362 lines
11 KiB
TeX
362 lines
11 KiB
TeX
\chapter{Strong connectivity}
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\index{strongly connected graph}
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In a directed graph,
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the edges can be traversed in one direction only,
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so even if the graph is connected,
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this does not guarantee that there would be
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a path from a node to another node.
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For this reason, it is meaningful to define a new concept
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that requires more than connectivity.
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A graph is \key{strongly connected}
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if there is a path from any node to all
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other nodes in the graph.
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For example, in the following picture,
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the left graph is strongly connected
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while the right graph is not.
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,1) {$1$};
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\node[draw, circle] (2) at (3,1) {$2$};
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\node[draw, circle] (3) at (1,-1) {$3$};
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\node[draw, circle] (4) at (3,-1) {$4$};
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\path[draw,thick,->] (1) -- (2);
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\path[draw,thick,->] (2) -- (4);
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\path[draw,thick,->] (4) -- (3);
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\path[draw,thick,->] (3) -- (1);
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\node[draw, circle] (1b) at (6,1) {$1$};
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\node[draw, circle] (2b) at (8,1) {$2$};
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\node[draw, circle] (3b) at (6,-1) {$3$};
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\node[draw, circle] (4b) at (8,-1) {$4$};
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\path[draw,thick,->] (1b) -- (2b);
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\path[draw,thick,->] (2b) -- (4b);
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\path[draw,thick,->] (4b) -- (3b);
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\path[draw,thick,->] (1b) -- (3b);
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\end{tikzpicture}
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\end{center}
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The right graph is not strongly connected
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because, for example, there is no path
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from node 2 to node 1.
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\index{strongly connected component}
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\index{component graph}
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The \key{strongly connected components}
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of a graph divide the graph into strongly connected
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parts that are as large as possible.
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The strongly connected components form an
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acyclic \key{component graph} that represents
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the deep structure of the original graph.
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For example, for the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,->] (2) -- (1);
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\path[draw,thick,->] (1) -- (3);
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\path[draw,thick,->] (3) -- (2);
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\path[draw,thick,->] (2) -- (4);
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\path[draw,thick,->] (3) -- (5);
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\path[draw,thick,->] (4) edge [bend left] (6);
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\path[draw,thick,->] (6) edge [bend left] (4);
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\path[draw,thick,->] (4) -- (5);
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\path[draw,thick,->] (5) -- (7);
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\path[draw,thick,->] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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the strongly connected components are as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,->] (2) -- (1);
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\path[draw,thick,->] (1) -- (3);
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\path[draw,thick,->] (3) -- (2);
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\path[draw,thick,->] (2) -- (4);
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\path[draw,thick,->] (3) -- (5);
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\path[draw,thick,->] (4) edge [bend left] (6);
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\path[draw,thick,->] (6) edge [bend left] (4);
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\path[draw,thick,->] (4) -- (5);
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\path[draw,thick,->] (5) -- (7);
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\path[draw,thick,->] (6) -- (7);
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\draw [red,thick,dashed,line width=2pt] (-0.5,2.5) rectangle (-3.5,-0.5);
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\draw [red,thick,dashed,line width=2pt] (-4.5,2.5) rectangle (-7.5,1.5);
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\draw [red,thick,dashed,line width=2pt] (-4.5,0.5) rectangle (-5.5,-0.5);
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\draw [red,thick,dashed,line width=2pt] (-6.5,0.5) rectangle (-7.5,-0.5);
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\end{tikzpicture}
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\end{center}
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The corresponding component graph is as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (-3,1) {$B$};
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\node[draw, circle] (2) at (-6,2) {$A$};
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\node[draw, circle] (3) at (-5,0) {$D$};
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\node[draw, circle] (4) at (-7,0) {$C$};
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\path[draw,thick,->] (1) -- (2);
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\path[draw,thick,->] (1) -- (3);
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\path[draw,thick,->] (2) -- (3);
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\path[draw,thick,->] (2) -- (4);
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\path[draw,thick,->] (3) -- (4);
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\end{tikzpicture}
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\end{center}
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The components are $A=\{1,2\}$,
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$B=\{3,6,7\}$, $C=\{4\}$ and $D=\{5\}$.
