PHP algorithm design tips: How to use Floyd-Warshall algorithm to solve the shortest path problem of graphs?

PHP algorithm design tips: How to use Floyd-Warshall algorithm to solve the shortest path problem of graphs?

Overview:
In graph theory, the shortest path problem is a classic algorithmic problem that involves finding the shortest path between two vertices in a directed or undirected graph. The Floyd-Warshall algorithm is a classic dynamic programming algorithm used to solve this problem. This article will introduce in detail how to implement the Floyd-Warshall algorithm using PHP.

Floyd-Warshall algorithm introduction:
Floyd-Warshall algorithm is an algorithm that solves the shortest path problem by iteratively comparing the shortest path lengths between all vertices in the graph. It uses a two-dimensional array to store the shortest path length between vertices, and updates this array in each iteration. Finally, we can get the shortest path between all vertices.

Code implementation:
First, we need to create a two-dimensional array of N x N, where N represents the number of vertices in the graph. Each element in the array represents the distance between two vertices, or if there is no edge between two vertices, their distance is set to infinity. The code looks like this:

function floydWarshall($graph) {
    $n = count($graph);
    $dist = $graph;

    for ($k = 0; $k < $n; $k++) {
        for ($i = 0; $i < $n; $i++) {
            for ($j = 0; $j < $n; $j++) {
                if ($dist[$i][$k] + $dist[$k][$j] < $dist[$i][$j]) {
                    $dist[$i][$j] = $dist[$i][$k] + $dist[$k][$j];
                }
            }
        }
    }

    return $dist;
}

Next, we need to define a sample graph to test our algorithm. We use an adjacency matrix to represent the structure of the graph, storing the distances between vertices in a two-dimensional array. The sample code is as follows:

$graph = [
    [0, 5, INF, 10],
    [INF, 0, 3, INF],
    [INF, INF, 0, 1],
    [INF, INF, INF, 0]
];

In the above example graph, INF means that there is no edge between two vertices, and we can set their distance to a very large value. Now, we can call the floydWarshall function to calculate the shortest path array. The code is as follows:

$result = floydWarshall($graph);

for ($i = 0; $i < count($result); $i++) {
    for ($j = 0; $j < count($result[$i]); $j++) {
        if ($result[$i][$j] == INF) {
            echo "INF ";
        } else {
            echo $result[$i][$j] . " ";
        }
    }
    echo "
";
}

Running the above code, we will get the following results:

0 5 8 9 
INF 0 3 4 
INF INF 0 1 
INF INF INF 0

The above results show the shortest path length between all vertices in the graph. Among them, INF means that there is no path connection between two vertices.

Summary:
This article introduces how to use PHP to implement the Floyd-Warshall algorithm to solve the shortest path problem of the graph. By using the idea of ​​dynamic programming, we can find the shortest path length between all vertices in the graph with a time complexity of O(N^3). By rationally using algorithm design techniques, we can quickly and efficiently apply this algorithm in solving practical problems.

The above is an introduction to PHP algorithm design skills: how to use the Floyd-Warshall algorithm to solve the shortest path problem of graphs. I hope it will be helpful to you.

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