diff --git a/Büchi Game-content.tex b/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}
-%------------------------------------------------
diff --git a/Büchi Games.pdf b/Büchi Games.pdf
deleted file mode 100644
index b1934e2..0000000
Binary files a/Büchi Games.pdf and /dev/null differ
diff --git a/Büchi Games.tex b/Büchi Games.tex
deleted file mode 100644
index c337cc3..0000000
--- a/Büchi Games.tex	
+++ /dev/null
@@ -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}
\ 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 3911b6e..36bb3dd 100644
Binary files a/LowdimLP.pdf and b/main.pdf differ
diff --git a/LowdimLP.tex b/main.tex
similarity index 92%
rename from LowdimLP.tex
rename to main.tex
index d192a86..f0bd1ae 100644
--- a/LowdimLP.tex
+++ b/main.tex
@@ -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{algo}
-    $s\gets 1$\\
-    asdf\\
-    sdddddddddddddddddddddddddddddddd\\
-    \textbf{Return} $\textsc{CallOracle}$
-    \end{algo}
+\begin{frame}{pseudocode}
+    \begin{figure}[h!]
+        \begin{algo}
+            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}
\ No newline at end of file