552], "score": 0.83, "content": "e=", "type": "inline_equation", "height": 10, "width": 19}], "index": 14}, {"bbox": [134, 552, 481, 567], "spans": [{"bbox": [134, 553, 167, 565], "score": 0.93, "content": "(u_{1},u_{2})", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [167, 554, 225, 567], "score": 1.0, "content": " crossing over", "type": "text"}, {"bbox": [226, 553, 236, 565], "score": 0.88, "content": "V_{j}", "type": "inline_equation", "height": 12, "width": 10}, {"bbox": [237, 554, 257, 567], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [257, 552, 368, 565], "score": 0.93, "content": "v a r_{1}(u_{1})\\,<\\,j\\,<\\,v a r_{1}(u_{2})", "type": "inline_equation", "height": 13, "width": 111}, {"bbox": [368, 554, 481, 567], "score": 1.0, "content": " then we can include all the", "type": "text"}], "index": 15}, {"bbox": [134, 565, 330, 579], "spans": [{"bbox": [134, 566, 183, 579], "score": 1.0, "content": "values from", "type": "text"}, {"bbox": [183, 565, 196, 578], "score": 0.89, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [196, 566, 277, 579], "score": 1.0, "content": " into a valid domain", "type": "text"}, {"bbox": [278, 566, 326, 578], "score": 0.93, "content": "V D_{j}\\leftarrow D_{j}", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [327, 566, 330, 579], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 14.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [133, 529, 482, 579]}, {"type": "text", "bbox": [134, 580, 481, 615], "lines": [{"bbox": [149, 579, 480, 592], "spans": [{"bbox": [149, 580, 199, 592], "score": 1.0, "content": "We refer to ", "type": "text"}, {"bbox": [199, 582, 205, 590], "score": 0.69, "content": "e", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [206, 580, 312, 592], "score": 1.0, "content": " as a long edge of length", "type": "text"}, {"bbox": [313, 579, 402, 592], "score": 0.92, "content": "v a r_{1}(u_{2})\\,-\\,v a r_{1}(u_{1})", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [402, 580, 480, 592], "score": 1.0, "content": ". Note that it skips", "type": "text"}], "index": 17}, {"bbox": [134, 591, 481, 605], "spans": [{"bbox": [134, 591, 212, 604], "score": 0.91, "content": "v a r(u_{2})-v a r(u_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [212, 592, 481, 605], "score": 1.0, "content": " Boolean variables, and therefore compactly represents the part of a", "type": "text"}], "index": 18}, {"bbox": [135, 602, 293, 615], "spans": [{"bbox": [135, 604, 221, 615], "score": 1.0, "content": "solution space of size", "type": "text"}, {"bbox": [222, 602, 289, 613], "score": 0.92, "content": "2^{v a r\\left(u_{2}\\right)-v a r\\left(u_{1}\\right)}", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [289, 604, 293, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [134, 579, 481, 615]}, {"type": "text", "bbox": [134, 618, 481, 665], "lines": [{"bbox": [148, 618, 482, 632], "spans": [{"bbox": [148, 618, 264, 632], "score": 1.0, "content": "For the remaining variables", "type": "text"}, {"bbox": [264, 620, 274, 629], "score": 0.82, "content": "x_{i}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [275, 618, 444, 632], "score": 1.0, "content": ", whose valid domain was not copied by ", "type": "text"}, {"bbox": [444, 618, 482, 629], "score": 0.85, "content": "C V D\\mathrm{~-~}", "type": "inline_equation", "height": 11, "width": 38}], "index": 20}, {"bbox": [134, 630, 480, 642], "spans": [{"bbox": [134, 630, 220, 642], "score": 1.0, "content": "Skipped, we execute", "type": "text"}, {"bbox": [220, 630, 275, 642], "score": 0.93, "content": "C V D(B,x_{i})", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [275, 630, 411, 642], "score": 1.0, "content": " from Fig. 6. There, for each value", "type": "text"}, {"bbox": [412, 630, 417, 641], "score": 0.84, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [418, 630, 468, 642], "score": 1.0, "content": " in a domain ", "type": "text"}, {"bbox": [468, 630, 480, 641], "score": 0.89, "content": "D_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}], "index": 21}, {"bbox": [133, 642, 481, 655], "spans": [{"bbox": [133, 642, 322, 655], "score": 1.0, "content": "we check whether it can be part of the domain", "type": "text"}, {"bbox": [323, 642, 335, 653], "score": 0.88, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [335, 642, 429, 655], "score": 1.0, "content": ". The key idea is that if", "type": "text"}, {"bbox": [430, 642, 460, 653], "score": 0.94, "content": "j\\in D_{i}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [461, 642, 481, 655], "score": 1.0, "content": " then", "type": "text"}], "index": 22}, {"bbox": [133, 654, 480, 667], "spans": [{"bbox": [133, 654, 191, 667], "score": 1.0, "content": "there must be ", "type": "text"}, {"bbox": [191, 654, 221, 664], "score": 0.91, "content": "u\\in V_{i}", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [221, 654, 366, 667], "score": 1.0, "content": " such that traversing the BDD from ", "type": "text"}, {"bbox": [366, 656, 373, 663], "score": 0.74, "content": "u", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [374, 654, 474, 667], "score": 1.0, "content": " with binary encoding of", "type": "text"}, {"bbox": [475, 654, 480, 665], "score": 0.81, "content": "j", "type": "inline_equation", "height": 11, "width": 5}], "index": 23}], "index": 21.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [133, 618, 482, 667]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_5.png | images/0704.1394_5.png |
|
0704.1394.pdf | 6 | $$C V D-S k i p p e d(B)$$
1: for each $$i=0$$ to $$n-1$$
2: $$L[i]\gets i+1$$
3: $$T\gets T$$ opologicalSor $$t(B)$$
4: for each $$k=0$$ to $$|T|-1$$
5: $$u_{1}\gets T[k],\;\;i_{1}\gets v a r_{1}(u_{1})$$
6: for each $$u_{2}\in A d j a c e n t[u_{1}]$$
7: $$L[i_{1}]\leftarrow m a x\{L[i_{1}],v a r_{1}(u_{2})\}$$
8: $$S\gets\{\},\;\;s\gets0$$
9: for $$i=0$$ to $$n-2$$
10: if $$i+1<L[s]$$
11: $$L[s]\leftarrow m a x\{L[s],L[i+1]\}$$
12: else
13: if s + 1 < L[s] S ←S ∪{s}
14: s ←i + 1
15: for each j ∈S
16: for i = j to L[j]
17: V Di ←Di
Fig. 5. In lines 1-7 the $$L[i]$$ array is created to record longest edge $$\boldsymbol{e}\,=\,(u_{1},u_{2})$$ orig-
inating from the $$V_{i}$$ layer, i.e. $$L[i]\,=\,m a x\{v a r_{1}(u^{\prime})\:\mid\:\exists(u,u^{\prime})\:\in\:E.v a r_{1}(u)\,=\,i\}$$ .
