Implementation method of maximum flow algorithm in PHP

Maximum flow algorithm implementation method in PHP

The maximum flow algorithm is one of the classic problems in graph theory. It is used to solve the problem of maximum flow in the network. In a network flow, each edge has a capacity limit, and each node has an incoming and outgoing flow. The goal of the maximum flow algorithm is to find the maximum value of the flow in the network, that is, the maximum value of the total flow of the edges passing through the network.

In PHP, we can use a method called Ford-Fulkerson algorithm to solve the maximum flow problem. The following will introduce how to implement the Ford-Fulkerson algorithm using PHP.

First, we need to define a graph class to represent the graph in the network flow problem. Each node in the graph has a unique identifier and an adjacency list. In PHP, we can use arrays to represent adjacency lists. The following is a simple graph class definition:

class Graph {
  private $graph;

  public function __construct() {
    $this->graph = array();
  }

  public function addEdge($u, $v, $w) {
    if (!isset($this->graph[$u])) {
      $this->graph[$u] = array();
    }
    $this->graph[$u][$v] = $w;
  }

  public function getAdjacencyList($u) {
    return isset($this->graph[$u]) ? $this->graph[$u] : array();
  }
}

Next, we can define a maximum flow algorithm class to implement the Ford-Fulkerson algorithm. This class stores graph information and contains a method for calculating the maximum flow. The following is a simple maximum flow algorithm class definition:

class FordFulkerson {
  private $graph;
  private $visited;
  private $source;
  private $sink;

  public function __construct(Graph $graph, $source, $sink) {
    $this->graph = $graph;
    $this->visited = array();
    $this->source = $source;
    $this->sink = $sink;
  }

  public function getMaxFlow() {
    $maxFlow = 0;

    while ($path = $this->findPath()) {
      $minCapacity = PHP_INT_MAX;

      for ($i = 1; $i < count($path); $i++) {
        $u = $path[$i - 1];
        $v = $path[$i];

        $capacity = $this->graph->getAdjacencyList($u)[$v];
        $minCapacity = min($minCapacity, $capacity);
      }

      for ($i = 1; $i < count($path); $i++) {
        $u = $path[$i - 1];
        $v = $path[$i];

        $this->graph->getAdjacencyList($u)[$v] -= $minCapacity;
        $this->graph->getAdjacencyList($v)[$u] += $minCapacity;
      }

      $maxFlow += $minCapacity;
    }

    return $maxFlow;
  }

  private function findPath($u = null) {
    if ($u === null) {
      $u = $this->source;
    }

    if ($u == $this->sink) {
      return [$this->sink];
    }

    $this->visited[$u] = true;

    foreach ($this->graph->getAdjacencyList($u) as $v => $capacity) {
      if (!$this->visited[$v] && $capacity > 0) {
        $path = $this->findPath($v);

        if ($path) {
          array_unshift($path, $u);
          return $path;
        }
      }
    }

    return null;
  }
}

Finally, we can use the graph class and maximum flow algorithm class defined above to solve network flow problems. The following is a usage example:

$graph = new Graph();

$graph->addEdge("s", "A", 10);
$graph->addEdge("s", "B", 5);
$graph->addEdge("A", "C", 15);
$graph->addEdge("B", "C", 10);
$graph->addEdge("B", "D", 20);
$graph->addEdge("C", "D", 5);
$graph->addEdge("C", "t", 20);
$graph->addEdge("D", "t", 10);

$fordFulkerson = new FordFulkerson($graph, "s", "t");
$maxFlow = $fordFulkerson->getMaxFlow();

echo "The maximum flow in the network is: " . $maxFlow;

In the above example, we define a directed graph, where "s" represents the source node, "t" represents the sink node, and other nodes are represented by letters and on the edges Respective capacity values ​​are marked. The maximum flow rate calculated using the Ford-Fulkerson algorithm will be printed.

Through the above examples and codes, we can implement the maximum flow algorithm in PHP and solve network flow problems. This algorithm has wide applications in scenarios such as network optimization and resource allocation, and is very helpful in understanding and solving related problems.

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