794 lines
26 KiB
TeX
794 lines
26 KiB
TeX
\chapter{Shortest paths}
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\index{shortest path}
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Finding the shortest path between two nodes
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of a graph
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is an important problem that has many
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applications in practice.
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For example, a natural problem in a road network
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is to calculate the length of the shorthest route
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between two cities, given the lengths of the roads.
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In an unweighted graph, the length of a path equals
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the number of edges in the path and we can
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simply use breadth-first search to find
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the shortest path.
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However, in this chapter we concentrate on
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weighted graphs,
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and more sophisticated algorithms
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are needed
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for finding shortest paths.
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\section{Bellman–Ford algorithm}
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\index{Bellman–Ford algorithm}
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The \key{Bellman–Ford algorithm} finds the
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shortest paths from a starting node to all
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other nodes in the graph.
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The algorithm can process all kinds of graphs,
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provided that the graph does not contain a
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cycle with negative length.
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If the graph contains a negative cycle,
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the algorithm can detect this.
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The algorithm keeps track of distances
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from the starting node to other nodes.
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Initially, the distance to the starting node is 0
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and the distance to all other nodes in infinite.
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The algorithm reduces the distances by finding
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edges that shorten the paths until it is not
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possible to reduce any distance.
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\subsubsection{Example}
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Let us consider how the Bellman–Ford algorithm
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works in the following graph:
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\begin{center}
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\begin{tikzpicture}
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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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\node[color=red] at (1,3+0.55) {$0$};
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\node[color=red] at (4,3+0.55) {$\infty$};
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\node[color=red] at (1,1-0.55) {$\infty$};
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\node[color=red] at (4,1-0.55) {$\infty$};
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\node[color=red] at (6,2-0.55) {$\infty$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
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\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
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\end{tikzpicture}
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\end{center}
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Each node in the graph is assigned a distance.
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Initially, the distance to the starting node is 0,
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and the distance to all other nodes is infinite.
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The algorithm searches for edges that reduce the
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distances.
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First, all edges from node 1 reduce the distances:
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\begin{center}
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\begin{tikzpicture}
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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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\node[color=red] at (1,3+0.55) {$0$};
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\node[color=red] at (4,3+0.55) {$2$};
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\node[color=red] at (1,1-0.55) {$3$};
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\node[color=red] at (4,1-0.55) {$7$};
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\node[color=red] at (6,2-0.55) {$\infty$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
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\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
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\path[draw=red,thick,->,line width=2pt] (1) -- (2);
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\path[draw=red,thick,->,line width=2pt] (1) -- (3);
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\path[draw=red,thick,->,line width=2pt] (1) -- (4);
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\end{tikzpicture}
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\end{center}
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After this, edges
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$2 \rightarrow 5$ and $3 \rightarrow 4$
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reduce the distances:
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\begin{center}
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\begin{tikzpicture}
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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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\node[color=red] at (1,3+0.55) {$0$};
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\node[color=red] at (4,3+0.55) {$2$};
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\node[color=red] at (1,1-0.55) {$3$};
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\node[color=red] at (4,1-0.55) {$1$};
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\node[color=red] at (6,2-0.55) {$7$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
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\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
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\path[draw=red,thick,->,line width=2pt] (2) -- (5);
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\path[draw=red,thick,->,line width=2pt] (3) -- (4);
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\end{tikzpicture}
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\end{center}
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Finally, there is one more change:
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\begin{center}
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\begin{tikzpicture}
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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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\node[color=red] at (1,3+0.55) {$0$};
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\node[color=red] at (4,3+0.55) {$2$};
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\node[color=red] at (1,1-0.55) {$3$};
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\node[color=red] at (4,1-0.55) {$1$};
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\node[color=red] at (6,2-0.55) {$3$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
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\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (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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After this, no edge can reduce any distance.
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This means that the distances are final
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and we have successfully
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calculated the shortest distance
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from the starting node to all other nodes.
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For example, the smallest distance 3
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from node 1 to node 5 corresponds to
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the following path:
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\begin{center}
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\begin{tikzpicture}
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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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\node[color=red] at (1,3+0.55) {$0$};
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\node[color=red] at (4,3+0.55) {$2$};
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\node[color=red] at (1,1-0.55) {$3$};
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\node[color=red] at (4,1-0.55) {$1$};
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\node[color=red] at (6,2-0.55) {$3$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:3] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:$-2$] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:3] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:2] {} (5);
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\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (4);
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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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\subsubsection{Implementation}
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The following implementation of the
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Bellman–Ford algorithm finds the shortest distances
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from a node $x$ to all other nodes in the graph.
