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luku11.tex
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luku11.tex
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\chapter{Basics of graphs}
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Many programming problems can be solved by
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modeling the problem as a graph problem
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and using an appropriate graph algorithm.
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A typical example of a graph is a network
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of roads and cities in a country.
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Sometimes, though, the graph is hidden
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in the problem and it may be difficult to detect it.
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This part of the book discusses graph algorithms,
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especially focusing on topics that
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are important in competitive programming.
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In this chapter, we go through concepts
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related to graphs,
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and study different ways to represent graphs in algorithms.
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\section{Graph terminology}
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\index{graph}
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\index{node}
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\index{edge}
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A \key{graph} consists of \key{nodes}
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and \key{edges} between them.
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In this book,
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the variable $n$ denotes the number of nodes
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in a graph, and the variable $m$ denotes
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the number of edges.
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The nodes are numbered
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using integers $1,2,\ldots,n$.
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For example, the following graph consists of 5 nodes and 7 edges:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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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,-] (1) -- (4);
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\path[draw,thick,-] (3) -- (4);
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\path[draw,thick,-] (2) -- (4);
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\path[draw,thick,-] (2) -- (5);
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\path[draw,thick,-] (4) -- (5);
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\end{tikzpicture}
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\end{center}
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\index{path}
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A \key{path} leads from node $a$ to node $b$
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through edges of the graph.
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The \key{length} of a path is the number of
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edges in it.
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For example, the above graph contains
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the path $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$
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from node 1 to node 5:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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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,-] (1) -- (4);
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\path[draw,thick,-] (3) -- (4);
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\path[draw,thick,-] (2) -- (4);
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\path[draw,thick,-] (2) -- (5);
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\path[draw,thick,-] (4) -- (5);
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\path[draw=red,thick,->,line width=2pt] (1) -- (3);
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\path[draw=red,thick,->,line width=2pt] (3) -- (4);
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\path[draw=red,thick,->,line width=2pt] (4) -- (5);
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\end{tikzpicture}
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\end{center}
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\index{cycle}
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A path is a \key{cycle} if the first and last
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node is the same.
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For example, the above graph contains
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the cycle $1 \rightarrow 3 \rightarrow 4 \rightarrow 1$.
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A path is \key{simple} if each node appears
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at most once in the path.
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%
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% \begin{itemize}
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% \item $1 \rightarrow 2 \rightarrow 5$ (length 2)
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% \item $1 \rightarrow 4 \rightarrow 5$ (length 2)
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% \item $1 \rightarrow 2 \rightarrow 4 \rightarrow 5$ (length 3)
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% \item $1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ (length 3)
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% \item $1 \rightarrow 4 \rightarrow 2 \rightarrow 5$ (length 3)
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% \item $1 \rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 5$ (length 4)
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% \end{itemize}
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\subsubsection{Connectivity}
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\index{connected graph}
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A graph is \key{connected} if there is path
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between any two nodes.
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For example, the following graph is connected:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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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,-] (3) -- (4);
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\path[draw,thick,-] (2) -- (4);
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\end{tikzpicture}
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\end{center}
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The following graph is not connected,
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because it is not possible to get
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from node 4 to any other node:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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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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\end{tikzpicture}
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\end{center}
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\index{component}
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The connected parts of a graph are
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called its \key{components}.
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For example, the following graph
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contains three components:
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$\{1,\,2,\,3\}$,
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$\{4,\,5,\,6,\,7\}$ and
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$\{8\}$.
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\begin{center}
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\begin{tikzpicture}[scale=0.8]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (6) at (6,1) {$6$};
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\node[draw, circle] (7) at (9,1) {$7$};
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\node[draw, circle] (4) at (6,3) {$4$};
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\node[draw, circle] (5) at (9,3) {$5$};
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\node[draw, circle] (8) at (11,2) {$8$};
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\path[draw,thick,-] (1) -- (2);
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\path[draw,thick,-] (2) -- (3);
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\path[draw,thick,-] (1) -- (3);
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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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\path[draw,thick,-] (6) -- (4);
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\end{tikzpicture}
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\end{center}
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\index{tree}
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A \key{tree} is a connected graph
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that consists of $n$ nodes and $n-1$ edges.