The execution time is dominated by $$T o p o l o g i c a l S o r t(B)$$ which can be implemented
as depth first search in $$O(|E|+|V|)=O(|E|)$$ time. In lines 8-14, the overlapping long
segments have been merged in $$O(n)$$ steps. Finally, in lines 15-17 the valid domains
have been copied in $$O(n)$$ steps. Hence, the total running time is $$O(|E|+n)$$ .
Fig. 6. Classical CVD algorithm. $$e n c(j)$$ denotes the binary encoding of number $$j$$ to $$k_{i}$$
values $$v_{0},\ldots,v_{k_{i}-1}$$ . If $$T r a v e r s e(u,j)$$ from Fig. 7 ends in a node different then $$T_{0}$$ ,
then $$j\in V D_{i}$$ .
| <p>$$C V D-S k i p p e d(B)$$
1: for each $$i=0$$ to $$n-1$$
2: $$L[i]\gets i+1$$
3: $$T\gets T$$ opologicalSor $$t(B)$$
4: for each $$k=0$$ to $$|T|-1$$
5: $$u_{1}\gets T[k],\;\;i_{1}\gets v a r_{1}(u_{1})$$
6: for each $$u_{2}\in A d j a c e n t[u_{1}]$$
7: $$L[i_{1}]\leftarrow m a x\{L[i_{1}],v a r_{1}(u_{2})\}$$
8: $$S\gets\{\},\;\;s\gets0$$
9: for $$i=0$$ to $$n-2$$
10: if $$i+1<L[s]$$
11: $$L[s]\leftarrow m a x\{L[s],L[i+1]\}$$
12: else
13: if s + 1 < L[s] S ←S ∪{s}
14: s ←i + 1
15: for each j ∈S
16: for i = j to L[j]
17: V Di ←Di</p>
<p>Fig. 5. In lines 1-7 the $$L[i]$$ array is created to record longest edge $$\boldsymbol{e}\,=\,(u_{1},u_{2})$$ orig-
inating from the $$V_{i}$$ layer, i.e. $$L[i]\,=\,m a x\{v a r_{1}(u^{\prime})\:\mid\:\exists(u,u^{\prime})\:\in\:E.v a r_{1}(u)\,=\,i\}$$ .
The execution time is dominated by $$T o p o l o g i c a l S o r t(B)$$ which can be implemented
as depth first search in $$O(|E|+|V|)=O(|E|)$$ time. In lines 8-14, the overlapping long
segments have been merged in $$O(n)$$ steps. Finally, in lines 15-17 the valid domains
have been copied in $$O(n)$$ steps. Hence, the total running time is $$O(|E|+n)$$ .</p>
<p>Fig. 6. Classical CVD algorithm. $$e n c(j)$$ denotes the binary encoding of number $$j$$ to $$k_{i}$$
values $$v_{0},\ldots,v_{k_{i}-1}$$ . If $$T r a v e r s e(u,j)$$ from Fig. 7 ends in a node different then $$T_{0}$$ ,
then $$j\in V D_{i}$$ .</p>
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In lines 1-7 the $$L[i]$$ array is created to record longest edge $$\\boldsymbol{e}\\,=\\,(u_{1},u_{2})$$ orig-\ninating from the $$V_{i}$$ layer, i.e. $$L[i]\\,=\\,m a x\\{v a r_{1}(u^{\\prime})\\:\\mid\\:\\exists(u,u^{\\prime})\\:\\in\\:E.v a r_{1}(u)\\,=\\,i\\}$$ .\nThe execution time is dominated by $$T o p o l o g i c a l S o r t(B)$$ which can be implemented\nas depth first search in $$O(|E|+|V|)=O(|E|)$$ time. In lines 8-14, the overlapping long\nsegments have been merged in $$O(n)$$ steps. Finally, in lines 15-17 the valid domains\nhave been copied in $$O(n)$$ steps. Hence, the total running time is $$O(|E|+n)$$ .", "block_type": "text", "index": 2}, {"type": "interline_equation", "coordinates": [137, 487, 306, 587], "content": "", "block_type": "interline_equation", "index": 3}, {"type": "text", "coordinates": [134, 597, 481, 634], "content": "Fig. 6. Classical CVD algorithm. $$e n c(j)$$ denotes the binary encoding of number $$j$$ to $$k_{i}$$\nvalues $$v_{0},\\ldots,v_{k_{i}-1}$$ . 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162, 330], "content": "16:", "score": 1.0, "index": 44}, {"type": "text", "coordinates": [176, 319, 257, 331], "content": "for i = j to L[j]", "score": 1.0, "index": 45}, {"type": "text", "coordinates": [144, 331, 162, 341], "content": "17:", "score": 1.0, "index": 46}, {"type": "text", "coordinates": [176, 329, 222, 342], "content": "V Di \u2190Di", "score": 1.0, "index": 47}, {"type": "text", "coordinates": [133, 351, 226, 365], "content": "Fig. 5. 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In lines 8-14, the overlapping long", "score": 1.0, "index": 63}, {"type": "text", "coordinates": [133, 399, 262, 412], "content": "segments have been merged in ", "score": 1.0, "index": 64}, {"type": "inline_equation", "coordinates": [262, 399, 285, 411], "content": "O(n)", "score": 0.9, "index": 65}, {"type": "text", "coordinates": [285, 399, 482, 412], "content": " steps. Finally, in lines 15-17 the valid domains", "score": 1.0, "index": 66}, {"type": "text", "coordinates": [133, 412, 215, 424], "content": "have been copied in", "score": 1.0, "index": 67}, {"type": "inline_equation", "coordinates": [215, 411, 237, 423], "content": "O(n)", "score": 0.92, "index": 68}, {"type": "text", "coordinates": [238, 412, 392, 424], "content": " steps. Hence, the total running time is", "score": 1.0, "index": 69}, {"type": "inline_equation", "coordinates": [392, 411, 440, 423], "content": "O(|E|+n)", "score": 0.93, "index": 70}, {"type": "text", "coordinates": [441, 412, 444, 424], "content": ".", "score": 1.0, "index": 71}, {"type": "interline_equation", "coordinates": [137, 487, 306, 587], "content": "\\begin{array}{r l r}&{C V D(B,x_{i})}\\\\ &{1:}&{V D_{i}\\gets\\{\\}}\\\\ &{2:}&{\\mathsf{f o r~e a c h~}j=0\\;\\;\\mathsf{t o~}\\;|D_{i}|-1}\\\\ &{3:}&{\\mathsf{f o r~e a c h~}k=0\\;\\;\\mathsf{t o}\\;|I n_{i}|-1}\\\\ &{4:}&{\\mathsf{u t}\\gets I n_{i}[k]}\\\\ &{5:}&{\\mathsf{u^{\\prime}}\\gets T r a v e r s e(u,j)}\\\\ &{6:}&{\\mathsf{i f}\\;\\;\\mathsf{u^{\\prime}}\\neq T_{0}}\\\\ &{7:}&{V D_{i}\\gets V D_{i}\\cup\\{j\\}}\\\\ &{8:}&{\\mathsf{R e t u r n}}\\end{array}", "score": 0.53, "index": 72}, {"type": "text", "coordinates": [133, 597, 266, 612], "content": "Fig. 6. Classical CVD algorithm.", "score": 1.0, "index": 73}, {"type": "inline_equation", "coordinates": [266, 598, 295, 609], "content": "e n c(j)", "score": 0.82, "index": 74}, {"type": "text", "coordinates": [295, 597, 453, 612], "content": " denotes the binary encoding of number", "score": 1.0, "index": 75}, {"type": "inline_equation", "coordinates": [453, 599, 460, 609], "content": "j", "score": 0.84, "index": 76}, {"type": "text", "coordinates": [460, 597, 470, 612], "content": " to", "score": 1.0, "index": 77}, {"type": "inline_equation", "coordinates": [471, 598, 480, 609], "content": "k_{i}", "score": 0.87, "index": 78}, {"type": "text", "coordinates": [133, 610, 161, 624], "content": "values", "score": 1.0, "index": 79}, {"type": "inline_equation", "coordinates": [162, 612, 217, 622], "content": "v_{0},\\ldots,v_{k_{i}-1}", "score": 0.89, "index": 80}, {"type": "text", "coordinates": [217, 610, 231, 624], "content": ". If", "score": 1.0, "index": 81}, {"type": "inline_equation", "coordinates": [231, 610, 296, 622], "content": "T r a v e r s e(u,j)", "score": 0.54, "index": 82}, {"type": "text", "coordinates": [297, 610, 466, 624], "content": " from Fig. 7 ends in a node different then", "score": 1.0, "index": 83}, {"type": "inline_equation", "coordinates": [466, 610, 478, 621], "content": "T_{0}", "score": 0.86, "index": 84}, {"type": "text", "coordinates": [478, 610, 481, 624], "content": ",", "score": 1.0, "index": 85}, {"type": "text", "coordinates": [133, 621, 153, 635], "content": "then", "score": 1.0, "index": 86}, {"type": "inline_equation", "coordinates": [153, 622, 190, 633], "content": "j\\in V D_{i}", "score": 0.91, "index": 87}, {"type": "text", "coordinates": [191, 621, 194, 635], "content": ".", "score": 1.0, "index": 88}] | [] | [{"type": "block", "coordinates": [137, 487, 306, 587], "content": "", "caption": ""}, {"type": "inline", "coordinates": [144, 144, 226, 155], "content": "C V D-S k i p p e d(B)", "caption": ""}, {"type": "inline", "coordinates": [208, 155, 230, 164], "content": "i=0", "caption": ""}, {"type": "inline", "coordinates": [250, 155, 273, 165], "content": "n-1", "caption": ""}, {"type": "inline", "coordinates": [171, 165, 221, 177], "content": "L[i]\\gets i+1", "caption": ""}, {"type": "inline", "coordinates": [160, 177, 188, 186], "content": "T\\gets T", "caption": ""}, {"type": "inline", "coordinates": [245, 177, 262, 187], "content": "t(B)", "caption": ""}, {"type": "inline", "coordinates": [208, 188, 232, 197], "content": "k=0", "caption": ""}, {"type": "inline", "coordinates": [252, 188, 281, 198], "content": "|T|-1", "caption": ""}, {"type": "inline", "coordinates": [171, 199, 286, 210], "content": "u_{1}\\gets T[k],\\;\\;i_{1}\\gets v a r_{1}(u_{1})", "caption": ""}, {"type": "inline", "coordinates": [219, 210, 293, 220], "content": "u_{2}\\in A d j a c e n t[u_{1}]", "caption": ""}, {"type": "inline", "coordinates": [181, 221, 302, 232], "content": "L[i_{1}]\\leftarrow m a x\\{L[i_{1}],v a r_{1}(u_{2})\\}", "caption": ""}, {"type": "inline", "coordinates": [160, 232, 225, 243], "content": "S\\gets\\{\\},\\;\\;s\\gets0", "caption": ""}, {"type": "inline", "coordinates": [181, 243, 204, 252], "content": "i=0", "caption": ""}, {"type": "inline", "coordinates": [223, 244, 246, 252], "content": "n-2", "caption": ""}, {"type": "inline", "coordinates": [192, 253, 241, 264], "content": "i+1<L[s]", "caption": ""}, {"type": "inline", "coordinates": [187, 264, 296, 276], "content": "L[s]\\leftarrow m a x\\{L[s],L[i+1]\\}", "caption": ""}, {"type": "inline", "coordinates": [227, 351, 244, 363], "content": "L[i]", "caption": ""}, {"type": "inline", "coordinates": [404, 351, 457, 363], "content": "\\boldsymbol{e}\\,=\\,(u_{1},u_{2})", "caption": ""}, {"type": "inline", "coordinates": [203, 363, 213, 374], "content": "V_{i}", "caption": ""}, {"type": "inline", "coordinates": [255, 363, 478, 375], "content": "L[i]\\,=\\,m a x\\{v a r_{1}(u^{\\prime})\\:\\mid\\:\\exists(u,u^{\\prime})\\:\\in\\:E.v a r_{1}(u)\\,=\\,i\\}", "caption": ""}, {"type": "inline", "coordinates": [279, 376, 369, 387], "content": "T o p o l o g i c a l S o r t(B)", "caption": ""}, {"type": "inline", "coordinates": [224, 387, 320, 399], "content": "O(|E|+|V|)=O(|E|)", "caption": ""}, {"type": "inline", "coordinates": [262, 399, 285, 411], "content": "O(n)", "caption": ""}, {"type": "inline", "coordinates": [215, 411, 237, 423], "content": "O(n)", "caption": ""}, {"type": "inline", "coordinates": [392, 411, 440, 423], "content": "O(|E|+n)", "caption": ""}, {"type": "inline", "coordinates": [266, 598, 295, 609], "content": "e n c(j)", "caption": ""}, {"type": "inline", "coordinates": [453, 599, 460, 609], "content": "j", "caption": ""}, {"type": "inline", "coordinates": [471, 598, 480, 609], "content": "k_{i}", "caption": ""}, {"type": "inline", "coordinates": [162, 