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Büchi Game-content.tex
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\definecolor{lightblue}{rgb}{0.67,0.87,0.9}
%------------------------------------------------
\section{Motivation \& References}
%------------------------------------------------
\begin{frame}{Motivation \& References}
Motivation: Reachability and Büchi games are important in system verification and testing. Computing the winning set of Büchi games is a central problem in computer aided verification with a large number of applications.\\~
References: \\~
\footnotesize{
\begin{thebibliography}{99}
\bibitem[Smith, 2012]{p1} John Smith (2012)
\newblock Title of the publication
\newblock \emph{Journal Name} 12(3), 45 -- 678.
\end{thebibliography}
}
\end{frame}
\section{Reachability Game}
\begin{frame}{Reachability Game}
A reachability game is a 2-player (namely P0 and P1) game on a directed finite graph.\\~
Game graph: directed graph $G(\{V_0\cup V_1\},E)$.($\{V_0,V_1\}$is a partition of $V$) \\~
Target set: target set is $T\subseteq \{V_0\cup V_1\}$.\\~
A play $P$ is a (finite or infinite) path in the game graph beginning at the initial vertex $s$. If $v\in V_0$, P0 moves along an outgoing edge of v. Otherwise, P1 takes the move.\\~
Definition of winning: P0 wins if $T\cap P \neq \emptyset$, otherwise P1 wins.\\~
Memoryless strategy: a strategy for P0 is a mapping $\alpha : V_0 \rightarrow V$ that defines how P0 should extend the current play.
\end{frame}
%frame 5
\begin{frame}{Example for Reachability Game}
Rectangle vertices are in $V_1$, circles are in $V_0$;\\
Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.45\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\node[shape=circle,draw=red] (1) at (0,0) {1};
\node[shape=rectangle,draw=black] (2) at (0,4) {2};
\node[shape=rectangle,draw=black] (3) at (2.5,4) {3};
\node[shape=circle,draw=black] (4) at (5,0) {4};
\node[shape=circle,draw=blue] (5) at (2.5,0) {5};
\node[shape=rectangle,draw=black] (6) at (5,4) {6} ;
\path [->] (2) edge[thick] node[] {} (1);
\path [->] (2) edge[thick] node[] {} (4);
\path [->] (6) edge[thick,bend left=15] node[] {} (4);
\path [->] (4) edge[thick,bend left=15] node[] {} (6);
\path [->] (5) edge[thick,bend left=15] node[] {} (6);
\path [->] (6) edge[thick,bend right=15] node[] {} (1);
\path [->] (5) edge[thick] node[] {} (3);
\path [->] (3) edge[thick,bend right=20] node[] {} (1);
\end{tikzpicture}
\end{adjustbox}
\column{.5\textwidth} % Right column and width
A winning play for P0 is $\{5,3,1\}$\\~
\end{columns}
\end{frame}
%frame 6
\begin{frame}{Algorithm for Reachability Game}
\begin{figure}
\centering
\begin{adjustbox}{width=0.3\textwidth}
\begin{tikzpicture}
% \draw (0,0) node {};
\draw[fill=lightblue] (6,0) ellipse (28pt and 20pt);
\draw[] (7,0) ellipse (58pt and 40pt);
\node[shape=circle,draw=lightblue](t) at (6,0) {$T$};