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There is a unique path
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between any two nodes in a tree.
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For example, the following graph is a tree:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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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,-] (1) -- (4);
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\path[draw,thick,-] (2) -- (5);
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\path[draw,thick,-] (2) -- (4);
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%\path[draw,thick,-] (4) -- (5);
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\end{tikzpicture}
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\end{center}
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\subsubsection{Edge directions}
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\index{directed graph}
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A graph is \key{directed}
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if the edges can be traversed
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in one direction only.
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For example, the following graph is directed:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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\path[draw,thick,->,>=latex] (1) -- (2);
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\path[draw,thick,->,>=latex] (2) -- (4);
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\path[draw,thick,->,>=latex] (2) -- (5);
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\path[draw,thick,->,>=latex] (4) -- (5);
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\path[draw,thick,->,>=latex] (4) -- (1);
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\path[draw,thick,->,>=latex] (3) -- (1);
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\end{tikzpicture}
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\end{center}
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The above graph contains
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the path $3 \rightarrow 1 \rightarrow 2 \rightarrow 5$
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from node $3$ to node $5$,
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but there is no path from node $5$ to node $3$.
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\subsubsection{Edge weights}
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\index{weighted graph}
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In a \key{weighted} graph, each edge is assigned
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a \key{weight}.
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Often the weights are interpreted as edge lengths.
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For example, the following graph is weighted:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:5] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:1] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:7] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:7] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:3] {} (5);
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\end{tikzpicture}
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\end{center}
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The length of a path in a weighted graph
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is the sum of edge weights on the path.
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For example, in the above graph,
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the length of the path
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$1 \rightarrow 2 \rightarrow 5$ is $12$
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and the length of the path
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$1 \rightarrow 3 \rightarrow 4 \rightarrow 5$ is $11$.
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The latter path is the \key{shortest} path from node $1$ to node $5$.
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\subsubsection{Neighbors and degrees}
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\index{neighbor}
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\index{degree}
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Two nodes are \key{neighbors} or \key{adjacent}
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if there is an edge between them.
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The \key{degree} of a node
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is the number of its neighbors.
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For example, in the following graph,
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the neighbors of node 2 are 1, 4 and 5,
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so its degree is 3.
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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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,-] (1) -- (4);
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\path[draw,thick,-] (3) -- (4);
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\path[draw,thick,-] (2) -- (4);
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\path[draw,thick,-] (2) -- (5);
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%\path[draw,thick,-] (4) -- (5);
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\end{tikzpicture}
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\end{center}
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The sum of degrees in a graph is always $2m$,
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where $m$ is the number of edges,
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because each edge
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increases the degree of exactly two nodes by one.
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For this reason, the sum of degrees is always even.
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\index{regular graph}
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\index{complete graph}
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A graph is \key{regular} if the
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degree of every node is a constant $d$.
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A graph is \key{complete} if the
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degree of every node is $n-1$, i.e.,
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the graph contains all possible edges
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between the nodes.
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\index{indegree}
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\index{outdegree}
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In a directed graph, the \key{indegree}
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of a node is the number of edges
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that end at the node,
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and the \key{outdegree} of a node
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is the number of edges that start at the node.
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For example, in the following graph,
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the indegree of node 2 is 2
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and the outdegree of the node is 1.
