使用 Prim 算法的最大生成树

给定无向加权图G ，任务是使用 Prim 算法找到图的最大生成树

Prims algorithm is a Greedy algorithm which can be used to find the Minimum Spanning Tree (MST) as well as the Maximum Spanning Tree of a Graph.

例子：

Input: graph[V][V] = {{0, 2, 0, 6, 0},  {2, 0, 3, 8, 5}, {0, 3, 0, 0, 7}, {6, 8, 0, 0, 9}, {0, 5, 7, 9, 0}}
Output:
The total weight of the Maximum Spanning tree is 30.
Edges   Weight
3 – 1        8
4 – 2        7
0 – 3        6
3 – 4        9
Explanation:
Choosing other edges won’t result in maximum spanning tree.

最大生成树：

给定一个无向加权图，最大生成树是具有最大权重的生成树。它可以使用 Prim 算法轻松计算。这里的目标是在所有可能的生成树中找到权重最大的生成树。

Prim 算法：

Prim 的算法是一种贪心算法，其工作原理是生成树必须连接所有顶点。该算法的工作原理是从任意起始顶点一次一个顶点构建树，并添加从树到另一个顶点的最昂贵的可能连接，这将为我们提供最大生成树 (MST) 。

请按照以下步骤解决问题：

• 初始化一个访问过的布尔数据类型数组，以跟踪到目前为止访问过的顶点。用false初始化所有值。
• 初始化一个数组weights[] ，表示连接该顶点的最大权重。用某个最小值初始化所有值。
• 初始化数组parent[] ，以跟踪最大生成树。
• 分配一些大值，作为第一个顶点和父节点的权重为-1 ，以便首先选择它并且没有父节点。
• 从所有未访问的顶点中，选择具有最大权重的顶点v并将其标记为已访问。
• 更新v的所有未访问的相邻顶点的权重。要更新权重，请遍历v 的所有未访问邻居。对于每一个相邻的顶点的x，如果V之间的边缘和重量x大于V的前面的值时，更新的v的与权重的值。

下面是上述算法的实现：

C++

// C++ program for the above algorithm
 
#include 
using namespace std;
#define V 5
 
// Function to find index of max-weight
// vertex from set of unvisited vertices
int findMaxVertex(bool visited[], int weights[])
{
 
    // Stores the index of max-weight vertex
    // from set of unvisited vertices
    int index = -1;
 
    // Stores the maximum weight from
    // the set of unvisited vertices
    int maxW = INT_MIN;
 
    // Iterate over all possible
    // nodes of a graph
    for (int i = 0; i < V; i++) {
 
        // If the current node is unvisited
        // and weight of current vertex is
        // greater than maxW
        if (visited[i] == false
            && weights[i] > maxW) {
 
            // Update maxW
            maxW = weights[i];
 
            // Update index
            index = i;
        }
    }
    return index;
}
 
// Utility function to find the maximum
// spanning tree of graph
void printMaximumSpanningTree(int graph[V][V],
                              int parent[])
{
 
    // Stores total weight of
    // maximum spanning tree
    // of a graph
    int MST = 0;
 
    // Iterate over all possible nodes
    // of a graph
    for (int i = 1; i < V; i++) {
 
        // Update MST
        MST += graph[i][parent[i]];
    }
 
    cout << "Weight of the maximum Spanning-tree "
         << MST << '\n'
         << '\n';
 
    cout << "Edges \tWeight\n";
 
    // Print the Edges and weight of
    // maximum spanning tree of a graph
    for (int i = 1; i < V; i++) {
        cout << parent[i] << " - " << i << " \t"
             << graph[i][parent[i]] << " \n";
    }
}
 
// Function to find the maximum spanning tree
void maximumSpanningTree(int graph[V][V])
{
 
    // visited[i]:Check if vertex i
    // is visited or not
    bool visited[V];
 
    // weights[i]: Stores maximum weight of
    // graph to connect an edge with i
    int weights[V];
 
    // parent[i]: Stores the parent node
    // of vertex i
    int parent[V];
 
    // Initialize weights as -INFINITE,
    // and visited of a node as false
    for (int i = 0; i < V; i++) {
        visited[i] = false;
        weights[i] = INT_MIN;
    }
 
    // Include 1st vertex in
    // maximum spanning tree
    weights[0] = INT_MAX;
    parent[0] = -1;
 