612, 217, 622], "content": "v_{0},\\ldots,v_{k_{i}-1}", "caption": ""}, {"type": "inline", "coordinates": [231, 610, 296, 622], "content": "T r a v e r s e(u,j)", "caption": ""}, {"type": "inline", "coordinates": [466, 610, 478, 621], "content": "T_{0}", "caption": ""}, {"type": "inline", "coordinates": [153, 622, 190, 633], "content": "j\\in V D_{i}", "caption": ""}] | [] | [612.0, 792.0] | [{"type": "text", "text": "$C V D-S k i p p e d(B)$ \n1: for each $i=0$ to $n-1$ \n2: $L[i]\\gets i+1$ \n3: $T\\gets T$ opologicalSor $t(B)$ \n4: for each $k=0$ to $|T|-1$ \n5: $u_{1}\\gets T[k],\\;\\;i_{1}\\gets v a r_{1}(u_{1})$ \n6: for each $u_{2}\\in A d j a c e n t[u_{1}]$ \n7: $L[i_{1}]\\leftarrow m a x\\{L[i_{1}],v a r_{1}(u_{2})\\}$ \n8: $S\\gets\\{\\},\\;\\;s\\gets0$ \n9: for $i=0$ to $n-2$ \n10: if $i+1<L[s]$ \n11: $L[s]\\leftarrow m a x\\{L[s],L[i+1]\\}$ \n12: else \n13: if s + 1 < L[s] S \u2190S \u222a{s} \n14: s \u2190i + 1 \n15: for each j \u2208S \n16: for i = j to L[j] \n17: V Di \u2190Di ", "page_idx": 6}, {"type": "text", "text": "Fig. 5. In lines 1-7 the $L[i]$ array is created to record longest edge $\\boldsymbol{e}\\,=\\,(u_{1},u_{2})$ originating from the $V_{i}$ layer, i.e. $L[i]\\,=\\,m a x\\{v a r_{1}(u^{\\prime})\\:\\mid\\:\\exists(u,u^{\\prime})\\:\\in\\:E.v a r_{1}(u)\\,=\\,i\\}$ . The execution time is dominated by $T o p o l o g i c a l S o r t(B)$ which can be implemented as depth first search in $O(|E|+|V|)=O(|E|)$ time. In lines 8-14, the overlapping long segments have been merged in $O(n)$ steps. Finally, in lines 15-17 the valid domains have been copied in $O(n)$ steps. Hence, the total running time is $O(|E|+n)$ . ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}&{C V D(B,x_{i})}\\\\ &{1:}&{V D_{i}\\gets\\{\\}}\\\\ &{2:}&{\\mathsf{f o r~e a c h~}j=0\\;\\;\\mathsf{t o~}\\;|D_{i}|-1}\\\\ &{3:}&{\\mathsf{f o r~e a c h~}k=0\\;\\;\\mathsf{t o}\\;|I n_{i}|-1}\\\\ &{4:}&{\\mathsf{u t}\\gets I n_{i}[k]}\\\\ &{5:}&{\\mathsf{u^{\\prime}}\\gets T r a v e r s e(u,j)}\\\\ &{6:}&{\\mathsf{i f}\\;\\;\\mathsf{u^{\\prime}}\\neq T_{0}}\\\\ &{7:}&{V D_{i}\\gets V D_{i}\\cup\\{j\\}}\\\\ &{8:}&{\\mathsf{R e t u r n}}\\end{array}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "Fig. 6. Classical CVD algorithm. $e n c(j)$ denotes the binary encoding of number $j$ to $k_{i}$ values $v_{0},\\ldots,v_{k_{i}-1}$ . If $T r a v e r s e(u,j)$ from Fig. 7 ends in a node different then $T_{0}$ , then $j\\in V D_{i}$ . 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"type": "text"}, {"bbox": [261, 285, 314, 298], "score": 1.0, "content": "S \u2190S \u222a{s}", "type": "text"}], "index": 13, "is_list_start_line": true}, {"bbox": [144, 297, 228, 308], "spans": [{"bbox": [144, 297, 163, 308], "score": 1.0, "content": "14:", "type": "text"}, {"bbox": [187, 298, 228, 308], "score": 1.0, "content": "s \u2190i + 1", "type": "text"}], "index": 14, "is_list_start_line": true}, {"bbox": [144, 308, 239, 319], "spans": [{"bbox": [144, 309, 163, 319], "score": 1.0, "content": "15:", "type": "text"}, {"bbox": [164, 308, 239, 319], "score": 1.0, "content": " for each j \u2208S", "type": "text"}], "index": 15, "is_list_start_line": true}, {"bbox": [144, 319, 257, 331], "spans": [{"bbox": [144, 320, 162, 330], "score": 1.0, "content": "16:", "type": "text"}, {"bbox": [176, 319, 257, 331], "score": 1.0, "content": "for i = j to L[j]", "type": "text"}], "index": 16, "is_list_start_line": true}, {"bbox": [144, 329, 222, 342], "spans": [{"bbox": [144, 331, 162, 341], 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In lines 1-7 the", "type": "text"}, {"bbox": [227, 351, 244, 363], "score": 0.93, "content": "L[i]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [244, 351, 404, 365], "score": 1.0, "content": " array is created to record longest edge ", "type": "text"}, {"bbox": [404, 351, 457, 363], "score": 0.92, "content": "\\boldsymbol{e}\\,=\\,(u_{1},u_{2})", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [458, 351, 480, 365], "score": 1.0, "content": " orig-", "type": "text"}], "index": 18}, {"bbox": [133, 363, 481, 377], "spans": [{"bbox": [133, 364, 202, 377], "score": 1.0, "content": "inating from the", "type": "text"}, {"bbox": [203, 363, 213, 374], "score": 0.86, "content": "V_{i}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [213, 364, 255, 377], "score": 1.0, "content": " layer, i.e.", "type": "text"}, {"bbox": [255, 363, 478, 375], "score": 0.89, "content": "L[i]\\,=\\,m a x\\{v a r_{1}(u^{\\prime})\\:\\mid\\:\\exists(u,u^{\\prime})\\:\\in\\:E.v a r_{1}(u)\\,=\\,i\\}", "type": "inline_equation", "height": 12, "width": 223}, {"bbox": [478, 364, 481, 377], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [134, 376, 482, 388], "spans": [{"bbox": [134, 376, 279, 388], "score": 1.0, "content": "The execution time is dominated by", "type": "text"}, {"bbox": [279, 376, 369, 387], "score": 0.32, "content": "T o p o l o g i c a l S o r t(B)", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [370, 376, 482, 388], "score": 1.0, "content": " which can be implemented", "type": "text"}], "index": 20}, {"bbox": [133, 387, 481, 401], "spans": [{"bbox": [133, 388, 223, 401], "score": 1.0, "content": "as depth first search in", "type": "text"}, {"bbox": [224, 387, 320, 399], "score": 0.92, "content": "O(|E|+|V|)=O(|E|)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [320, 388, 481, 401], "score": 1.0, "content": " time. 