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$1$};
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\node[draw, circle] (2) at (4,3) {$2$};
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\node[draw, circle] (3) at (1,1) {$3$};
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\node[draw, circle] (4) at (4,1) {$4$};
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\node[draw, circle] (5) at (6,2) {$5$};
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\path[draw,thick,->,>=latex] (1) -- (2);
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\path[draw,thick,->,>=latex] (1) -- (3);
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\path[draw,thick,->,>=latex] (1) -- (4);
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\path[draw,thick,->,>=latex] (3) -- (4);
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\path[draw,thick,->,>=latex] (2) -- (4);
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\path[draw,thick,<-,>=latex] (2) -- (5);
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\end{tikzpicture}
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\end{center}
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\subsubsection{Colorings}
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\index{coloring}
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\index{bipartite graph}
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In a \key{coloring} of a graph,
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each node is assigned a color so that
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no adjacent nodes have the same color.
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A graph is \key{bipartite} if
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it is possible to color it using two colors.
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It turns out that a graph is bipartite
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exactly when it does not contain a cycle
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with an odd number of edges.
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For example, the graph
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle] (1) at (1,3) {$2$};
|
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\node[draw, circle] (2) at (4,3) {$3$};
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\node[draw, circle] (3) at (1,1) {$5$};
|
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\node[draw, circle] (4) at (4,1) {$6$};
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\node[draw, circle] (5) at (-2,1) {$4$};
|
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\node[draw, circle] (6) at (-2,3) {$1$};
|
||||
\path[draw,thick,-] (1) -- (2);
|
||||
\path[draw,thick,-] (1) -- (3);
|
||||
\path[draw,thick,-] (3) -- (4);
|
||||
\path[draw,thick,-] (2) -- (4);
|
||||
\path[draw,thick,-] (3) -- (6);
|
||||
\path[draw,thick,-] (5) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
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is bipartite, because it can be colored as follows:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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\node[draw, circle, fill=blue!40] (1) at (1,3) {$2$};
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\node[draw, circle, fill=red!40] (2) at (4,3) {$3$};
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||||
\node[draw, circle, fill=red!40] (3) at (1,1) {$5$};
|
||||
\node[draw, circle, fill=blue!40] (4) at (4,1) {$6$};
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||||
\node[draw, circle, fill=red!40] (5) at (-2,1) {$4$};
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||||
\node[draw, circle, fill=blue!40] (6) at (-2,3) {$1$};
|
||||
\path[draw,thick,-] (1) -- (2);
|
||||
\path[draw,thick,-] (1) -- (3);
|
||||
\path[draw,thick,-] (3) -- (4);
|
||||
\path[draw,thick,-] (2) -- (4);
|
||||
\path[draw,thick,-] (3) -- (6);
|
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\path[draw,thick,-] (5) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
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However, the graph
|
||||
\begin{center}
|
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\begin{tikzpicture}[scale=0.9]
|
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\node[draw, circle] (1) at (1,3) {$2$};
|
||||
\node[draw, circle] (2) at (4,3) {$3$};
|
||||
\node[draw, circle] (3) at (1,1) {$5$};
|
||||
\node[draw, circle] (4) at (4,1) {$6$};
|
||||
\node[draw, circle] (5) at (-2,1) {$4$};
|
||||
\node[draw, circle] (6) at (-2,3) {$1$};
|
||||
\path[draw,thick,-] (1) -- (2);
|
||||
\path[draw,thick,-] (1) -- (3);
|
||||
\path[draw,thick,-] (3) -- (4);
|
||||
\path[draw,thick,-] (2) -- (4);
|
||||
\path[draw,thick,-] (3) -- (6);
|
||||
\path[draw,thick,-] (5) -- (6);
|
||||
\path[draw,thick,-] (1) -- (6);
|
||||
\end{tikzpicture}
|
||||
\end{center}
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||||
is not bipartite, because it is not possible to color
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||||
the following cycle of three nodes using two colors:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$2$};
|
||||
\node[draw, circle] (2) at (4,3) {$3$};
|
||||
\node[draw, circle] (3) at (1,1) {$5$};
|
||||
\node[draw, circle] (4) at (4,1) {$6$};
|
||||
\node[draw, circle] (5) at (-2,1) {$4$};
|
||||
\node[draw, circle] (6) at (-2,3) {$1$};
|
||||
\path[draw,thick,-] (1) -- (2);
|
||||
\path[draw,thick,-] (1) -- (3);
|
||||
\path[draw,thick,-] (3) -- (4);
|
||||
\path[draw,thick,-] (2) -- (4);
|
||||
\path[draw,thick,-] (3) -- (6);
|
||||
\path[draw,thick,-] (5) -- (6);
|
||||
\path[draw,thick,-] (1) -- (6);
|
||||
|
||||
\path[draw=red,thick,-,line width=2pt] (1) -- (3);
|
||||
\path[draw=red,thick,-,line width=2pt] (3) -- (6);
|
||||
\path[draw=red,thick,-,line width=2pt] (6) -- (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\subsubsection{Simplicity}
|
||||
|
||||
\index{simple graph}
|
||||
|
||||
A graph is \key{simple}
|
||||
if no edge starts and ends at the same node,
|
||||
and there are no multiple
|
||||
edges between two nodes.
