Kyudan/arXiv_latex · Datasets at Hugging Face

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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{)}&<R_{1}\times\epsilon\times w^{}\times(\underbrace{-S_{R}^{% \prime}+S_{L}^{\prime}}_{\leq 0})\leq 0\end{array}
\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<B\xrightarrow{}0<0
(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<k-1\xrightarrow[]{}i<\frac{k}{2}-\frac{1}{2}% \xrightarrow[\text{since }k\text{ is even}]{}i\leq\frac{k}{2}-1
\|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}})

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{)}&<R_{1}\times\epsilon\times w^{*}\times(\underbrace{-S_{R}^{% \prime}+S_{L}^{\prime}}_{\leq 0})\leq 0\end{array}

\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<B\xrightarrow{}0<0

(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<k-1\xrightarrow[]{}i<\frac{k}{2}-\frac{1}{2}% \xrightarrow[\text{since }k\text{ is even}]{}i\leq\frac{k}{2}-1

|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}})