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v_{1},v_{2}\in\overline{V_{m}}
|\Delta E|^{\prime}=w^{*}
1\leq j\leq k
{\mathcal{R}}
\pi\left(G^{\prime}\right)\leq\pi(G)-d
\displaystyle\underbrace{n_{R}\times|y-C|\times k^{\prime}}_{\text{between the% subpath of }w_{2}\text{ and the vertices in }u}+\underbrace{n_{R}\times|y-B-C% |\times k}_{{\text{between the subpath of }w_{2}\text{ and the vertices in }v}}+
\operatorname{min}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M_{R}^{*}))
E_{m}=\{e^{*}\}
\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|\xrightarrow{n_{R}=0}
P
v_{i}\in V_{L}
M_{R}^{*}\xleftarrow[]{}\emptyset
\alpha_{1}=A-x+x-A-B<B\xrightarrow{}0<2B
M_{R}
u,v\in V
n_{L}=3
\left|\sum_{k=i}^{j-2}(w_{k}+\epsilon_{k})-w^{*}-\sum_{k=i}^{j-2}w_{k}\right|=% \left|w^{*}-\sum_{k=i}^{j-2}\epsilon_{k}\right|
n_{L}(|x-A|+|x-A-B|)
\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}(i+1)\;w_{i+1}% -(k-i)\;w_{i+1}\big{)}
w^{\prime}(e_{i})=w(e_{i})+w(e^{*})
S_{R}\xleftarrow{}\sum_{\forall e_{i}\in E_{R}}R_{i}
u,v\in G_{1}
x\geq A
\Delta({\text{MARK\_RIGHT}})
\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))\leq\mathcal{% E}(M)
{\mathcal{L}}-(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})% =\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}
u_{j+1}
T_{j}^{R}
\varphi(x)=x/\alpha-\beta
\mathcal{E}=\mathcal{E}_{R}
S_{R}-S_{L}\leq R_{i}
e_{3}
a\geq 0
V_{L}
{n_{L}}
n_{R}
n_{L}\geq 0
c_{2}\geq\epsilon
{n^{2}_{L}}\times((i+1)({\mathcal{L}}-i-1)-i({\mathcal{L}}-i))={n^{2}_{L}}% \times(i{\mathcal{L}}-i^{2}-i+{\mathcal{L}}-i-1-i{\mathcal{L}}+i^{2})={n^{2}_{% L}}\times({\mathcal{L}}-2i-1)
e^{*}_{k}=(u_{j},u_{j+1})
|\Delta E|=|\overline{V_{m}}|(w^{*}_{1}+\dots+w^{*}_{k})=(n-2k)(w^{*}_{1}+% \dots+w^{*}_{k})
e_{1},e_{2},\dots,e_{k}
-\left|\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|
\overline{E_{m}}
e^{*}\in E
\epsilon_{1}+\epsilon_{2}=B
n_{L}+n_{R}=n-2
c^{\prime}_{1}+c^{\prime}_{2}=1
\mathcal{E}_{L}^{(j)},j\neq\frac{k}{2}
L_{i}\geq S_{L}-S_{R}
j>0
\Delta_{1}({\text{MARK\_LEFT}})
L_{i}\times R_{j}\times w^{*}
e_{j}\neq e_{1}
x-A
j\leq\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}
|a|=a
2\times L_{i}\times(S_{LM})
n_{L}+n_{R}=|\overline{V_{m}}|=n-(k+k^{\prime})
M_{L}^{*}\xleftarrow[]{}\emptyset
\frac{\mathcal{E}_{L}}{n_{L}}
\displaystyle(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})(% {n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({\mathcal{R}}))+\frac{{% \mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}({n^{2}_{L}}\times({% \mathcal{L}}-1)+{n_{L}}{n_{R}}(-{\mathcal{R}}))
\displaystyle\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}w_{1}+w_{1}+w_{2}+w_{1}+w% _{2}+w_{3}+\dots+w_{1}+w_{2}+\dots+w_{k}\big{)}
\mathcal{W}^{\prime}(E^{(v_{1},u_{5})})=\epsilon_{1}+\epsilon_{2}+\epsilon_{2}% +\epsilon_{3}+\epsilon_{4}
v_{i}\in\overline{V_{m}}
e_{4}=(v_{2},v_{6})
u\in\overline{V_{m}}
{T_{i}^{L}},i\in\{1,\dots,{\mathcal{L}}\}
\operatorname{CONTRACTION}
{{\mathcal{L}}\choose{2}}\times n_{L}\times n_{L}\times 2w^{*}={n^{2}_{L}}% \times{\mathcal{L}}({\mathcal{L}}-1)\times w^{*}
\displaystyle n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|
e_{3}=(v_{2},v_{5})
\mathcal{E}(M_{R}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error % associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{R}}R_{i}% \times w^{*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{% MARK\_RIGHT}})\text{'s that transform }M_{0}\text{ into }M_{R}^{*}}
{\mathcal{L}}=2
\displaystyle\geq\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{% i}\times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{L}% }L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})
\mathcal{E}_{LR}=n_{L}\times n_{R}\times|x+y-w_{0}-w_{1}-\dots-w_{k+1}|
w_{j}
\displaystyle 0
\Delta({\text{MARK\_LEFT}})
j\xleftarrow{}j
M_{0}
x=A+\epsilon_{1}
\Delta_{v_{i},u_{j}}+\Delta_{v_{i},u_{j+1}}\leq\left|w^{*}_{k}-\mathcal{W}^{% \prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|-\left|w^{% *}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime}(E^{(v_{i},u_{j}% )})\right|\leq 0
L_{i}=|\{v|v\in{T_{i}^{L}}\}|
R_{j}=|\{v|v\in{T_{j}^{R}}\}|
E^{\prime}=E-e^{*}
e^{\prime}=e_{1}
n_{L}
E_{R}=\{(v,w)|(v,w)\in E,w\neq u\}
e^{*}_{k}=e^{*}_{3}=(u_{5},u_{6})
(S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_{RU})\leq 0\xrightarrow[]{S_{LM}+S_{LU}=S_{L}}% S_{L}-L_{i}+S_{RM}-S_{RU}\leq 0
S_{R}^{\prime}=S_{R}-R_{1}
\mathcal{E}_{1}=|x-w_{0}|+\dots+|w-w_{0}-\dots-w_{k}|
S_{LM}>0
\Delta({\text{MARK\_LEFT}})\leq 0\text{ if }{n_{L}}\times({\mathcal{L}}-1)\leq% {n_{R}}({\mathcal{R}}-2j)\xrightarrow[]{\text{Rearranging the terms}}j\leq% \frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}
\displaystyle\underbrace{\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R_{i}\times w^% {*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{MARK\_RIGHT% }})\text{'s by }\epsilon_{2}}
k\geq k^{\prime}
c^{\prime}_{i}=0
w_{i+1}
(v_{1},v_{4})