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The code assumes that the graph is stored
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as adjacency lists in an array
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\begin{lstlisting}
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vector<pair<int,int>> v[N];
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\end{lstlisting}
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as pairs of the form $(x,w)$:
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there is an edge to node $x$ with weight $w$.
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The algorithm consists of $n-1$ rounds,
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and on each round the algorithm goes through
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all edges in the graph and tries to
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reduce the distances.
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The algorithm builds an array \texttt{e}
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that will contain the distance from $x$
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to all nodes in the graph.
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The initial value $10^9$ means infinity.
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\begin{lstlisting}
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for (int i = 1; i <= n; i++) e[i] = 1e9;
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e[x] = 0;
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for (int i = 1; i <= n-1; i++) {
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for (int a = 1; a <= n; a++) {
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for (auto b : v[a]) {
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e[b.first] = min(e[b.first],e[a]+b.second);
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}
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}
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}
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\end{lstlisting}
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The time complexity of the algorithm is $O(nm)$,
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because the algorithm consists of $n-1$ rounds and
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iterates through all $m$ edges during a round.
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If there are no negative cycles in the graph,
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all distances are final after $n-1$ rounds,
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because each shortest path can contain at most $n-1$ edges.
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In practice, the final distances can usually
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be found much faster than in $n-1$ rounds.
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Thus, a possible way to make the algorithm more efficient
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is to stop the algorithm if no distance
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can be reduced during a round.
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\subsubsection{Negative cycle}
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\index{negative cycle}
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The Bellman–Ford algorithm can be also used to
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check if the graph contains a cycle with negative length.
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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 (0,0) {$1$};
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\node[draw, circle] (2) at (2,1) {$2$};
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\node[draw, circle] (3) at (2,-1) {$3$};
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\node[draw, circle] (4) at (4,0) {$4$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:$3$] {} (2);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:$1$] {} (4);
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\path[draw,thick,-] (1) -- node[font=\small,label=below:$5$] {} (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=right:$2$] {} (3);
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\end{tikzpicture}
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\end{center}
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\noindent
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contains a negative cycle
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$2 \rightarrow 3 \rightarrow 4 \rightarrow 2$
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with length $-4$.
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If the graph contains a negative cycle,
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we can shorten a path that contains the cycle
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infinitely many times by repeating the cycle
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again and again.
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Thus, the concept of a shortest path
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is not meaningful in this situation.
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A negative cycle can be detected
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using the Bellman–Ford algorithm by
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running the algorithm for $n$ rounds.
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If the last round reduces any distance,
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the graph contains a negative cycle.
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Note that this algorithm searches for
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a negative cycle in the whole graph
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regardless of the starting node.
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\subsubsection{SPFA algorithm}
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\index{SPFA algorithm}
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The \key{SPFA algorithm} (''Shortest Path Faster Algorithm'')
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is a variant of the Bellman–Ford algorithm,
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that is often more efficient than the original algorithm.
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It does not go through all the edges on each round,
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but instead, it chooses the edges to be examined
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in a more intelligent way.
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The algorithm maintains a queue of nodes that might
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be used for reducing the distances.
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First, the algorithm adds the starting node $x$
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to the queue.
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Then, the algorithm always processes the
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first node in the queue, and when an edge
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$a \rightarrow b$ reduces a distance,
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node $b$ is added to the end of the queue.
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The following implementation uses a
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\texttt{queue} structure \texttt{q}.
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In addition, the array \texttt{z} indicates
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if a node is already in the queue,
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in which case the algorithm does not add
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the node to the queue again.
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\begin{lstlisting}
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for (int i = 1; i <= n; i++) e[i] = 1e9;
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e[x] = 0;
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q.push(x);
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while (!q.empty()) {
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int a = q.front(); q.pop();
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z[a] = 0;
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for (auto b : v[a]) {
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if (e[a]+b.second < e[b.first]) {
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e[b.first] = e[a]+b.second;
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if (!z[b]) {q.push(b); z[b] = 1;}
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}
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}
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}
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\end{lstlisting}
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The efficiency of the SPFA algorithm depends
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on the structure of the graph:
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the algorithm is often very efficient,
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but its worst case time complexity is still
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$O(nm)$ and it is possible to create inputs
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that make the algorithm as slow as the
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standard Bellman–Ford algorithm.