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Hence, the total running time is", "type": "text"}, {"bbox": [392, 411, 440, 423], "score": 0.93, "content": "O(|E|+n)", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [441, 412, 444, 424], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 20.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [133, 351, 482, 424]}, {"type": "interline_equation", "bbox": [137, 487, 306, 587], "lines": [{"bbox": [137, 487, 306, 587], "spans": [{"bbox": [137, 487, 306, 587], "score": 0.53, "content": "\\begin{array}{r l r}&{C V D(B,x_{i})}\\\\ &{1:}&{V D_{i}\\gets\\{\\}}\\\\ &{2:}&{\\mathsf{f o r~e a c h~}j=0\\;\\;\\mathsf{t o~}\\;|D_{i}|-1}\\\\ &{3:}&{\\mathsf{f o r~e a c h~}k=0\\;\\;\\mathsf{t o}\\;|I n_{i}|-1}\\\\ &{4:}&{\\mathsf{u t}\\gets I n_{i}[k]}\\\\ &{5:}&{\\mathsf{u^{\\prime}}\\gets T r a v e r s e(u,j)}\\\\ &{6:}&{\\mathsf{i f}\\;\\;\\mathsf{u^{\\prime}}\\neq T_{0}}\\\\ &{7:}&{V D_{i}\\gets V D_{i}\\cup\\{j\\}}\\\\ &{8:}&{\\mathsf{R e t u r n}}\\end{array}", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [134, 597, 481, 634], "lines": [{"bbox": [133, 597, 480, 612], "spans": [{"bbox": [133, 597, 266, 612], "score": 1.0, "content": "Fig. 6. Classical CVD algorithm.", "type": "text"}, {"bbox": [266, 598, 295, 609], "score": 0.82, "content": "e n c(j)", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [295, 597, 453, 612], "score": 1.0, "content": " denotes the binary encoding of number", "type": "text"}, {"bbox": [453, 599, 460, 609], "score": 0.84, "content": "j", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [460, 597, 470, 612], "score": 1.0, "content": " to", "type": "text"}, {"bbox": [471, 598, 480, 609], "score": 0.87, "content": "k_{i}", "type": "inline_equation", "height": 11, "width": 9}], "index": 25}, {"bbox": [133, 610, 481, 624], "spans": [{"bbox": [133, 610, 161, 624], "score": 1.0, "content": "values", "type": "text"}, {"bbox": [162, 612, 217, 622], "score": 0.89, "content": "v_{0},\\ldots,v_{k_{i}-1}", "type": "inline_equation", "height": 10, "width": 55}, {"bbox": [217, 610, 231, 624], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [231, 610, 296, 622], "score": 0.54, "content": "T r a v e r s e(u,j)", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [297, 610, 466, 624], "score": 1.0, "content": " from Fig. 7 ends in a node different then", "type": "text"}, {"bbox": [466, 610, 478, 621], "score": 0.86, "content": "T_{0}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [478, 610, 481, 624], "score": 1.0, "content": ",", "type": "text"}], "index": 26}, {"bbox": [133, 621, 194, 635], "spans": [{"bbox": [133, 621, 153, 635], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [153, 622, 190, 633], "score": 0.91, "content": "j\\in V D_{i}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [191, 621, 194, 635], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [133, 597, 481, 635]}]} | ./arxiv_full_mineru_outputs_20/images/0704.1394_6.png | images/0704.1394_6.png |
|
0704.1394.pdf | 7 | will lead to a node other than $$T_{0}$$ , because then there is at least one satisfying path to $$T_{1}$$
allowing $$x_{i}=j$$ .
T raverse(u,j)
1: $$i\leftarrow v a r_{1}(u)$$
2: $$v_{0},\ldots,v_{k_{i}-1}\leftarrow\ e n c(j)$$
3: $$s\gets v a r_{2}(u)$$
4: if $$M a r k e d[u]=j$$ return $$T_{0}$$
5: $$M a r k e d[u]\gets j$$
6: while $$s\leq k_{i}-1$$
7: if $$v a r_{1}(u)>i$$ return $${\boldsymbol u}$$
8: if $$v_{s}=0\;\;u\leftarrow l o w(u)$$
10: else $$u\gets h i g h(u)$$
12: if $$M a r k e d[u]=j$$ return $$T_{0}$$
13: $$M a r k e d[u]\leftarrow j$$
14: $$s\gets v a r_{2}(u)$$
Fig. 7. For fixed $$u\in V,i=v a r_{1}(u)$$ , T raverse $$(u,j)$$ iterates through $$V_{i}$$ and returns
the node in which the traversal ends up.
When traversing with $$T r a v e r s e(u,j)$$ we mark the already traversed nodes $$u_{t}$$ with
$$j$$ $$,\;M a r k e d[u_{t}]\;\gets\;j$$ and prevent processing them again in the future $$j$$ -traversals
$$T r a v e r s e(u^{\prime},j)$$ . Namely, if $$T r a v e r s e(u,j)$$ reached $$T_{0}$$ node through $$u_{t}$$ , then any
other traversal $$\boldsymbol{T r a v e r s e}(u^{\prime},j)$$ reaching $$u_{t}$$ must as well end up in $$T_{0}$$ . Therefore, for
every value $$j\in D_{i}$$ , every node $$u\in V_{i}$$ is traversed at most once, leading to worst case
running time complexity of $$O(|V_{i}|\cdot|D_{i}|)$$ . Hence, the total running time for all variables
is $$\textstyle O(\sum_{i=0}^{n-1}|V_{i}|\cdot|D_{i}|)$$ .
The total worst-case running time for the two $$C V D$$ algorithms is therefore $$O(\sum_{i=0}^{n-1}|V_{i}|$$
$$\begin{array}{r}{|D_{i}|+|E|+n)=O(\sum_{i=0}^{n-1}|V_{i}|\cdot|D_{i}|+n).}\end{array}$$ .