\node[shape=circle,draw=white](o) at (7,2.4) {};
\node[shape=circle,draw=black] (1) at (7.8,0.5) {};
\node[shape=rectangle,draw=black] (2) at (8.1,-0.5) {};
\path [->] (1) edge[thick] node[] {} (t);
\path [->] (1) edge[thick] node[] {} (o);
\path [->] (2) edge[thick,bend left=15] node[] {} (t);
\path [->] (2) edge[thick,bend right=15] node[] {} (t);
\node[shape=circle,draw opacity=0](txt) at (7,-1) {Rank 1};
\end{tikzpicture}
\end{adjustbox}
\end{figure}
\begin{itemize}
\item if $s$ is in $T$, P0 wins;
\item if $s\in V_0$ and $s$ has at least one outgoing edge to $u\in T$, P0 wins in one step;
\item if $s\in V_1$ and all of $s$'s outgoing edges go to $u\in T$, P0 wins in one step;
\end{itemize}
\end{frame}
%frame 7
\begin{frame}{Algorithm for Reachability Game}
We defined Rank 0 and Rank 1 already, now we define Rank i.\\
$R_i:=\{v\in V|$ P0 can force a visit from v to a vertex in $T$ in i steps$\}$\\~
Define Reachability set of $T$ for P0, $\Reach(T,0) := \bigcup_{i=1}^{n-1}R_i$\\~
A vertex $v\in R_i$: \\~
if $v \in V_0$ and there is an edge $e(v,u)\quad u\in R_{i-1}$;\\~
if $v \in V_1$ and for every edge $e(v,u)$ we have $u\in \bigcup_{j=0}^{i-1} R_j$;\\
\end{frame}
%frame 8
\begin{frame}{Algorithm for Reachability Game}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.35\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\node[shape=circle,draw=red] (1) at (0,0) {1};
\node[shape=circle,draw=red] (2) at (2,0) {2};
\node[shape=circle,draw=blue] (3) at (4,0) {3};
\node[shape=rectangle,draw=black] (4) at (0,3) {4};
\node[shape=rectangle,draw=black] (5) at (2,3) {5};
\node[shape=rectangle,draw=black] (6) at (4,3) {6} ;
\path [->] (3) edge[thick] node[] {} (6);
\path [->] (3) edge[thick] node[] {} (5);
\path [->] (5) edge[thick] node[] {} (1);
\path [->] (5) edge[thick] node[] {} (2);
\path [->] (4) edge[thick] node[] {} (1);
\path [->] (4) edge[thick] node[] {} (3);
\path [->] (6) edge[thick, bend right=30] node[] {} (4);
\end{tikzpicture}
\end{adjustbox}
\column{.6\textwidth} % Right column and width
\begin{itemize}
\item $R_0=\{1,2\}$;
\item $R_1=\{5\}$;
\item $R_2=\{3\}$;
\item $R_3=\{4\}$;
\item $R_4=\{6\}$;
\end{itemize}
For simplicity, denote $u\in R_k$ by Rank[$u$]=k.
\end{columns}
\end{frame}
% frame 9
\begin{frame}{An O(m) Algorithm for Reachability Game}
\begin{algorithm}[H]
\scriptsize
\SetAlgoLined
\KwData{game graph $G$, target set $T$}
\KwResult{Rank[$|V|$]}
Q:= an empty queue\;
Rank[$|V|$],count[$|V|$]:= all 0s array\;
Q.push({T})\;
\While{Q is not empty}{
$u$:=Q.front,Q.pop\;
\For{$e(v,u)\in E$}{
\uIf{$v\in V_0$ and $v$ has not been visited}
{Rank[$v$]:=Rank[$u$]+1; Q.push($\{v\}$)}
\ElseIf{$v\in V_1$}{
count[v]:=count[v]+1\;
\lIf{count[v]=Out Degree of $v$}{
Rank[$v$]:=Rank[$u$]+1; Q.push($\{v\}$)
}
}
}
}
\caption{Reachability for P0}
\end{algorithm}
Every edge is used at most once.