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\section{Dijkstra's algorithm}
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\index{Dijkstra's algorithm}
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\key{Dijkstra's algorithm} finds the shortest
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paths from the starting node to all other nodes,
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like the Bellman–Ford algorithm.
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The benefit in Dijsktra's algorithm is that
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it is more efficient and can be used for
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processing large graphs.
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However, the algorithm requires that there
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are no negative weight edges in the graph.
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Like the Bellman–Ford algorithm,
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Dijkstra's algorithm maintains distances
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for the nodes and reduces them during the algorithm.
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Dijkstra's algorithm is efficient, because
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it only processes
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each edge in the graph once, using the fact
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that there are no negative edges.
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\subsubsection{Example}
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Let us consider how Dijkstra's algorithm
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works in the following graph when the
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starting node is node 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) {3};
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\node[draw, circle] (2) at (4,3) {4};
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\node[draw, circle] (3) at (1,1) {2};
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\node[draw, circle] (4) at (4,1) {1};
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\node[draw, circle] (5) at (6,2) {5};
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\node[color=red] at (1,3+0.6) {$\infty$};
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\node[color=red] at (4,3+0.6) {$\infty$};
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\node[color=red] at (1,1-0.6) {$\infty$};
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\node[color=red] at (4,1-0.6) {$0$};
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\node[color=red] at (6,2-0.6) {$\infty$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
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\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
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\end{tikzpicture}
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\end{center}
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Like in the Bellman–Ford algorithm,
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intially the distance to the starting node is 0
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and the distance to all other nodes is infinite.
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At each step, Dijkstra's algorithm selects a node
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that has not been processed yet and whose distance
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is as small as possible.
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The first such node is node 1 with distance 0.
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When a node is selected, the algorithm
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goes through all edges that start at the node
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and reduces the distances using them:
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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) {3};
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\node[draw, circle] (2) at (4,3) {4};
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\node[draw, circle] (3) at (1,1) {2};
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\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
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\node[draw, circle] (5) at (6,2) {5};
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\node[color=red] at (1,3+0.6) {$\infty$};
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\node[color=red] at (4,3+0.6) {$9$};
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\node[color=red] at (1,1-0.6) {$5$};
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\node[color=red] at (4,1-0.6) {$0$};
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\node[color=red] at (6,2-0.6) {$1$};
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\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
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\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
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\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
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\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
|
||
\path[draw=red,thick,->,line width=2pt] (4) -- (2);
|
||
\path[draw=red,thick,->,line width=2pt] (4) -- (3);
|
||
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
The edges from node 1 reduced distances to
|
||
nodes 2, 4 and 5, whose distances are now 5, 9 and 1.
|
||
|
||
The next node to be processed is node 5 with distance 1:
|
||
\begin{center}
|
||
\begin{tikzpicture}
|
||
\node[draw, circle] (1) at (1,3) {3};
|
||
\node[draw, circle] (2) at (4,3) {4};
|
||
\node[draw, circle] (3) at (1,1) {2};
|
||
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
|
||
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};
|
||
|
||
\node[color=red] at (1,3+0.6) {$\infty$};
|
||
\node[color=red] at (4,3+0.6) {$3$};
|
||
\node[color=red] at (1,1-0.6) {$5$};
|
||
\node[color=red] at (4,1-0.6) {$0$};
|
||
\node[color=red] at (6,2-0.6) {$1$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
|
||
\path[draw=red,thick,->,line width=2pt] (5) -- (2);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
After this, the next node is node 4:
|
||
\begin{center}
|
||
\begin{tikzpicture}[scale=0.9]
|
||
\node[draw, circle] (1) at (1,3) {3};
|
||
\node[draw, circle, fill=lightgray] (2) at (4,3) {4};
|
||
\node[draw, circle] (3) at (1,1) {2};
|
||
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
|
||
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};
|
||
|
||
\node[color=red] at (1,3+0.6) {$9$};
|
||
\node[color=red] at (4,3+0.6) {$3$};
|
||
\node[color=red] at (1,1-0.6) {$5$};
|
||
\node[color=red] at (4,1-0.6) {$0$};
|
||
\node[color=red] at (6,2-0.6) {$1$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
|
||
\path[draw=red,thick,->,line width=2pt] (2) -- (1);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
|
||
A remarkable property in Dijkstra's algorithm is that
|
||
whenever a node is selected, its distance is final.
|
||
For example, at this point of the algorithm,
|
||
the distances 0, 1 and 3 are the final distances
|
||
to nodes 1, 5 and 4.