# References
1. Jensen, R.M.: CLab: A $$C++$$ library for fast backtrack-free interactive product configuration.
http://www.itu.dk/people/rmj/clab/ (2007)
2. Raskin, J.: The Humane Interface. Addison Wesley (2000)
3. Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in
dynamic CSPs-application to configuration. Artificial Intelligence 1-2 (2002) 199–234
ftp://fpt.irit.fr/pub/IRIT/RPDMP/Configuration/.
4. Madsen, J.N.: Methods for interactive constraint satisfaction. Master’s thesis, Department
of Computer Science, University of Copenhagen (2003)
5. Hadzic, T., Subbarayan, S., Jensen, R.M., Andersen, H.R., Møller, J., Hulgaard, H.: Fast
backtrack-free product configuration using a precompiled solution space representation. In:
PETO Conference, DTU-tryk (2004) 131–138
6. Møller, J., Andersen, H.R., Hulgaard, H.: Product configuration over the internet. In: Pro-
ceedings of the 6th INFORMS Conference on Information Systems and Technology. (2002)
7. Configit Software A/S. http://www.configit-software.com (online)
| <p>will lead to a node other than $$T_{0}$$ , because then there is at least one satisfying path to $$T_{1}$$
allowing $$x_{i}=j$$ .</p>
<p>T raverse(u,j)
1: $$i\leftarrow v a r_{1}(u)$$
2: $$v_{0},\ldots,v_{k_{i}-1}\leftarrow\ e n c(j)$$
3: $$s\gets v a r_{2}(u)$$
4: if $$M a r k e d[u]=j$$ return $$T_{0}$$
5: $$M a r k e d[u]\gets j$$
6: while $$s\leq k_{i}-1$$
7: if $$v a r_{1}(u)>i$$ return $${\boldsymbol u}$$
8: if $$v_{s}=0\;\;u\leftarrow l o w(u)$$
10: else $$u\gets h i g h(u)$$
12: if $$M a r k e d[u]=j$$ return $$T_{0}$$
13: $$M a r k e d[u]\leftarrow j$$
14: $$s\gets v a r_{2}(u)$$</p>
<p>Fig. 7. For fixed $$u\in V,i=v a r_{1}(u)$$ , T raverse $$(u,j)$$ iterates through $$V_{i}$$ and returns
the node in which the traversal ends up.</p>
<p>When traversing with $$T r a v e r s e(u,j)$$ we mark the already traversed nodes $$u_{t}$$ with
$$j$$ $$,\;M a r k e d[u_{t}]\;\gets\;j$$ and prevent processing them again in the future $$j$$ -traversals
$$T r a v e r s e(u^{\prime},j)$$ . Namely, if $$T r a v e r s e(u,j)$$ reached $$T_{0}$$ node through $$u_{t}$$ , then any
other traversal $$\boldsymbol{T r a v e r s e}(u^{\prime},j)$$ reaching $$u_{t}$$ must as well end up in $$T_{0}$$ . Therefore, for
every value $$j\in D_{i}$$ , every node $$u\in V_{i}$$ is traversed at most once, leading to worst case
running time complexity of $$O(|V_{i}|\cdot|D_{i}|)$$ . Hence, the total running time for all variables
is $$\textstyle O(\sum_{i=0}^{n-1}|V_{i}|\cdot|D_{i}|)$$ .
The total worst-case running time for the two $$C V D$$ algorithms is therefore $$O(\sum_{i=0}^{n-1}|V_{i}|$$</p>
<p>$$\begin{array}{r}{|D_{i}|+|E|+n)=O(\sum_{i=0}^{n-1}|V_{i}|\cdot|D_{i}|+n).}\end{array}$$ .</p>
<h1>References</h1>
<p>1. Jensen, R.M.: CLab: A $$C++$$ library for fast backtrack-free interactive product configuration.
http://www.itu.dk/people/rmj/clab/ (2007)
2. Raskin, J.: The Humane Interface. Addison Wesley (2000)
3. Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in
dynamic CSPs-application to configuration. Artificial Intelligence 1-2 (2002) 199–234
ftp://fpt.irit.fr/pub/IRIT/RPDMP/Configuration/.
4. Madsen, J.N.: Methods for interactive constraint satisfaction. Master’s thesis, Department
of Computer Science, University of Copenhagen (2003)
5. Hadzic, T., Subbarayan, S., Jensen, R.M., Andersen, H.R., Møller, J., Hulgaard, H.: Fast
backtrack-free product configuration using a precompiled solution space representation. In:
PETO Conference, DTU-tryk (2004) 131–138
6. Møller, J., Andersen, H.R., Hulgaard, H.: Product configuration over the internet. In: Pro-
ceedings of the 6th INFORMS Conference on Information Systems and Technology. (2002)
7. Configit Software A/S. http://www.configit-software.com (online)</p>
| [{"type": "text", "coordinates": [133, 116, 481, 141], "content": "will lead to a node other than $$T_{0}$$ , because then there is at least one satisfying path to $$T_{1}$$\nallowing $$x_{i}=j$$ .", "block_type": "text", "index": 1}, {"type": "text", "coordinates": [144, 159, 307, 303], "content": "T raverse(u,j)\n1: $$i\\leftarrow v a r_{1}(u)$$\n2: $$v_{0},\\ldots,v_{k_{i}-1}\\leftarrow\\ e n c(j)$$\n3: $$s\\gets v a r_{2}(u)$$\n4: if $$M a r k e d[u]=j$$ return $$T_{0}$$\n5: $$M a r k e d[u]\\gets j$$\n6: while $$s\\leq k_{i}-1$$\n7: if $$v a r_{1}(u)>i$$ return $${\\boldsymbol u}$$\n8: if $$v_{s}=0\\;\\;u\\leftarrow l o w(u)$$\n10: else $$u\\gets h i g h(u)$$\n12: if $$M a r k e d[u]=j$$ return $$T_{0}$$\n13: $$M a r k e d[u]\\leftarrow j$$\n14: $$s\\gets v a r_{2}(u)$$", "block_type": "text", "index": 2}, {"type": "text", "coordinates": [133, 313, 482, 337], "content": "Fig. 7. For fixed $$u\\in V,i=v a r_{1}(u)$$ , T raverse $$(u,j)$$ iterates through $$V_{i}$$ and returns\nthe node in which the traversal ends up.", "block_type": "text", "index": 3}, {"type": "text", "coordinates": [132, 359, 499, 457], "content": "When traversing with $$T r a v e r s e(u,j)$$ we mark the already traversed nodes $$u_{t}$$ with\n$$j$$ $$,\\;M a r k e d[u_{t}]\\;\\gets\\;j$$ and prevent processing them again in the future $$j$$ -traversals\n$$T r a v e r s e(u^{\\prime},j)$$ . Namely, if $$T r a v e r s e(u,j)$$ reached $$T_{0}$$ node through $$u_{t}$$ , then any\nother traversal $$\\boldsymbol{T r a v e r s e}(u^{\\prime},j)$$ reaching $$u_{t}$$ must as well end up in $$T_{0}$$ . Therefore, for\nevery value $$j\\in D_{i}$$ , every node $$u\\in V_{i}$$ is traversed at most once, leading to worst case\nrunning time complexity of $$O(|V_{i}|\\cdot|D_{i}|)$$ . Hence, the total running time for all variables\nis $$\\textstyle O(\\sum_{i=0}^{n-1}|V_{i}|\\cdot|D_{i}|)$$ .