$\alpha_{2}=|y-C|+|y-B-C|$

$v_{j}\in V_{R}$

$\displaystyle{n^{2}_{L}}\times{\mathcal{L}}({\mathcal{L}}-1)-{n^{2}_{R}}\times% {\mathcal{R}}({\mathcal{R}}-1)$

$u_{6}$

$\alpha_{2}=B$

$w_{1}+w_{2}=\sum_{k=1}^{2}w_{k}$

$R_{i}$

$k+1$

$\mathcal{E}_{L}^{(\frac{k}{2})}+(\frac{k}{2}+1)\;\epsilon-(\frac{k}{2})\;% \epsilon>\mathcal{E}_{L}^{(\frac{k}{2})}$

$w^{\prime}(e)=w(e)$

$i<{\mathcal{L}}$

$\mathcal{C},\;\mathcal{C}:V_{s}\rightarrow\mathbb{N},$

$v_{6}$

$\Delta_{1}({\text{UNMARK\_LEFT}})\geq 0$

$L_{1}\times R_{1}\times w^{*}$

$v_{4}$

$\mathcal{W}(E^{(v_{1},u_{5})})=w_{1}+w_{2}+w_{3}+w_{4}$

$k=k^{\prime}=1$

$n_{L}=0$

$w(e_{i})$

$\Delta(\text{UNMARK\_RIGHT})\leq 0$

$\Delta({\text{MARK\_LEFT}})=L_{i}\times\bigg{(}S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_% {RU}\bigg{)}$

$\alpha_{1}=A-x+A+B-x

$c_{1}+c_{2}<1+\epsilon=c_{1}+c_{2}$

$M^{*}$

$\mathcal{E}(M)$

$n_{L}+n_{R}=n-(k+k^{\prime})$

$\mathcal{O}(|V|)$

$e^{\prime}$

$S_{L}-S_{R}\leq L_{i}$

$B\times k^{\prime}\times(n-(k+k^{\prime}))$

$e\in S$

$1\leq i\leq{\mathcal{L}}$

$e_{2}$

$\mathcal{E}(M^{\prime})\leq\mathcal{E}(M)$

$-S_{L}+L_{i}+S_{R}\geq 0\xrightarrow{}L_{i}\geq S_{L}-S_{R}$

$\displaystyle n_{L}\times k^{\prime}\times B$

$e_{j}$

$\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i=\frac{{% \mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i$

$\displaystyle\pi^{\prime}_{v_{i},u_{j}}$

$w^{\prime}\left(e_{i}\right)=w\left(e_{i}\right)+c_{i}w\left(e^{*}\right)$

$i

$w^{\prime}$

$u_{1}$

$u,v\in V,$

$E_{m}$

$\displaystyle(k-k^{\prime})\times n_{L}\times|x-A|+n_{L}\times k^{\prime}% \times\big{(}|x-A|+|x-A-B|\big{)}$

$\displaystyle\underbrace{|w_{2}-(w_{2}+\epsilon_{2})|}_{\text{between }u\text{% and }v_{1}}+\underbrace{|w_{2}+w^{*}-(w_{2}+\epsilon_{2})|}_{\text{between }u% \text{ and }v_{2}}=|\epsilon_{2}|+|w^{*}-\epsilon_{2}|=|\epsilon_{2}|+|% \epsilon_{2}-w^{*}|$

$n_{1}=n_{L}$

$T^{L}_{1}$

$-\epsilon\times w^{*}$

$V_{m}$

$B$

$V$

$|\Delta E|\geq B(n-2)+n_{L}n_{R}|x+y-A-B-C|\geq B(n-2)=|\overline{V_{m}}|B$

$e^{*}=(u_{1},v_{1})$

$e=(u,w_{2})$

$A\leq x\leq A+B$

$T=(V,E)$

$\epsilon_{1}$

$|V|$

$\mathcal{E}_{L}^{(i)}$

$c^{\prime}_{i}=c_{i}+\epsilon$

$R_{1}\times R_{2}\times w^{*}$

$e^{*}=(u_{1},v_{1}),w^{*}=w(e^{*})$

$|z|-|x|\leq|z-x|$

$|x|=-x$

${\mathcal{R}}\xleftarrow[]{}|E_{R}|$

$V_{L}=\{v_{3},v_{4}\}$

$\displaystyle|\Delta E|=\underbrace{n_{L}\times|x-A|\times k}_{\text{between % the subpath of }w_{1}\text{ and the vertices in }v}+\underbrace{n_{L}\times|x-% A-B|\times k^{\prime}}_{\text{between the subpath of }w_{1}\text{ and the % vertices in }u}+$

$\displaystyle\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}({n% ^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(-{\mathcal{R}}))$

$|w^{*}|-\left|w^{*}-\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|\leq\left|\sum_{k% =n_{1}+1}^{j-2}\epsilon_{k}\right|$

$\left|w^{*}-\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$

$0\leq j\leq k$

$e^{*}=(v_{n_{1}+1},v_{n_{1}+2}$

$L_{i}\times L_{j}\times 2w^{*}$

$i=\{1,\dots,{\mathcal{R}}\}$

$e_{i}\in\overline{E_{m}}=E-E_{m}$

$R_{i}\times S_{LU}$

$\overline{V_{m}}$

$c_{j}$

$R_{1}$

${n^{2}_{L}}\times 2\bigg{(}{i+1\choose 2}-{i\choose 2}\bigg{)}={n^{2}_{L}}% \times 2i$

$i=\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$

$\begin{array}[]{cc}\Delta({\text{MARK\_LEFT}})\leq 0\xrightarrow[]{}{n^{2}_{L}% }\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{\mathcal{R}})&\leq 0\\ \end{array}$

$\displaystyle n_{L}\times\big{(}\sum_{j=0}^{i+1}j\;w_{j}+\sum_{j=i+2}^{k}(k+1-% j)\;w_{j}\big{)}$

$\{e_{1},\dots,e_{k}\}$

$e^{\prime}\neq e^{*}$

$w_{1}+\epsilon_{1}+w_{2}+\epsilon_{2}=\sum_{k=1}^{2}(w_{k}+\epsilon_{k})$

$e^{*}_{k}=(u_{j},u_{j+1})\in E_{m}$

$n_{L}=n_{R}$

$\Delta(\text{MARK\_LEFT})\leq 0$

$e^{*}_{j}$

$\alpha_{1}

$|w^{*}|-\left|\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$

$i+1$

$\Delta_{v_{i},u_{j+1}}=\mathcal{E}^{v_{i},u_{j+1}}_{2}-\mathcal{E}^{v_{i},u_{j% +1}}_{1}=-\left|w^{*}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{% \prime}(E^{(v_{i},u_{j})})\right|$

$v_{i},v_{j}\in V_{L}(i

$w^{\prime}:E\rightarrow\mathbb{R}_{\geq 0}$

$x=\epsilon+\sum_{j=0}^{\frac{k}{2}}w_{j}$