|
||
|
||
After this, the algorithm processes the two
|
||
remaining nodes, and the final distances are as follows:
|
||
|
||
\begin{center}
|
||
\begin{tikzpicture}[scale=0.9]
|
||
\node[draw, circle, fill=lightgray] (1) at (1,3) {3};
|
||
\node[draw, circle, fill=lightgray] (2) at (4,3) {4};
|
||
\node[draw, circle, fill=lightgray] (3) at (1,1) {2};
|
||
\node[draw, circle, fill=lightgray] (4) at (4,1) {1};
|
||
\node[draw, circle, fill=lightgray] (5) at (6,2) {5};
|
||
|
||
\node[color=red] at (1,3+0.6) {$7$};
|
||
\node[color=red] at (4,3+0.6) {$3$};
|
||
\node[color=red] at (1,1-0.6) {$5$};
|
||
\node[color=red] at (4,1-0.6) {$0$};
|
||
\node[color=red] at (6,2-0.6) {$1$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:6] {} (2);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
|
||
\subsubsection{Negative edges}
|
||
|
||
The efficiency of Dijkstra's algorithm is
|
||
based on the fact that the graph does not
|
||
contain negative edges.
|
||
If there is a negative edge,
|
||
the algorithm may give incorrect results.
|
||
As an example, consider the following graph:
|
||
|
||
\begin{center}
|
||
\begin{tikzpicture}[scale=0.9]
|
||
\node[draw, circle] (1) at (0,0) {$1$};
|
||
\node[draw, circle] (2) at (2,1) {$2$};
|
||
\node[draw, circle] (3) at (2,-1) {$3$};
|
||
\node[draw, circle] (4) at (4,0) {$4$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:2] {} (2);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:3] {} (4);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=below:6] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:$-5$] {} (4);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
\noindent
|
||
The shortest path from node 1 to node 4 is
|
||
$1 \rightarrow 3 \rightarrow 4$
|
||
and its length is 1.
|
||
However, Dijkstra's algorithm
|
||
finds the path $1 \rightarrow 2 \rightarrow 4$
|
||
by following the minimum weight edges.
|
||
The algorithm does not take into account that
|
||
on the lower path, the weight $-5$
|
||
compensates the previous large weight $6$.
|
||
|
||
\subsubsection{Implementation}
|
||
|
||
The following implementation of Dijkstra's algorithm
|
||
calculates the minimum distances from a node $x$
|
||
to all other nodes.
|
||
The graph is stored in an array \texttt{v}
|
||
as adjacency lists like in the Bellman–Ford algorithm.
|
||
|
||
An efficient implementation of Dijkstra's algorithm
|
||
requires that it is possible to efficiently find the
|
||
minimum distance node that has not been processed.
|
||
An appropriate data structure for this is a priority queue
|
||
that contains the nodes ordered by their distances.
|
||
Using a priority queue, the next node to be processed
|
||
can be retrieved in logarithmic time.
|
||
|
||
In the following implementation,
|
||
the priority queue contains pairs whose first
|
||
element is the current distance of the node and second
|
||
element is the identifier of the node.
|
||
\begin{lstlisting}
|
||
priority_queue<pair<int,int>> q;
|
||
\end{lstlisting}
|
||
A small difficulty is that in Dijkstra's algorithm,
|
||
we should find the node with the \emph{minimum} distance,
|
||
while the C++ priority queue finds the \emph{maximum}
|
||
element as default.
|
||
An easy solution is to use \emph{negative} distances,
|
||
which allows us to directly use the C++ priority queue.
|
||
|
||
The code keeps track of processed nodes
|
||
in the array \texttt{z},
|
||
and maintains estimated distances in array \texttt{e}.
|
||
Initially, the distance to the starting node is 0,
|
||
and the distance to all other nodes is $10^9$ (infinite).
|
||
|
||
\begin{lstlisting}
|
||
for (int i = 1; i <= n; i++) e[i] = 1e9;
|
||
e[x] = 0;
|
||
q.push({0,x});
|
||
while (!q.empty()) {
|
||
int a = q.top().second; q.pop();
|
||
if (z[a]) continue;
|
||
z[a] = 1;
|
||
for (auto b : v[a]) {
|
||
if (e[a]+b.second < e[b]) {
|
||
e[b] = e[a]+b.second;
|
||
q.push({-e[b],b});
|
||
}
|
||
}
|
||
}
|
||
\end{lstlisting}
|
||
|
||
The time complexity of the above implementation is
|
||
$O(n+m \log m)$ because the algorithm goes through
|
||
all nodes in the graph, and adds for each edge
|
||
at most one estimated distance to the priority queue.