\nThe total worst-case running time for the two $$C V D$$ algorithms is therefore $$O(\\sum_{i=0}^{n-1}|V_{i}|$$", "block_type": "text", "index": 4}, {"type": "text", "coordinates": [132, 445, 501, 472], "content": "$$\\begin{array}{r}{|D_{i}|+|E|+n)=O(\\sum_{i=0}^{n-1}|V_{i}|\\cdot|D_{i}|+n).}\\end{array}$$ .", "block_type": "text", "index": 5}, {"type": "title", "coordinates": [134, 487, 191, 502], "content": "References", "block_type": "title", "index": 6}, {"type": "text", "coordinates": [136, 512, 483, 666], "content": "1. Jensen, R.M.: CLab: A $$C++$$ library for fast backtrack-free interactive product configuration.\nhttp://www.itu.dk/people/rmj/clab/ (2007)\n2. Raskin, J.: The Humane Interface. Addison Wesley (2000)\n3. Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in\ndynamic CSPs-application to configuration. Artificial Intelligence 1-2 (2002) 199\u2013234\nftp://fpt.irit.fr/pub/IRIT/RPDMP/Configuration/.\n4. Madsen, J.N.: Methods for interactive constraint satisfaction. Master\u2019s thesis, Department\nof Computer Science, University of Copenhagen (2003)\n5. Hadzic, T., Subbarayan, S., Jensen, R.M., Andersen, H.R., M\u00f8ller, J., Hulgaard, H.: Fast\nbacktrack-free product configuration using a precompiled solution space representation. In:\nPETO Conference, DTU-tryk (2004) 131\u2013138\n6. M\u00f8ller, J., Andersen, H.R., Hulgaard, H.: Product configuration over the internet. In: Pro-\nceedings of the 6th INFORMS Conference on Information Systems and Technology. (2002)\n7. 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192, 249], "content": "if ", "score": 1.0, "index": 26}, {"type": "inline_equation", "coordinates": [192, 237, 240, 248], "content": "v a r_{1}(u)>i", "score": 0.7, "index": 27}, {"type": "text", "coordinates": [241, 237, 281, 249], "content": " return ", "score": 1.0, "index": 28}, {"type": "inline_equation", "coordinates": [281, 239, 288, 246], "content": "{\\boldsymbol u}", "score": 0.54, "index": 29}, {"type": "text", "coordinates": [144, 249, 157, 258], "content": "8:", "score": 1.0, "index": 30}, {"type": "text", "coordinates": [176, 249, 191, 259], "content": "if", "score": 1.0, "index": 31}, {"type": "inline_equation", "coordinates": [192, 248, 271, 258], "content": "v_{s}=0\\;\\;u\\leftarrow l o w(u)", "score": 0.58, "index": 32}, {"type": "text", "coordinates": [144, 259, 163, 270], "content": "10:", "score": 1.0, "index": 33}, {"type": "text", "coordinates": [176, 258, 203, 270], "content": "else ", "score": 1.0, "index": 34}, {"type": "inline_equation", "coordinates": [204, 259, 255, 270], "content": "u\\gets h i g h(u)", "score": 0.54, "index": 35}, {"type": "text", "coordinates": [144, 271, 162, 281], "content": "12:", "score": 1.0, "index": 36}, {"type": "text", "coordinates": [176, 270, 193, 282], "content": "if ", "score": 1.0, "index": 37}, {"type": "inline_equation", "coordinates": [193, 270, 255, 281], "content": "M a r k e d[u]=j", "score": 0.59, "index": 38}, {"type": "text", "coordinates": [255, 270, 295, 282], "content": " return ", "score": 1.0, "index": 39}, {"type": "inline_equation", "coordinates": [295, 270, 307, 280], "content": "T_{0}", "score": 0.78, "index": 40}, {"type": "text", "coordinates": [144, 282, 163, 291], "content": "13:", "score": 1.0, "index": 41}, {"type": "inline_equation", "coordinates": [176, 281, 240, 292], "content": "M a r k e d[u]\\leftarrow j", "score": 0.61, "index": 42}, {"type": "text", "coordinates": [144, 293, 162, 303], "content": "14:", "score": 1.0, "index": 43}, {"type": "inline_equation", "coordinates": [177, 292, 227, 303], "content": "s\\gets v a r_{2}(u)", "score": 0.63, "index": 44}, {"type": "text", "coordinates": [133, 313, 201, 326], "content": "Fig. 7. For fixed", "score": 1.0, "index": 45}, {"type": "inline_equation", "coordinates": [201, 313, 285, 325], "content": "u\\in V,i=v a r_{1}(u)", "score": 0.88, "index": 46}, {"type": "text", "coordinates": [285, 313, 331, 326], "content": ", T raverse", "score": 1.0, "index": 47}, {"type": "inline_equation", "coordinates": [331, 313, 354, 325], "content": "(u,j)", "score": 0.7, "index": 48}, {"type": "text", "coordinates": [354, 313, 422, 326], "content": " iterates through ", "score": 1.0, "index": 49}, {"type": "inline_equation", "coordinates": [422, 313, 432, 324], "content": "V_{i}", "score": 0.88, "index": 50}, {"type": "text", "coordinates": [433, 313, 481, 326], "content": " and returns", "score": 1.0, "index": 51}, {"type": "text", "coordinates": [133, 325, 293, 338], "content": "the node in which the traversal ends up.", "score": 1.0, "index": 52}, {"type": "text", "coordinates": [148, 360, 236, 372], "content": "When traversing with", "score": 1.0, "index": 53}, {"type": "inline_equation", "coordinates": [236, 359, 301, 371], "content": "T r a v e r s e(u,j)", "score": 0.83, "index": 54}, {"type": "text", "coordinates": [302, 360, 450, 372], "content": " we mark the already traversed nodes", "score": 1.0, "index": 55}, {"type": "inline_equation", "coordinates": [450, 361, 460, 370], "content": "u_{t}", "score": 0.86, "index": 56}, {"type": "text", "coordinates": [461, 360, 481, 372], "content": " with", "score": 1.0, "index": 57}, {"type": "inline_equation", "coordinates": [133, 372, 139, 383], "content": "j", "score": 0.32, "index": 58}, {"type": "inline_equation", "coordinates": [140, 371, 224, 383], "content": ",\\;M a r k e d[u_{t}]\\;\\gets\\;j", "score": 0.78, "index": 59}, {"type": "text", "coordinates": [224, 372, 433, 384], "content": " and prevent processing them again in the future ", "score": 1.0, "index": 60}, {"type": "inline_equation", "coordinates": [433, 372, 439, 383], "content": "j", "score": 0.83, "index": 61}, {"type": "text", "coordinates": [439, 372, 481, 384], "content": "-traversals", "score": 1.0, "index": 62}, {"type": "inline_equation", "coordinates": [134, 383, 201, 395], "content": "T r a v e r s e(u^{\\prime},j)", "score": 0.87, "index": 63}, {"type": "text", "coordinates": [201, 383, 253, 397], "content": ". 