Step by Step solution about How to resolve the algorithm Floyd-Warshall algorithm step by step in the PHP programming language

The provided PHP code demonstrates the implementation of the Floyd-Warshall algorithm to find the shortest paths between all pairs of vertices in a weighted graph.

It starts by creating a 10x10 adjacency matrix ($graph), where each element represents the weight of the edge between the corresponding vertices. Initially, all weights are set to a large value (9999999) to represent no direct connection.

The code then randomly generates edge weights between vertex 0 and all other vertices (line 11). These weights represent the initial distances between vertex 0 and the rest of the graph.

Next, the Floyd-Warshall algorithm is implemented in the nested loops starting at line 15. For each vertex k, it iterates over all pairs of vertices i and j. If the current path from i to j through k is shorter than the previously recorded path, it updates the weight in the adjacency matrix to reflect the new shorter path.

This process continues until the algorithm converges, ensuring that the final adjacency matrix contains the shortest paths between all pairs of vertices in the graph.

Finally, the code prints the updated adjacency matrix ($graph), which now contains the shortest paths between all vertices.

<?php
$graph = array();
for ($i = 0; $i < 10; ++$i) {
    $graph[] = array();
    for ($j = 0; $j < 10; ++$j)
        $graph[$i][] = $i == $j ? 0 : 9999999;
}

for ($i = 1; $i < 10; ++$i) {
    $graph[0][$i] = $graph[$i][0] = rand(1, 9);
}

for ($k = 0; $k < 10; ++$k) {
    for ($i = 0; $i < 10; ++$i) {
        for ($j = 0; $j < 10; ++$j) {
            if ($graph[$i][$j] > $graph[$i][$k] + $graph[$k][$j])
                $graph[$i][$j] = $graph[$i][$k] + $graph[$k][$j];
        }
    }
}

print_r($graph);
?>