\end{frame}
%frame 10
\begin{frame}{Type}
$T_1,T_2,...,T_k$ are disjoint subsets of $V$, now we want to compute Reachability of each one of them.\\~\\
\begin{columns}
\column{.45\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw (0,0) node {};
%\draw[fill=lightblue] (1,0) ellipse (28pt and 20pt);
\draw[] (2,1) ellipse (78pt and 60pt);
\node[shape=circle,draw=lightblue,fill =lightblue](t) at (1,0) {$T_1$};
\node[shape=circle,draw=lightblue,fill =lightblue](t) at (2.2,0) {$T_2$};
\node[shape=circle,draw=lightblue,fill =lightblue](t) at (3.4,0) {$T_3$};
% \node[shape=circle,draw=white](o) at (2,2.4) {};
%\node[shape=circle,draw=black] (1) at (2.8,0.5) {};
% \node[shape=rectangle,draw=black] (2) at (3.1,-0.5) {};
\end{tikzpicture}
\end{adjustbox}
\column{.5\textwidth} % Right column and width
\textbf{Definition} A type of vertex $x$ is a tuple $(y_1, \ldots, y_k)$, where each $x_i \in \{0,1\}$, such that $y_i=1$ iff $x$ is in $\Reach (T_i, 0)$.
\end{columns}
\end{frame}
%frame 11
\begin{frame}{Compute Types}
\begin{itemize}
\item Run reachability algorithm for every $T_i$, $O(km)$;
\item Compute simultaneously.
% \begin{itemize}
% \item similar to Algorithm 1, updating Ranks is replaced with updating Types.
% \item one update needs $O(k)$, $O(m)$ times of update is needed. $O(km)$
% \end{itemize}
\item Can it be done in linear or nearly linear time?
\end{itemize}
\end{frame}
%frame 12
\begin{frame}{Minimum Base}
The minimum base of $T$ is the minimum subset of $T$ which can generate the same Reachability set as $T$.\\~
Computing the minimum base is NP-hard.\\~
\begin{problem}[Set cover]
Given a set $S$ of n elements, a collections $S_1,S_2,...,S_m$ of subsets of $S$, and a number K, does there exists a collection of at most k of these sets whose union is equal to all of $S$.
\end{problem}
\end{frame}
%frame 13
\begin{frame}{Minimum Base}
\textbf{Proof}:\\
We prove that the decision problem for minimum base is NP-Complete.\\
The decision problem $L$ is can we find a base with at most k vertices.\\
\begin{enumerate}
\item $L$ is in NP.
\item set cover problem(which is NP-Complete) can be reduced to $L$ in polynomial time.
\begin{itemize}
\item Construct a Reachability game graph $G(V_0,E)$. There are $m$ vertices in $T$ representing $m$ subsets in set cover problem, $n$ vertices not in $T$ representing $n$ elements in $S$.
\item If subset $S_i$ contains element $x_j$, connect an edge from vertex representing $S_i$ to vertex representing $x_j$ in $T$.
\end{itemize}
\end{enumerate}
%So $L$ is NP-Complete. The minimum base problem is NP-Hard.
\end{frame}
\begin{frame}{Minimum Base}
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw (0,0) node {};
\node[shape=circle,draw=lightblue,fill =lightblue](t1) at (4,0) {$S_1$};
\node[shape=circle,draw=lightblue,fill =lightblue](t2) at (5.2,0) {$S_2$};
\node[shape=circle,draw=lightblue,fill =lightblue](t3) at (6.4,0) {$S_3$};
\node[shape=circle,draw=black](x1) at (3,2) {$x_1$};
\node[shape=circle,draw=black](x2) at (5,2) {$x_2$};
\node[shape=circle,draw=black](x3) at (7,2) {$x_3$};
\node[shape=circle,draw=black](x4) at (9,2) {$\ldots$};
\path [->] (x1) edge[thick] node[] {} (t1);
\path [->] (x2) edge[thick] node[] {} (t1);
\path [->] (x1) edge[thick] node[] {} (t2);
\path [->] (x3) edge[thick] node[] {} (t2);
\end{tikzpicture}
\end{adjustbox}\\
$S_1=\{x_1,x_2\}$\\
$S_2=\{x_1,x_3\}$\\~
So $L$ is NP-Complete. The minimum base problem is NP-Hard.