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", "page_idx": 7}, {"type": "text", "text": "T raverse(u,j) \n1: $i\\leftarrow v a r_{1}(u)$ \n2: $v_{0},\\ldots,v_{k_{i}-1}\\leftarrow\\ e n c(j)$ \n3: $s\\gets v a r_{2}(u)$ \n4: if $M a r k e d[u]=j$ return $T_{0}$ \n5: $M a r k e d[u]\\gets j$ \n6: while $s\\leq k_{i}-1$ \n7: if $v a r_{1}(u)>i$ return ${\\boldsymbol u}$ \n8: if $v_{s}=0\\;\\;u\\leftarrow l o w(u)$ \n10: else $u\\gets h i g h(u)$ \n12: if $M a r k e d[u]=j$ return $T_{0}$ \n13: $M a r k e d[u]\\leftarrow j$ \n14: $s\\gets v a r_{2}(u)$ ", "page_idx": 7}, {"type": "text", "text": "Fig. 7. For fixed $u\\in V,i=v a r_{1}(u)$ , T raverse $(u,j)$ iterates through $V_{i}$ and returns the node in which the traversal ends up. ", "page_idx": 7}, {"type": "text", "text": "When traversing with $T r a v e r s e(u,j)$ we mark the already traversed nodes $u_{t}$ with \n$j$ $,\\;M a r k e d[u_{t}]\\;\\gets\\;j$ and prevent processing them again in the future $j$ -traversals \n$T r a v e r s e(u^{\\prime},j)$ . Namely, if $T r a v e r s e(u,j)$ reached $T_{0}$ node through $u_{t}$ , then any \nother traversal $\\boldsymbol{T r a v e r s e}(u^{\\prime},j)$ reaching $u_{t}$ must as well end up in $T_{0}$ . Therefore, for \nevery value $j\\in D_{i}$ , every node $u\\in V_{i}$ is traversed at most once, leading to worst case \nrunning time complexity of $O(|V_{i}|\\cdot|D_{i}|)$ . Hence, the total running time for all variables \nis $\\textstyle O(\\sum_{i=0}^{n-1}|V_{i}|\\cdot|D_{i}|)$ . The total worst-case running time for the two $C V D$ algorithms is therefore $O(\\sum_{i=0}^{n-1}|V_{i}|$ ", "page_idx": 7}, {"type": "text", "text": "$\\begin{array}{r}{|D_{i}|+|E|+n)=O(\\sum_{i=0}^{n-1}|V_{i}|\\cdot|D_{i}|+n).}\\end{array}$ . ", "page_idx": 7}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "1. Jensen, R.M.: CLab: A $C++$ library for fast backtrack-free interactive product configuration. http://www.itu.dk/people/rmj/clab/ (2007) \n2. Raskin, J.: The Humane Interface. Addison Wesley (2000) \n3. Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in dynamic CSPs-application to configuration. Artificial Intelligence 1-2 (2002) 199\u2013234 ftp://fpt.irit.fr/pub/IRIT/RPDMP/Configuration/. \n4. Madsen, J.N.: Methods for interactive constraint satisfaction. Master\u2019s thesis, Department of Computer Science, University of Copenhagen (2003) \n5. Hadzic, T., Subbarayan, S., Jensen, R.M., Andersen, H.R., M\u00f8ller, J., Hulgaard, H.: Fast backtrack-free product configuration using a precompiled solution space representation. In: PETO Conference, DTU-tryk (2004) 131\u2013138 \n6. M\u00f8ller, J., Andersen, H.R., Hulgaard, H.: Product configuration over the internet. In: Proceedings of the 6th INFORMS Conference on Information Systems and Technology. (2002) \n7. Configit Software A/S. http://www.configit-software.com (online) \n8. Tsang, E.: Foundations of Constraint Satisfaction. Academic Press (1993) \n9. Dechter, R.: Constraint Processing. Morgan Kaufmann (2003) \n10. Subbarayan, S., Jensen, R.M., Hadzic, T., Andersen, H.R., Hulgaard, H., M\u00f8ller, J.: Comparing two implementations of a complete and backtrack-free interactive configurator. In: CP\u201904 CSPIA Workshop. (2004) 97\u2013111 \n11. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers 8 (1986) 677\u2013691 \n12. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design. Springer (1998) \n13. Wegener, I.: Branching Programs and Binary Decision Diagrams. 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|
0704.1394.pdf | 8 | 8. Tsang, E.: Foundations of Constraint Satisfaction. Academic Press (1993)
9. Dechter, R.: Constraint Processing. Morgan Kaufmann (2003)
10. Subbarayan, S., Jensen, R.M., Hadzic, T., Andersen, H.R., Hulgaard, H., Møller, J.: Com-
paring two implementations of a complete and backtrack-free interactive configurator. In:
CP’04 CSPIA Workshop. (2004) 97–111
11. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions
on Computers 8 (1986) 677–691
12. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design. Springer (1998)
13. Wegener, I.: Branching Programs and Binary Decision Diagrams. Society for Industrial and
Applied Mathematics (SIAM) (2000)
| <p>8. Tsang, E.: Foundations of Constraint Satisfaction. Academic Press (1993)
9. Dechter, R.: Constraint Processing. Morgan Kaufmann (2003)
10. Subbarayan, S., Jensen, R.M., Hadzic, T., Andersen, H.R., Hulgaard, H., Møller, J.: Com-
paring two implementations of a complete and backtrack-free interactive configurator. In:
CP’04 CSPIA Workshop. (2004) 97–111
11. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions
on Computers 8 (1986) 677–691
12. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design. Springer (1998)
13. Wegener, I.: Branching Programs and Binary Decision Diagrams. Society for Industrial and
Applied Mathematics (SIAM) (2000)</p>
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