$E^{(v_{1},u_{5})}=\{e_{1},e^{*}_{1},e_{2},e_{3},e^{*}_{2},e_{4}\}$
$x=\sum_{k=i}^{n_{1}}\epsilon_{k}$
$v_{i},v_{j}\in V_{R}$
$G^{\prime}$
$c_{i+1},\dots,c_{k}$
$S_{R}-R_{i}-S_{L}\leq 0$
$n\geq 3$
$\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|% }_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})\text{ % because }c_{i}+c_{j}\leq 1}-\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w% ^{*})|}_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*})\text{ because }c_{i}+c_{% j}\leq 1}=\epsilon\times w^{*}$
$w_{i}=w(e_{i})\;\forall i\in\{0,\dots,k+1\}$
$u_{1}\in T_{1}^{L},u_{2}\in T_{2}^{L}$
$\pi$
$\displaystyle\mathcal{E}_{L}^{(i+1)}=$
$i\geq\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$
$\displaystyle\mathcal{E}(M)=$
$n_{L}n_{R}|x+y-A-B-C|$
$V^{\prime}\subset V$
$R_{i}={n_{R}},\;1\leq i\leq{\mathcal{R}}$
$\pi_{v_{i},u_{j+1}}=\pi_{v_{i},u_{j}}+w^{*}_{k}\text{, }\pi^{\prime}_{v_{i},u_% {j}}=\pi^{\prime}_{v_{i},u_{j+1}}\text{, and }\pi^{\prime\prime}_{v_{i},u_{j}}% =\pi^{\prime\prime}_{v_{i},u_{j+1}}$
$u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{L}$
$\epsilon=0.4$
$e\in E_{m}$
$c_{i}$
$e_{i},w_{i}=w(e_{i})$
$\displaystyle\mathcal{E}^{(x $V_{m}=\{u_{1},\dots,u_{2k}\}$ $\displaystyle\underbrace{n_{L}\times n_{R}\times|x+y-A-B-C|}_{\text{between %
the subpath of }w_{1}\text{ and }w_{2}}$ $\mathcal{E}^{v_{i},u_{j}}_{2}=\left|\pi^{\prime\prime}_{v_{i},u_{j}}-\pi_{v_{i%
},u_{j}}\right|=\left|w^{*}_{k}\right|$ $c_{i}>0$ $d$ $x $\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}-\frac{\mathcal{E}_{L}}{n_{L}}\leq
0$ $\mathcal{E}^{v_{i},u_{j+1}}_{2}=0$ $\displaystyle\xrightarrow[]{}\mathcal{E}(M)\geq\mathcal{E}(M_{L}^{*})$ $|x+y-A-B-C|$ $E^{(u,v)}$ $x=w_{0}+w_{1}+\dots+w_{\frac{k}{2}}$ $w_{1}+w_{2}+w_{3}$ $j\xleftarrow[]{}0$ $S_{R}=S_{RU}+S_{RM}$ $v^{*}$ $c_{1}+c_{2}=0.6+0.8=1.4>1$ $S_{L}=\sum_{i=1}^{{\mathcal{L}}}L_{i}$ $c_{1}+c_{2}=1+\epsilon$ $w^{\prime}(e_{i})=w(e_{i})+w^{*}$ $\{e_{1},e_{3}\}$ $G^{{}^{\prime}}$ $i\xleftarrow{}i+1$ $\Delta(f)$ $\displaystyle n_{L}\times k^{\prime}\times\underbrace{\big{(}|x-A|+|x-A-B|\big%
{)}}_{\geq B\text{ (Lemma \ref{l12})}}+n_{R}\times k^{\prime}\times\underbrace%
{\big{(}|y-C|+|y-B-C|\big{)}}_{\geq B\text{ (Lemma \ref{l12})}}$ $i<\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ $\epsilon_{2}\in\{0,w^{*}\}$ $\displaystyle\mathcal{E}(M)$ $|V|=n$ $V_{L}=\{u|(u,v_{1})\in E^{\prime}\},{\mathcal{L}}=|V_{L}|$ $e_{k+1}$ $|x-A-B|$ $n_{L}+n_{R}=n-2=\left|\overline{V_{m}}\right|$ $T_{i}^{L}$ $x,z$ $i={\mathcal{L}}$ $\epsilon>0$ $G_{1}$ $v_{1}$ $E^{\prime}\subseteq E$ $S_{R}^{\prime}\geq S_{L}^{\prime}$ $M$ $e^{*}$ $u$ $u_{1}\in T_{i}^{R},u_{2}\in T_{j}^{R}$ $y$ $\displaystyle|\Delta E|=$ $v_{i}$ $V_{L},V_{R}\subset V$ $\displaystyle=\mathcal{E}(M_{0})+(\epsilon_{1}+\epsilon_{2})\times\sum_{e_{i}%
\in E_{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\xrightarrow[S_{L}-L_{i}-S%
_{R}\leq 0]{\epsilon_{1}+\epsilon_{2}\leq 1}$ $\{u_{1},v_{1}\}$ $\Delta({\text{UNMARK\_RIGHT}})=R_{i}\times\bigg{(}-(S_{RM}-R_{i})-S_{RU}-S_{LM%
}+S_{LU}\bigg{)}$ $(u,v)$ $\alpha_{1}=|x-A|+|x-A-B|$ $|z|\leq|x|+|z-x|\xrightarrow[]{}|z|-|x|\leq|z-x|$ $B<0$ $n_{R}\geq 0$ $21-7>1=R_{2}$ $v^{*}=\{v_{2},v_{3},\dots,v_{k+2}\}$ $w:E\rightarrow\mathbb{R}_{\geq 0}$ $e\in E$ $i=\{1,\dots,{\mathcal{L}}\}$ $\Delta_{1}({\text{MARK\_LEFT}})<0$ $0-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$ $u,v\in G$ $\mathcal{E}_{R}=n_{R}\times\big{(}\underbrace{|y-w_{k+1}|}_{\text{between the %
vertices of }V_{R}\text{ and }v_{k+2}}+\underbrace{|y-w_{k+1}-w_{k}|}_{\text{%
between the vertices of }V_{R}\text{ and }v_{k+1}}+\dots+\underbrace{|y-w_{k+1%
}-w_{k}-\dots-w_{1}|}_{\text{between the vertices of }V_{R}\text{ and }v_{2}}%
\big{)}$ $V_{m}=\{v_{1},v_{2}\}$ $R_{i}\times R_{j}\times w^{*}$ $L_{i}\times L_{j}\times w^{*}$ $\displaystyle\underbrace{\mathcal{E}(M_{0})}_{\text{The error associated with %
the empty marking}}+\underbrace{\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}%
\times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{%
MARK\_LEFT}})\text{'s by
}\epsilon_{1}}$ $(u_{2i-1},u_{2i})$ $V_{R}=\{v_{i}|n_{1}+2\leq i\leq n_{2}+2\}$ $E_{R}$ $w:E\Rightarrow\mathbb{R}_{\geq 0}$ $x=A$ $\epsilon_{i}=0\;\;\forall i\neq n_{1}$