|
||||
Often we assume that graphs are simple.
|
||||
For example, the following graph is \emph{not} simple:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$2$};
|
||||
\node[draw, circle] (2) at (4,3) {$3$};
|
||||
\node[draw, circle] (3) at (1,1) {$5$};
|
||||
\node[draw, circle] (4) at (4,1) {$6$};
|
||||
\node[draw, circle] (5) at (-2,1) {$4$};
|
||||
\node[draw, circle] (6) at (-2,3) {$1$};
|
||||
|
||||
\path[draw,thick,-] (1) edge [bend right=20] (2);
|
||||
\path[draw,thick,-] (2) edge [bend right=20] (1);
|
||||
%\path[draw,thick,-] (1) -- (2);
|
||||
\path[draw,thick,-] (1) -- (3);
|
||||
\path[draw,thick,-] (3) -- (4);
|
||||
\path[draw,thick,-] (2) -- (4);
|
||||
\path[draw,thick,-] (3) -- (6);
|
||||
\path[draw,thick,-] (5) -- (6);
|
||||
|
||||
\tikzset{every loop/.style={in=135,out=190}}
|
||||
\path[draw,thick,-] (5) edge [loop left] (5);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
\section{Graph representation}
|
||||
|
||||
There are several ways to represent graphs
|
||||
in algorithms.
|
||||
The choice of a data structure
|
||||
depends on the size of the graph and
|
||||
the way the algorithm processes it.
|
||||
Next we will go through three common representations.
|
||||
|
||||
\subsubsection{Adjacency list representation}
|
||||
|
||||
\index{adjacency list}
|
||||
|
||||
In the adjacency list representation,
|
||||
each node $x$ in the graph is assigned an \key{adjacency list}
|
||||
that consists of nodes
|
||||
to which there is an edge from $x$.
|
||||
Adjacency lists are the most popular
|
||||
way to represent graphs, and most algorithms can be
|
||||
efficiently implemented using them.
|
||||
|
||||
A convenient way to store the adjacency lists is to declare
|
||||
an array of vectors as follows:
|
||||
\begin{lstlisting}
|
||||
vector<int> v[N];
|
||||
\end{lstlisting}
|
||||
|
||||
The constant $N$ is chosen so that all
|
||||
adjacency lists can be stored.
|
||||
For example, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
can be stored as follows:
|
||||
\begin{lstlisting}
|
||||
v[1].push_back(2);
|
||||
v[2].push_back(3);
|
||||
v[2].push_back(4);
|
||||
v[3].push_back(4);
|
||||
v[4].push_back(1);
|
||||
\end{lstlisting}
|
||||
|
||||
If the graph is undirected, it can be stored in a similar way,
|
||||
but each edge is added in both directions.
