From bf7f3bc6e64288659501449b3eaa14a605eef507 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 13 Nov 2024 00:07:30 +0800 Subject: [PATCH] add a new example --- Büchi Game-content.tex | 481 --------------------------------------- Büchi Games.pdf | Bin 226113 -> 0 bytes Büchi Games.tex | 27 --- beamerthemeSimple.sty | 12 +- LowdimLP.pdf => main.pdf | Bin 555062 -> 557202 bytes LowdimLP.tex => main.tex | 27 ++- 6 files changed, 25 insertions(+), 522 deletions(-) delete mode 100644 Büchi Game-content.tex delete mode 100644 Büchi Games.pdf delete mode 100644 Büchi Games.tex rename LowdimLP.pdf => main.pdf (79%) rename LowdimLP.tex => main.tex (92%) diff --git a/Büchi Game-content.tex b/Büchi Game-content.tex deleted file mode 100644 index 69c7e8b..0000000 --- a/Büchi Game-content.tex +++ /dev/null @@ -1,481 +0,0 @@ -\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} -%------------------------------------------------ diff --git a/Büchi Games.pdf b/Büchi Games.pdf deleted file mode 100644 index b1934e2c78121cbda442a73aae2dbffc1a999f3d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 226113 zcmb@t19YTKw>CU6Cbn&NoQZAQwmZhewrx*56Wg|JOl)V8{4+29b-wdH>pRa`>$_L4 z>gv1iUR}HPuDY&`B3BR*qX#gs!IDqU4=lq1IEk2u?2W8od3azMWlZhNoh^u108B*x zykHr{ENz@kfkcdAHipioBBsXnCZ@1_e6UW=KvP3oSohTiC0Y4CCb*7A^`|r`*2wHB zy-+L^%>Mb$LU^s!r!?PXao7A&eeuNwh0lcH$QV50^=-P$#RZ{Y?O0s=hQp$)W2mNM zOr|l06GecvmWC<3c;s)~qq8X9+0dB?F@bwq?K*V}7Rdeq^MfEih6FG&M1NFgi3rEL zke_GkSDV!6V%OlIqJ@c%SP17Oxn=C0879MvL4wH6iz!SH)wX}as;}=6Vhg4O*qxf` z$-kufWLl%yd*o+h-UBlj`r2!Ed7Bm^bXhKqr^BoD>T8~zTpm-83C>KJW-G%iRgIF4 z0_iku@&n2Gi>?8PT4&8oi@Tl;XPr%afeLx4G2+Ttji<`RuV!r-`w$Q_c1pZn$#H1u zktaDH1_*om$#)XP;brx2IMkU^ 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-\end{document} \ No newline at end of file diff --git a/beamerthemeSimple.sty b/beamerthemeSimple.sty index 9c235c4..ef52a36 100644 --- a/beamerthemeSimple.sty +++ b/beamerthemeSimple.sty @@ -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} diff --git a/LowdimLP.pdf b/main.pdf similarity index 79% rename from LowdimLP.pdf rename to main.pdf index 3911b6e9f059ace5a933c02153a9d205ed37b789..36bb3ddf0477388ad5498517b1352afa4dfbe4d1 100644 GIT binary patch delta 113708 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