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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})