|
||
|
||
\section{Floyd–Warshall algorithm}
|
||
|
||
\index{Floyd–Warshall algorithm}
|
||
|
||
The \key{Floyd–Warshall algorithm}
|
||
is an alternative way to approach the problem
|
||
of finding shortest paths.
|
||
Unlike the other algorihms in this chapter,
|
||
it finds all shortest paths between the nodes
|
||
in a single run.
|
||
|
||
The algorithm maintains a two-dimensional array
|
||
that contains distances between the nodes.
|
||
First, the distances are calculated only using
|
||
direct edges between the nodes.
|
||
After this the algorithm updates the distances
|
||
by using intermediate nodes in the paths.
|
||
|
||
\subsubsection{Example}
|
||
|
||
Let us consider how the Floyd–Warshall algorithm
|
||
works in the following graph:
|
||
|
||
\begin{center}
|
||
\begin{tikzpicture}[scale=0.9]
|
||
\node[draw, circle] (1) at (1,3) {$3$};
|
||
\node[draw, circle] (2) at (4,3) {$4$};
|
||
\node[draw, circle] (3) at (1,1) {$2$};
|
||
\node[draw, circle] (4) at (4,1) {$1$};
|
||
\node[draw, circle] (5) at (6,2) {$5$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
|
||
Initially, the distance from each node to itself is $0$,
|
||
and the distance between nodes $a$ and $b$ is $x$
|
||
if there is an edge between nodes $a$ and $b$ with weight $x$.
|
||
All other distances are infinite.
|
||
|
||
In this graph, the initial array is as follows:
|
||
\begin{center}
|
||
\begin{tabular}{r|rrrrr}
|
||
& 1 & 2 & 3 & 4 & 5 \\
|
||
\hline
|
||
1 & 0 & 5 & $\infty$ & 9 & 1 \\
|
||
2 & 5 & 0 & 2 & $\infty$ & $\infty$ \\
|
||
3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\
|
||
4 & 9 & $\infty$ & 7 & 0 & 2 \\
|
||
5 & 1 & $\infty$ & $\infty$ & 2 & 0 \\
|
||
\end{tabular}
|
||
\end{center}
|
||
\vspace{10pt}
|
||
The algorithm consists of consecutive rounds.
|
||
On each round, the algorithm selects a new node
|
||
that can act as an intermediate node in paths from now on,
|
||
and the algorithm reduces the distances in the array
|
||
using this node.
|
||
|
||
On the first round, node 1 is the intermediate node.
|
||
There is a new path between nodes 2 and 4
|
||
with length 14, because node 1 connects them.
|
||
There is also a new path
|
||
between nodes 2 and 5 with length 6.
|
||
|
||
\begin{center}
|
||
\begin{tabular}{r|rrrrr}
|
||
& 1 & 2 & 3 & 4 & 5 \\
|
||
\hline
|
||
1 & 0 & 5 & $\infty$ & 9 & 1 \\
|
||
2 & 5 & 0 & 2 & \textbf{14} & \textbf{6} \\
|
||
3 & $\infty$ & 2 & 0 & 7 & $\infty$ \\
|
||
4 & 9 & \textbf{14} & 7 & 0 & 2 \\
|
||
5 & 1 & \textbf{6} & $\infty$ & 2 & 0 \\
|
||
\end{tabular}
|
||
\end{center}
|
||
\vspace{10pt}
|
||
|
||
On the second round, node 2 is the intermediate node.
|
||
This creates new paths between nodes 1 and 3
|
||
and between nodes 3 and 5:
|
||
|
||
\begin{center}
|
||
\begin{tabular}{r|rrrrr}
|
||
& 1 & 2 & 3 & 4 & 5 \\
|
||
\hline
|
||
1 & 0 & 5 & \textbf{7} & 9 & 1 \\
|
||
2 & 5 & 0 & 2 & 14 & 6 \\
|
||
3 & \textbf{7} & 2 & 0 & 7 & \textbf{8} \\
|
||
4 & 9 & 14 & 7 & 0 & 2 \\
|
||
5 & 1 & 6 & \textbf{8} & 2 & 0 \\
|
||
\end{tabular}
|
||
\end{center}
|
||
\vspace{10pt}
|
||
|
||
On the third round, node 3 is the intermediate round.