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A component graph is an acyclic, directed graph,
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so it is easier to process than the original graph.
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Since the graph does not contain cycles,
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we can always construct a topological sort and
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use dynamic programming techniques like those
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presented in Chapter 16.
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\section{Kosaraju's algorithm}
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\index{Kosaraju's algorithm}
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\key{Kosaraju's algorithm}\footnote{According to \cite{aho83},
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S. R. Kosaraju invented this algorithm in 1978
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but did not publish it. In 1981, the same algorithm was rediscovered
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and published by M. Sharir \cite{sha81}.} is an efficient
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method for finding the strongly connected components
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of a directed graph.
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The algorithm performs two depth-first searches:
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the first search constructs a list of nodes
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according to the structure of the graph,
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and the second search forms the strongly connected components.
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\subsubsection{Search 1}
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The first phase of Kosaraju's algorithm constructs
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a list of nodes in the order in which a
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depth-first search processes them.
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The algorithm goes through the nodes,
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and begins a depth-first search at each
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unprocessed node.
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Each node will be added to the list
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after it has been processed.
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In the example graph, the nodes are processed
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in the following order:
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\node at (-7,2.75) {$1/8$};
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\node at (-5,2.75) {$2/7$};
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\node at (-3,2.75) {$9/14$};
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\node at (-7,-0.75) {$4/5$};
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\node at (-5,-0.75) {$3/6$};
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\node at (-3,-0.75) {$11/12$};
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\node at (-1,1.75) {$10/13$};
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\path[draw,thick,->] (2) -- (1);
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\path[draw,thick,->] (1) -- (3);
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\path[draw,thick,->] (3) -- (2);
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\path[draw,thick,->] (2) -- (4);
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\path[draw,thick,->] (3) -- (5);
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\path[draw,thick,->] (4) edge [bend left] (6);
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\path[draw,thick,->] (6) edge [bend left] (4);
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\path[draw,thick,->] (4) -- (5);
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\path[draw,thick,->] (5) -- (7);
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\path[draw,thick,->] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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The notation $x/y$ means that
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processing the node started
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at time $x$ and finished at time $y$.
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Thus, the corresponding list is as follows:
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\begin{tabular}{ll}
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\\
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node & processing time \\
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\hline
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4 & 5 \\
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5 & 6 \\
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2 & 7 \\
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1 & 8 \\
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6 & 12 \\
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7 & 13 \\
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3 & 14 \\
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\\
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\end{tabular}
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%
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% In the second phase of the algorithm,
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% the nodes will be processed
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% in reverse order: $[3,7,6,1,2,5,4]$.
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\subsubsection{Search 2}
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The second phase of the algorithm
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forms the strongly connected components
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of the graph.
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First, the algorithm reverses every
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edge in the graph.
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This guarantees that during the second search,
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we will always find strongly connected
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components that do not have extra nodes.
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After reversing the edges,
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the example graph is as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,<-] (2) -- (1);
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\path[draw,thick,<-] (1) -- (3);
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\path[draw,thick,<-] (3) -- (2);
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\path[draw,thick,<-] (2) -- (4);
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\path[draw,thick,<-] (3) -- (5);
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\path[draw,thick,<-] (4) edge [bend left] (6);
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\path[draw,thick,<-] (6) edge [bend left] (4);
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\path[draw,thick,<-] (4) -- (5);
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\path[draw,thick,<-] (5) -- (7);
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\path[draw,thick,<-] (6) -- (7);
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\end{tikzpicture}
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\end{center}
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After this, the algorithm goes through
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the list of nodes created by the first search,
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in \emph{reverse} order.
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If a node does not belong to a component,
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the algorithm creates a new component
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and starts a depth-first search
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that adds all new nodes found during the search
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to the new component.