\end{frame}
\section{Büchi Game}
%frame 14
\begin{frame}{Büchi Game}
\begin{definition}[Büchi Game]
A \textbf{Büchi game} is a game $\mathcal{G}=(G,s,T)$ where $G$ is the Reachability game graph, $V_i$ is an initial vertex, $T\subseteq V$ is the target set as in Reachability game.\\~
Play: The definition of play in Büchi Game is the same as in Reachability game.\\~
Definition of winning: We assume the play $P$ is infinite here.
if there exists infinite many vertices $v\in T$ in $P$, P0 wins. Otherwise P1 wins.
\end{definition}
\end{frame}
%frame 15
\begin{frame}{Example for Büchi Game}
\begin{columns}
\column{0.45\textwidth}
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\node[shape=rectangle,draw=black] (1) at (0,0) {1};
\node[shape=circle,draw=black] (2) at (1.5,0) {2};
\node[shape=circle,draw=black] (3) at (4.5,0) {3};
\node[shape=circle,draw=black] (4) at (0,3) {4};
\node[shape=rectangle,draw=red] (5) at (1.5,3) {5};
\node[shape=rectangle,draw=black] (6) at (3,3) {6} ;
\node[shape=rectangle,draw=red] (7) at (4.5,3) {7} ;
\path [->] (4) edge[thick] node[] {} (5);
\path [->] (5) edge[thick] node[] {} (6);
\path [->] (6) edge[thick] node[] {} (7);
\path [->] (3) edge[thick] node[] {} (7);
\path [->] (2) edge[thick] node[] {} (3);
\path [->] (1) edge[thick] node[] {} (2);
\path [->] (1) edge[thick] node[] {} (5);
\path [->] (5) edge[thick] node[] {} (2);
\path [->] (6) edge[thick] node[] {} (2);
\path [->] (7) edge[thick] node[] {} (2);
\path [->] (2) edge[thick,bend right=15] node[] {} (7);
\end{tikzpicture}
\end{adjustbox}
\column{0.5\textwidth}
% P0 can force a visit from any vertex in $S=\{2,3,5,6,7\}$ to any other vertex in $S$.\\~
P0 is always winning on this game graph.
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 1}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (6,2) -- (6,-2) -- cycle; % reach
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T$};
\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)$};
\end{tikzpicture}
\end{adjustbox}
\column{.35\textwidth} % Right column and width
If $v\notin \Reach(T,0)\cup T$, $v$ can not reach $T$, P0 will lose.\\~
Some vertices in $T$ can not reach $\Reach(T,0)\cup T$, P0 will also lose on these vertices.
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 1}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (6,2) -- (6,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (0,1) -- (0,2) -- (6,2) -- (6,1) -- cycle; % reach
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_2$};
\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$};
\node[shape=circle,draw opacity=0](txt) at (4,-1.7) {$\Reach(T_2,0)$};
\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$};
\end{tikzpicture}
\end{adjustbox}
\column{.37\textwidth} % Right column and width
$T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\~
Some vertices in $T_2$ can only reach $\Reach(T,0)\backslash \Reach(T_2,0)$\\~
We find $T_3=\{v\in T_2|v$ can't reach $T_2\cup \Reach(T_2,0)\}$
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 1}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (6,2) -- (6,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (0,1) -- (0,2) -- (6,2) -- (6,1) -- cycle;
\draw[thick,fill=red, fill opacity=0.3] (0,0.5) -- (0,1) -- (6,1) -- (6,0.5) -- cycle;
\draw[thick,fill=red, fill opacity=0.3] (0,0.25) -- (0,0.5) -- (6,0.5) -- (6,0.25) -- cycle;
\draw[thick,fill=red, fill opacity=0.3] (0,0.125) -- (0,0.25) -- (6,0.25) -- (6,0.125) -- cycle;
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$};
\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$};
\node[shape=circle,draw opacity=0](txt) at (2.1,-1.5) {Winning set for P0};
\end{tikzpicture}
\end{adjustbox}
\column{.35\textwidth} % Right column and width
We repeat this process until $T_k$ does not shrink.\\~
The remaining part of $T_k\cup \Reach(T_k,0)$ is the winning set for P0.
