$c^{\prime}_{i}=c_{j}$

$\mathcal{W}^{*}(E^{(v_{1},u_{5})})=w^{*}_{1}+w^{*}_{2}$

$|w_{1}+w^{*}+w_{2}+w^{*}-w_{1}-w_{2}|=2w^{*}$

$e_{i}\in E_{m},1\leq i\leq k$

$|w^{*}|$

$P^{{}^{\prime}}\subseteq P$

$P^{\prime}\subseteq P$

$k-i$

$c_{1}+c_{2}>1$

$c_{j}>0$

$e_{1},e_{2}$

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

$n_{R}(|y-C|+|y-B-C|)$

$\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))=\mathcal{E}(% M_{L}^{*})$

$i\xleftarrow{}i$

$\displaystyle\leq$

$\mathcal{E}_{L}=n_{L}\times\big{(}\underbrace{|x-w_{0}|}_{\text{between the % vertices of }V_{L}\text{ and }v_{2}}+\underbrace{|x-w_{0}-w_{1}|}_{\text{% between the vertices of }V_{L}\text{ and }v_{3}}+\dots+\underbrace{|x-w_{0}-w_% {1}-\dots-w_{k}|}_{\text{between the vertices of }V_{L}\text{ and }v_{k+2}}% \big{)}$

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

$M_{L}^{*}$

${n_{L}}\times{n_{L}}\times 2w^{*}$

$M^{\prime\prime}$

$f\in\mathcal{F}=\{{\text{MARK\_LEFT}},{\text{UNMARK\_LEFT}},{\text{MARK\_RIGHT% }},{\text{UNMARK\_RIGHT}}\}$

$7-21\leq 2=L_{2}$

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

$x=w_{0}+w_{1}+\dots+w_{i}$

$x>w_{0}+\dots+w_{k}$

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

$e_{2}=(v_{1},v_{4})$

$e_{1}$

$H$

$S_{L}-L_{i}-S_{R}\leq 0\xrightarrow[]{}S_{L}-S_{R}\leq L_{i}$

$j>\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$

$\mathcal{E}(M)=\mathcal{E}(M^{\prime})+\Delta_{1}({\text{MARK\_LEFT}})$

$w_{i}$

$u,v$

$v_{5}$

$n_{2}=n_{R}$

$u\in G_{2}$

$e_{i}\in E$

$\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}$

$\alpha_{2}$

$w(e^{\prime})\xleftarrow[]{}w(e^{\prime})+w(e)$

$V_{L}=\{v_{i}|1\leq i\leq n_{1}+1\}$

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

$n_{L}\times\big{(}(k-i)\;w_{i+1}\big{)}$

${T_{i}^{L}},i\in\{1,2\}$

$\mathcal{W}^{\prime}(E^{\prime})$

$e^{\prime}\in E_{m}$

$\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|$

$M^{*}\xleftarrow[]{}\operatorname{argmin}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M% _{R}^{*}))$

$u,v\in G_{2}$

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

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

$L_{i}=n_{L},\;1\leq i\leq{\mathcal{L}}$

$u\in G_{1}$

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

$\beta\geq 0$

${T_{j}^{R}},j\in\{1,\dots,{\mathcal{R}}\}$

$\Delta({\text{UNMARK\_LEFT}})$

$\displaystyle B\times k^{\prime}\times\underbrace{(n_{L}+n_{R})}_{=n-(k+k^{% \prime})}=B\times k^{\prime}\times(n-(k+k^{\prime}))$

$P^{\prime}\subset P$

$0\leq i\leq k$

$e_{i},e_{j}$

$C\leq y\leq B+C$

$c_{i}\neq c_{j}$

$\alpha_{1}=B$

$x-A-B$

$\displaystyle\xrightarrow{\text{setting }x=A+B,y=C}=$

$v\in V_{s}$

$\mathcal{E}_{1}$

$L_{1}$

$X<0$

$V_{s}$

$|w^{*}-(c_{2}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|-|w^{*}-(c_{% 2}\times w^{*}+c_{j}\times w^{*})|\leq\epsilon\times w^{*}$

$e^{*}\in S$

$c^{\prime}_{2}=c_{2}-\epsilon$

$f\in\mathcal{F}=\{\text{MARK\_LEFT},\text{UNMARK\_RIGHT}\}$

$S_{RM}$

$|w_{1}+w^{*}+w_{3}+w^{*}-w_{1}-w^{*}-w_{3}|=w^{*}$

$M_{L}^{*}\xleftarrow[]{}M_{L}^{*}\cup\{e_{i}\}$

$\begin{array}[]{cc}\Delta_{1}({\text{UNMARK\_RIGHT}})\leq R_{1}\times\epsilon% \times w^{*}\times\bigg{(}-S_{R}^{\prime}\underbrace{-L_{1}}_{<0}+S_{L}^{% \prime}\bigg{)}&

$\mathcal{E}^{v_{i},u_{j}}_{1}=\left|\pi_{v_{i},u_{j}}-\pi^{\prime}_{v_{i},u_{j% }}\right|=\left|\pi^{\prime}_{v_{i},u_{j}}-\pi_{v_{i},u_{j}}\right|=\left|% \mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|$

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

$(u,v),\;\;u\in V_{m},\;v\in\overline{V_{m}}$

$w_{1}$

$|\Delta E|\geq B(n_{L}+n_{R})+n_{L}n_{R}|x+y-A-B-C|$

$v^{\prime}_{1}$

$\operatorname{d}_{G^{\prime}}(u,v)\geq\varphi\left(\operatorname{d}_{G}(u,v)\right)$

$S\xleftarrow[]{}E_{m}$

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

$7-21\leq 3=L_{3}$

${n^{2}_{R}}\times 2\big{(}{j-1\choose 2}-{j\choose 2}\big{)}={n^{2}_{R}}\times% (-2(j-1))$

$\mathcal{E}_{LR}=0$

$w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in[0,w^{*}]$

$\displaystyle+$

$i+1-k+i<0\xrightarrow[]{}2i

$|x-A|$

$\alpha_{1}$

$V_{m}=\{v_{2},v_{3}\}$

$\Delta({\text{MARK\_LEFT}})={n^{2}_{L}}\times(2i+{\mathcal{L}}-2i-1)+{n_{L}}{n% _{R}}(j+j-{\mathcal{R}})={n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-% {\mathcal{R}})$