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In the example graph, the first component
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begins at node 3:
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,<-] (2) -- (1);
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\path[draw,thick,<-] (1) -- (3);
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\path[draw,thick,<-] (3) -- (2);
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\path[draw,thick,<-] (2) -- (4);
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\path[draw,thick,<-] (3) -- (5);
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\path[draw,thick,<-] (4) edge [bend left] (6);
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\path[draw,thick,<-] (6) edge [bend left] (4);
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\path[draw,thick,<-] (4) -- (5);
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\path[draw,thick,<-] (5) -- (7);
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\path[draw,thick,<-] (6) -- (7);
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\draw [red,thick,dashed,line width=2pt] (-0.5,2.5) rectangle (-3.5,-0.5);
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\end{tikzpicture}
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\end{center}
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Note that since all edges are reversed,
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the component does not ''leak'' to other parts in the graph.
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\begin{samepage}
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The next nodes in the list are nodes 7 and 6,
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but they already belong to a component,
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so the next new component begins at node 1:
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,<-] (2) -- (1);
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\path[draw,thick,<-] (1) -- (3);
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\path[draw,thick,<-] (3) -- (2);
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\path[draw,thick,<-] (2) -- (4);
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\path[draw,thick,<-] (3) -- (5);
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\path[draw,thick,<-] (4) edge [bend left] (6);
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\path[draw,thick,<-] (6) edge [bend left] (4);
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\path[draw,thick,<-] (4) -- (5);
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\path[draw,thick,<-] (5) -- (7);
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\path[draw,thick,<-] (6) -- (7);
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\draw [red,thick,dashed,line width=2pt] (-0.5,2.5) rectangle (-3.5,-0.5);
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\draw [red,thick,dashed,line width=2pt] (-4.5,2.5) rectangle (-7.5,1.5);
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%\draw [red,thick,dashed,line width=2pt] (-4.5,0.5) rectangle (-5.5,-0.5);
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%\draw [red,thick,dashed,line width=2pt] (-6.5,0.5) rectangle (-7.5,-0.5);
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\end{tikzpicture}
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\end{center}
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\end{samepage}
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\begin{samepage}
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Finally, the algorithm processes nodes 5 and 4
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that create the remaining strongly connected components:
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\begin{center}
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\begin{tikzpicture}[scale=0.9,label distance=-2mm]
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\node[draw, circle] (1) at (-1,1) {$7$};
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\node[draw, circle] (2) at (-3,2) {$3$};
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\node[draw, circle] (4) at (-5,2) {$2$};
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\node[draw, circle] (6) at (-7,2) {$1$};
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\node[draw, circle] (3) at (-3,0) {$6$};
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\node[draw, circle] (5) at (-5,0) {$5$};
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\node[draw, circle] (7) at (-7,0) {$4$};
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\path[draw,thick,<-] (2) -- (1);
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\path[draw,thick,<-] (1) -- (3);
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\path[draw,thick,<-] (3) -- (2);
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\path[draw,thick,<-] (2) -- (4);
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\path[draw,thick,<-] (3) -- (5);
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\path[draw,thick,<-] (4) edge [bend left] (6);
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\path[draw,thick,<-] (6) edge [bend left] (4);
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\path[draw,thick,<-] (4) -- (5);
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\path[draw,thick,<-] (5) -- (7);
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\path[draw,thick,<-] (6) -- (7);
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\draw [red,thick,dashed,line width=2pt] (-0.5,2.5) rectangle (-3.5,-0.5);
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\draw [red,thick,dashed,line width=2pt] (-4.5,2.5) rectangle (-7.5,1.5);
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\draw [red,thick,dashed,line width=2pt] (-4.5,0.5) rectangle (-5.5,-0.5);
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\draw [red,thick,dashed,line width=2pt] (-6.5,0.5) rectangle (-7.5,-0.5);
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\end{tikzpicture}
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\end{center}
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\end{samepage}
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The time complexity of the algorithm is $O(n+m)$,
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because the algorithm
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performs two depth-first searches.
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