    // Search for other (V-1) vertices
    // and build a tree
    for (int i = 0; i < V - 1; i++) {
 
        // Stores index of max-weight vertex
        // from a set of unvisited vertex
        int maxVertexIndex
            = findMaxVertex(visited, weights);
 
        // Mark that vertex as visited
        visited[maxVertexIndex] = true;
 
        // Update adjacent vertices of
        // the current visited vertex
        for (int j = 0; j < V; j++) {
 
            // If there is an edge between j
            // and current visited vertex and
            // also j is unvisited vertex
            if (graph[j][maxVertexIndex] != 0
                && visited[j] == false) {
 
                // If graph[v][x] is
                // greater than weight[v]
                if (graph[j][maxVertexIndex] > weights[j]) {
 
                    // Update weights[j]
                    weights[j] = graph[j][maxVertexIndex];
 
                    // Update parent[j]
                    parent[j] = maxVertexIndex;
                }
            }
        }
    }
 
    // Print maximum spanning tree
    printMaximumSpanningTree(graph, parent);
}
 
// Driver Code
int main()
{
 
    // Given graph
    int graph[V][V] = { { 0, 2, 0, 6, 0 },
                        { 2, 0, 3, 8, 5 },
                        { 0, 3, 0, 0, 7 },
                        { 6, 8, 0, 0, 9 },
                        { 0, 5, 7, 9, 0 } };
 
    // Function call
    maximumSpanningTree(graph);
 
    return 0;
}

Java

// Java program for the above algorithm
import java.io.*;
class GFG
{
  public static int V = 5;
 
  // Function to find index of max-weight
  // vertex from set of unvisited vertices
  static int findMaxVertex(boolean visited[],
                           int weights[])
  {
 
    // Stores the index of max-weight vertex
    // from set of unvisited vertices
    int index = -1;
 
    // Stores the maximum weight from
    // the set of unvisited vertices
    int maxW = Integer.MIN_VALUE;
 
    // Iterate over all possible
    // nodes of a graph
    for (int i = 0; i < V; i++)
    {
 
      // If the current node is unvisited
      // and weight of current vertex is
      // greater than maxW
      if (visited[i] == false && weights[i] > maxW)
      {
 
        // Update maxW
        maxW = weights[i];
 
        // Update index
        index = i;
      }
    }
    return index;
  }
 
  // Utility function to find the maximum
  // spanning tree of graph
  static void printMaximumSpanningTree(int graph[][],
                                       int parent[])
  {
 
    // Stores total weight of
    // maximum spanning tree
    // of a graph
    int MST = 0;
 
    // Iterate over all possible nodes
    // of a graph
    for (int i = 1; i < V; i++)
    {
 
      // Update MST
      MST += graph[i][parent[i]];
    }
 
    System.out.println("Weight of the maximum Spanning-tree "
                       + MST);
    System.out.println();
    System.out.println("Edges \tWeight");
 
    // Print the Edges and weight of
    // maximum spanning tree of a graph
    for (int i = 1; i < V; i++)
    {
      System.out.println(parent[i] + " - " + i + " \t"
                         + graph[i][parent[i]]);
    }
  }
 
  // Function to find the maximum spanning tree
  static void maximumSpanningTree(int[][] graph)
  {
 
    // visited[i]:Check if vertex i
    // is visited or not
    boolean[] visited = new boolean[V];
 
    // weights[i]: Stores maximum weight of
    // graph to connect an edge with i
    int[] weights = new int[V];
 
    // parent[i]: Stores the parent node
    // of vertex i
    int[] parent = new int[V];
 
    // Initialize weights as -INFINITE,
    // and visited of a node as false
    for (int i = 0; i < V; i++) {
      visited[i] = false;
      weights[i] = Integer.MIN_VALUE;
    }
 
    // Include 1st vertex in
    // maximum spanning tree
    weights[0] = Integer.MAX_VALUE;
    parent[0] = -1;
 
    // Search for other (V-1) vertices
    // and build a tree
    for (int i = 0; i < V - 1; i++) {
 
      // Stores index of max-weight vertex
      // from a set of unvisited vertex
      int maxVertexIndex
        = findMaxVertex(visited, weights);
 
      // Mark that vertex as visited
      visited[maxVertexIndex] = true;
 
      // Update adjacent vertices of
      // the current visited vertex
      for (int j = 0; j < V; j++) {
 