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 1}
\begin{itemize}
\item How to find $T_1$\\
$T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\
$T_1=\{v\in T|v$ can only reach $V\backslash \{T\cup \Reach(T,0)\}\}$\\
P1 wants to reach $V\backslash \{T\cup \Reach(T,0)\}\}$, P0 tries to avoid $V\backslash \{T\cup \Reach(T,0)\}\}$.\\
compute $\Reach(V\backslash \{T\cup \Reach(T,0)\}\},1)$
\item Time complexity\\
$O(m)$ to find $T_i$, at most $O(n)$ times. Worst-case $O(nm)$.
\end{itemize}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 2}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (5,2) -- (5,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (0,-2) -- (0,-0.5) -- (5,-0.5) -- (5,-2) -- cycle; % reach
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T$};
\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
\node[shape=circle,draw opacity=0](txt) at (2,-1.7) {$\Reach(C_0\cup C_1,1)$};
\end{tikzpicture}
\end{adjustbox}
\column{.35\textwidth} % Right column and width
Compute $C_0$ and $C_1$.\\~
$C_0$ is a set of vertices in $V_0\backslash T$ having all outgoing edges to vertices in $V\backslash T$.\\
$C_1$ is a set of vertices in $V_1\backslash T$ having an outgoing edge to vertices in $V\backslash T$.\\~
Compute $\Reach(C_0\cup C_1,1)$
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 2}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (5,2) -- (5,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (0,-2) -- (0,-0.5) -- (5,-0.5) -- (5,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (2,-2) -- (2,-0.5) -- (3,-0.5) -- (3,-2) -- cycle; % reach
\draw[thick,fill=purple, fill opacity=0.3] (3,-2) -- (3,-1) -- (8,-1) -- (8,-2) -- cycle; % reach
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T$};
\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$};
\node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$};
\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$};
\node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$};
\end{tikzpicture}
\end{adjustbox}
\column{.39\textwidth} % Right column and width
Some vertices in $\Reach(C_0\cup C_1,1)$ can "reach" $T_1$.($D$ in the left picture)\\~
Compute $\Reach(T_1\cup D,0)$.\\~
$E=\Reach(C_0\cup C_1,1)\backslash \{T_1\cup D \cup \Reach(T_1\cup D,0)\}$\\~
$\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the set of vertices which can't "reach" $T$.
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 2}
\begin{columns}[c] % The "c" option specifies centered vertical alignment while the "t" option is used for top vertical alignment
\column{.6\textwidth} % Left column and width
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\draw[thick, fill opacity=0.3] (0,-2) -- (0,2) -- (8,2) -- (8,-2) -- cycle; % G
\draw[thick,fill=blue, fill opacity=0.3] (0,-2) -- (0,2) -- (2,2) -- (2,-2) -- cycle; % T
\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (5,2) -- (5,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (0,-2) -- (0,-0.5) -- (5,-0.5) -- (5,-2) -- cycle; % reach
\draw[thick,fill=red, fill opacity=0.3] (2,-2) -- (2,-0.5) -- (3,-0.5) -- (3,-2) -- cycle; % reach
\draw[thick,fill=purple, fill opacity=0.3] (3,-2) -- (3,-1) -- (8,-1) -- (8,-2) -- cycle; % reach
\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T$};
\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$};
\node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$};
\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$};
\node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$};
\end{tikzpicture}
\end{adjustbox}
\column{.4\textwidth} % Right column and width
$S=\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the same as $V\backslash \{T\cup \Reach(T)\}$ in Algorithm 1.\\~
Then we can compute $\Reach(S,1)$ to delete some losing vertices for P0 in $T$.\\~
Repeat the same process on $G\backslash\{T\backslash \Reach(S,1)\}$
\end{columns}
\end{frame}
\begin{frame}{Algorithm for Büchi Game 2}
\begin{itemize}
\item Time complexity\\~
Finding $S$ needs $O(m)$ time.\\
Also in the worst case we need to compute $S$ $O(n)$ times. worst case $O(nm)$.