|
||||
|
||||
For a weighted graph, the structure can be extended
|
||||
as follows:
|
||||
|
||||
\begin{lstlisting}
|
||||
vector<pair<int,int>> v[N];
|
||||
\end{lstlisting}
|
||||
|
||||
If there is an edge from node $a$ to node $b$
|
||||
with weight $w$, the adjacency list of node $a$
|
||||
contains the pair $(b,w)$.
|
||||
For example, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
can be stored as follows:
|
||||
\begin{lstlisting}
|
||||
v[1].push_back({2,5});
|
||||
v[2].push_back({3,7});
|
||||
v[2].push_back({4,6});
|
||||
v[3].push_back({4,5});
|
||||
v[4].push_back({1,2});
|
||||
\end{lstlisting}
|
||||
|
||||
The benefit in using adjacency lists is that
|
||||
we can efficiently find the nodes to which
|
||||
we can move from a given node through an edge.
|
||||
For example, the following loop goes through all nodes
|
||||
to which we can move from node $s$:
|
||||
|
||||
\begin{lstlisting}
|
||||
for (auto u : v[s]) {
|
||||
// process node u
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
\subsubsection{Adjacency matrix representation}
|
||||
|
||||
\index{adjacency matrix}
|
||||
|
||||
An \key{adjacency matrix} is a two-dimensional array
|
||||
that indicates which edges the graph contains.
|
||||
We can efficiently check from an adjacency matrix
|
||||
if there is an edge between two nodes.
|
||||
The matrix can be stored as an array
|
||||
\begin{lstlisting}
|
||||
int v[N][N];
|
||||
\end{lstlisting}
|
||||
where each value $\texttt{v}[a][b]$ indicates
|
||||
whether the graph contains an edge from
|
||||
node $a$ to node $b$.
|
||||
If the edge is included in the graph,
|
||||
then $\texttt{v}[a][b]=1$,
|
||||
and otherwise $\texttt{v}[a][b]=0$.
|
||||
For example, the graph
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
can be represented as follows:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.7]
|
||||
\draw (0,0) grid (4,4);
|
||||
\node at (0.5,0.5) {1};
|
||||
\node at (1.5,0.5) {0};
|
||||
\node at (2.5,0.5) {0};
|
||||
\node at (3.5,0.5) {0};
|
||||
\node at (0.5,1.5) {0};
|
||||
\node at (1.5,1.5) {0};
|
||||
\node at (2.5,1.5) {0};
|
||||
\node at (3.5,1.5) {1};
|
||||
\node at (0.5,2.5) {0};
|
||||
\node at (1.5,2.5) {0};
|
||||
\node at (2.5,2.5) {1};
|
||||
\node at (3.5,2.5) {1};
|
||||
\node at (0.5,3.5) {0};
|
||||
\node at (1.5,3.5) {1};
|
||||
\node at (2.5,3.5) {0};
|
||||
\node at (3.5,3.5) {0};
|
||||
\node at (-0.5,0.5) {4};
|
||||
\node at (-0.5,1.5) {3};
|
||||
\node at (-0.5,2.5) {2};
|
||||
\node at (-0.5,3.5) {1};
|
||||
\node at (0.5,4.5) {1};
|
||||
\node at (1.5,4.5) {2};
|
||||
\node at (2.5,4.5) {3};
|
||||
\node at (3.5,4.5) {4};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
If the graph is directed, the adjacency matrix
|
||||
representation can be extended so that
|
||||
the matrix contains the weight of the edge
|
||||
if the edge exists.