|
||
There is a new path between nodes 2 and 4:
|
||
|
||
\begin{center}
|
||
\begin{tabular}{r|rrrrr}
|
||
& 1 & 2 & 3 & 4 & 5 \\
|
||
\hline
|
||
1 & 0 & 5 & 7 & 9 & 1 \\
|
||
2 & 5 & 0 & 2 & \textbf{9} & 6 \\
|
||
3 & 7 & 2 & 0 & 7 & 8 \\
|
||
4 & 9 & \textbf{9} & 7 & 0 & 2 \\
|
||
5 & 1 & 6 & 8 & 2 & 0 \\
|
||
\end{tabular}
|
||
\end{center}
|
||
\vspace{10pt}
|
||
|
||
The algorithm continues like this,
|
||
until all nodes have been intermediate nodes.
|
||
After the algorithm has finished, the array contains
|
||
the minimum distances between any two nodes:
|
||
|
||
\begin{center}
|
||
\begin{tabular}{r|rrrrr}
|
||
& 1 & 2 & 3 & 4 & 5 \\
|
||
\hline
|
||
1 & 0 & 5 & 7 & 3 & 1 \\
|
||
2 & 5 & 0 & 2 & 9 & 6 \\
|
||
3 & 7 & 2 & 0 & 7 & 8 \\
|
||
4 & 3 & 9 & 7 & 0 & 2 \\
|
||
5 & 1 & 6 & 8 & 2 & 0 \\
|
||
\end{tabular}
|
||
\end{center}
|
||
|
||
For example, the array indicates that the
|
||
shortest path between nodes 2 and 4 has length 8.
|
||
This corresponds to the following path:
|
||
|
||
\begin{center}
|
||
\begin{tikzpicture}[scale=0.9]
|
||
\node[draw, circle] (1) at (1,3) {$3$};
|
||
\node[draw, circle] (2) at (4,3) {$4$};
|
||
\node[draw, circle] (3) at (1,1) {$2$};
|
||
\node[draw, circle] (4) at (4,1) {$1$};
|
||
\node[draw, circle] (5) at (6,2) {$5$};
|
||
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=above:7] {} (2);
|
||
\path[draw,thick,-] (1) -- node[font=\small,label=left:2] {} (3);
|
||
\path[draw,thick,-] (3) -- node[font=\small,label=below:5] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=left:9] {} (4);
|
||
\path[draw,thick,-] (2) -- node[font=\small,label=above:2] {} (5);
|
||
\path[draw,thick,-] (4) -- node[font=\small,label=below:1] {} (5);
|
||
|
||
\path[draw=red,thick,->,line width=2pt] (3) -- (4);
|
||
\path[draw=red,thick,->,line width=2pt] (4) -- (5);
|
||
\path[draw=red,thick,->,line width=2pt] (5) -- (2);
|
||
\end{tikzpicture}
|
||
\end{center}
|
||
|
||
\subsubsection{Implementation}
|
||
|
||
The advantage of the
|
||
Floyd–Warshall algorithm that it is
|
||
easy to implement.
|
||
The following code constructs a
|
||
distance matrix \texttt{d} where $\texttt{d}[a][b]$
|
||
is the smallest distance between nodes $a$ and $b$.
|
||
First, the algorithm initializes \texttt{d}
|
||
using the adjacency matrix \texttt{v} of the graph
|
||
($10^9$ means infinity):
|
||
|
||
\begin{lstlisting}
|
||
for (int i = 1; i <= n; i++) {
|
||
for (int j = 1; j <= n; j++) {
|
||
if (i == j) d[i][j] = 0;
|
||
else if (v[i][j]) d[i][j] = v[i][j];
|
||
else d[i][j] = 1e9;
|
||
}
|
||
}
|
||
\end{lstlisting}
|
||
|
||
After this, the shortest paths can be found as follows:
|
||
|
||
\begin{lstlisting}
|
||
for (int k = 1; k <= n; k++) {
|
||
for (int i = 1; i <= n; i++) {
|
||
for (int j = 1; j <= n; j++) {
|
||
d[i][j] = min(d[i][j], d[i][k]+d[k][j]);
|
||
}
|
||
}
|
||
}
|
||
\end{lstlisting}
|
||
|
||
The time complexity of the algorithm is $O(n^3)$,
|
||
because it contains three nested loops
|
||
that go through the nodes in the graph.
|
||
|
||
Since the implementation of the Floyd–Warshall
|
||
algorithm is simple, the algorithm can be
|
||
a good choice even if it is only needed to find a
|
||
single shortest path in the graph.
|
||
However, the algorithm can only be used when the graph
|
||
is so small that a cubic time complexity is fast enough. |