\end{itemize}
\end{frame}
%------------------------------------------------

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Büchi Games.tex
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@@ -1,27 +0,0 @@
\documentclass{beamer}
\author{Yu Cong}
\title{Reachability and Büchi games}
\date{\today}
% \AtBeginSection[]{
% \frame{\frametitle{Outline}\tableofcontents[currentsection,
% subsectionstyle=show/show/shaded]}
% }
\usetheme{Simple}
% \useoutertheme{tree}
\DeclareMathOperator{\Reach}{Reach}
\begin{document}
\begin{frame}[plain]
% Print the title page as the first slide
\titlepage
\end{frame}
\begin{frame}[plain]{Overview}
% Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
\tableofcontents
\end{frame}
\input{Büchi Game-content.tex}
\end{document}

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@@ -25,13 +25,13 @@
\quad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\kill}
\def\end@lg{\end{tabbing}\end{minipage}}
\newenvironment{algorithm}
{\begin{tabular}{|l|}\hline\begin@lg}
{\end@lg\\\hline\end{tabular}}
% \newenvironment{algorithm}
% {\begin{tabular}{|l|}\hline\begin@lg}
% {\end@lg\\\hline\end{tabular}}
\newenvironment{algo}
{\begin{center}\begin{algorithm}}
{\end{algorithm}\end{center}}
% \newenvironment{algo}
% {\begin{center}\begin{algorithm}}
% {\end{algorithm}\end{center}}
% a color box
\RequirePackage[breakable, theorems, skins]{tcolorbox}

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LowdimLP.tex → main.tex
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@@ -1,7 +1,7 @@
\documentclass{beamer}
\author{Yu Cong}
\title[Minimizing sum of pwl convex function]{Minimizing the Sum of Piecewise Linear Convex Functions}
\title[template example]{Minimizing the Sum of Piecewise Linear Convex Functions}
\date{\today}
% \AtBeginSection[]{
@@ -10,6 +10,7 @@
% }
\usetheme{Simple}
\usepackage{algo}
% \useoutertheme{tree}
\begin{document}
@@ -234,13 +235,23 @@ However, observe that in our problem the piecewise linear convex function is not
\end{align*}
However, this is not possible for general pwl convex functions in $\R^d$.\footnote{see this \href{https://talldoor.uk/posts/2024-09-16-piecewise-linear.html}{blog post} for detail.}
\end{frame}
\begin{frame}{test algo}
\begin{frame}{pseudocode}
\begin{figure}[h!]
\begin{algo}
$s\gets 1$\\
asdf\\
sdddddddddddddddddddddddddddddddd\\
\textbf{Return} $\textsc{CallOracle}$
sort vertices in $G$ in such that $\deg(v_1)\geq \dots \geq \deg(v_n)$\\
for $i\in[n]$:\\
\quad for each vertex $u\in N(v_i)$:\\
\quad \quad let $U[v]=\emptyset$ for all $v$.\\
\quad \quad for each vertex $w\in N(u)$ that is not $v_i$:\\
\quad \quad \quad add $u$ to $U[w]$.\\
\quad for all vertex $w\in V$ that is not $v_i$:\\
\quad \quad if $|U[w]|\geq \ell$:\\ \quad
\quad \quad \textbf{output} tuple $(v_i,w,U[w])$\\
\quad $G=G-v_i$
\end{algo}
\caption{An $O(m\alpha(G))$ algorithm for finding all colored $K_{2,\ell'}$ for $\ell' \geq \ell$}
\label{figalg:malpha}
\end{figure}
\end{frame}
\end{document}