|
||||
Using this representation, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\begin{samepage}
|
||||
corresponds to the following matrix:
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.7]
|
||||
\draw (0,0) grid (4,4);
|
||||
\node at (0.5,0.5) {2};
|
||||
\node at (1.5,0.5) {0};
|
||||
\node at (2.5,0.5) {0};
|
||||
\node at (3.5,0.5) {0};
|
||||
\node at (0.5,1.5) {0};
|
||||
\node at (1.5,1.5) {0};
|
||||
\node at (2.5,1.5) {0};
|
||||
\node at (3.5,1.5) {5};
|
||||
\node at (0.5,2.5) {0};
|
||||
\node at (1.5,2.5) {0};
|
||||
\node at (2.5,2.5) {7};
|
||||
\node at (3.5,2.5) {6};
|
||||
\node at (0.5,3.5) {0};
|
||||
\node at (1.5,3.5) {5};
|
||||
\node at (2.5,3.5) {0};
|
||||
\node at (3.5,3.5) {0};
|
||||
\node at (-0.5,0.5) {4};
|
||||
\node at (-0.5,1.5) {3};
|
||||
\node at (-0.5,2.5) {2};
|
||||
\node at (-0.5,3.5) {1};
|
||||
\node at (0.5,4.5) {1};
|
||||
\node at (1.5,4.5) {2};
|
||||
\node at (2.5,4.5) {3};
|
||||
\node at (3.5,4.5) {4};
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\end{samepage}
|
||||
|
||||
The drawback in the adjacency matrix representation
|
||||
is that there are $n^2$ elements in the matrix
|
||||
and usually most of them are zero.
|
||||
For this reason, the representation cannot be used
|
||||
if the graph is large.
|
||||
|
||||
\subsubsection{Edge list representation}
|
||||
|
||||
\index{edge list}
|
||||
|
||||
An \key{edge list} contains all edges of a graph
|
||||
in some order.
|
||||
This is a convenient way to represent a graph
|
||||
if the algorithm processes all edges of the graph
|
||||
and it is not needed to find edges that start
|
||||
at a given node.
|
||||
|
||||
The edge list can be stored in a vector
|
||||
\begin{lstlisting}
|
||||
vector<pair<int,int>> v;
|
||||
\end{lstlisting}
|
||||
where each pair $(a,b)$ denotes that
|
||||
there is an edge from node $a$ to node $b$.
|
||||
Thus, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
can be represented as follows:
|
||||
\begin{lstlisting}
|
||||
v.push_back({1,2});
|
||||
v.push_back({2,3});
|
||||
v.push_back({2,4});
|
||||
v.push_back({3,4});
|
||||
v.push_back({4,1});
|
||||
\end{lstlisting}
|
||||
|
||||
\noindent
|
||||
If the graph is weighted, the structure can
|
||||
be extended as follows:
|
||||
\begin{lstlisting}
|
||||
vector<tuple<int,int,int>> v;
|
||||
\end{lstlisting}
|
||||
Each element in this list is of the
|
||||
form $(a,b,w)$, which means that there
|
||||
is an edge from node $a$ to node $b$ with weight $w$.
|
||||
For example, the graph
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=0.9]
|
||||
\node[draw, circle] (1) at (1,3) {$1$};
|
||||
\node[draw, circle] (2) at (3,3) {$2$};
|
||||
\node[draw, circle] (3) at (5,3) {$3$};
|
||||
\node[draw, circle] (4) at (3,1) {$4$};
|
||||
|
||||
\path[draw,thick,->,>=latex] (1) -- node[font=\small,label=above:5] {} (2);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=above:7] {} (3);
|
||||
\path[draw,thick,->,>=latex] (2) -- node[font=\small,label=left:6] {} (4);
|
||||
\path[draw,thick,->,>=latex] (3) -- node[font=\small,label=right:5] {} (4);
|
||||
\path[draw,thick,->,>=latex] (4) -- node[font=\small,label=left:2] {} (1);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\begin{samepage}
|
||||
can be represented as follows:
|
||||
\begin{lstlisting}
|
||||
v.push_back(make_tuple(1,2,5));
|
||||
v.push_back(make_tuple(2,3,7));
|
||||
v.push_back(make_tuple(2,4,6));
|
||||
v.push_back(make_tuple(3,4,5));
|
||||
v.push_back(make_tuple(4,1,2));
|
||||
\end{lstlisting}
|
||||
\end{samepage}
|
||||
Loading…
Add table
Add a link
Reference in a new issue