        // If there is an edge between j
        // and current visited vertex and
        // also j is unvisited vertex
        if (graph[j][maxVertexIndex] != 0
            && visited[j] == false) {
 
          // If graph[v][x] is
          // greater than weight[v]
          if (graph[j][maxVertexIndex]
              > weights[j]) {
 
            // Update weights[j]
            weights[j]
              = graph[j][maxVertexIndex];
 
            // Update parent[j]
            parent[j] = maxVertexIndex;
          }
        }
      }
    }
 
    // Print maximum spanning tree
    printMaximumSpanningTree(graph, parent);
  }
 
  // Driver Code
  public static void main(String[] args)
  {
 
    // Given graph
    int[][] graph = { { 0, 2, 0, 6, 0 },
                     { 2, 0, 3, 8, 5 },
                     { 0, 3, 0, 0, 7 },
                     { 6, 8, 0, 0, 9 },
                     { 0, 5, 7, 9, 0 } };
 
    // Function call
    maximumSpanningTree(graph);
  }
}
 
// This code is contributed by Dharanendra L V

Python3

# Python program for the above algorithm
import sys
V = 5;
 
# Function to find index of max-weight
# vertex from set of unvisited vertices
def findMaxVertex(visited, weights):
   
    # Stores the index of max-weight vertex
    # from set of unvisited vertices
    index = -1;
 
    # Stores the maximum weight from
    # the set of unvisited vertices
    maxW = -sys.maxsize;
 
    # Iterate over all possible
    # Nodes of a graph
    for i in range(V):
 
        # If the current Node is unvisited
        # and weight of current vertex is
        # greater than maxW
        if (visited[i] == False and weights[i] > maxW):
           
            # Update maxW
            maxW = weights[i];
 
            # Update index
            index = i;
    return index;
 
# Utility function to find the maximum
# spanning tree of graph
def printMaximumSpanningTree(graph, parent):
   
    # Stores total weight of
    # maximum spanning tree
    # of a graph
    MST = 0;
 
    # Iterate over all possible Nodes
    # of a graph
    for i in range(1, V):
       
        # Update MST
        MST += graph[i][parent[i]];
 
    print("Weight of the maximum Spanning-tree ", MST);
    print();
    print("Edges \tWeight");
 
    # Prthe Edges and weight of
    # maximum spanning tree of a graph
    for i in range(1, V):
        print(parent[i] , " - " , i , " \t" , graph[i][parent[i]]);
 
# Function to find the maximum spanning tree
def maximumSpanningTree(graph):
   
    # visited[i]:Check if vertex i
    # is visited or not
    visited = [True]*V;
 
    # weights[i]: Stores maximum weight of
    # graph to connect an edge with i
    weights = [0]*V;
 
    # parent[i]: Stores the parent Node
    # of vertex i
    parent = [0]*V;
 
    # Initialize weights as -INFINITE,
    # and visited of a Node as False
    for i in range(V):
        visited[i] = False;
        weights[i] = -sys.maxsize;
 
    # Include 1st vertex in
    # maximum spanning tree
    weights[0] = sys.maxsize;
    parent[0] = -1;
 
    # Search for other (V-1) vertices
    # and build a tree
    for i in range(V - 1):
 
        # Stores index of max-weight vertex
        # from a set of unvisited vertex
        maxVertexIndex = findMaxVertex(visited, weights);
 
        # Mark that vertex as visited
        visited[maxVertexIndex] = True;
 
        # Update adjacent vertices of
        # the current visited vertex
        for j in range(V):
 
            # If there is an edge between j
            # and current visited vertex and
            # also j is unvisited vertex
            if (graph[j][maxVertexIndex] != 0 and visited[j] == False):
 
                # If graph[v][x] is
                # greater than weight[v]
                if (graph[j][maxVertexIndex] > weights[j]):
                   
                    # Update weights[j]
                    weights[j] = graph[j][maxVertexIndex];
 
                    # Update parent[j]
                    parent[j] = maxVertexIndex;
 
    # Prmaximum spanning tree
    printMaximumSpanningTree(graph, parent);
 
# Driver Code
if __name__ == '__main__':
    # Given graph
    graph = [[0, 2, 0, 6, 0], [2, 0, 3, 8, 5], [0, 3, 0, 0, 7], [6, 8, 0, 0, 9],
                                                                 [0, 5, 7, 9, 0]];
 
    # Function call
    maximumSpanningTree(graph);
 
    # This code is contributed by 29AjayKumar

C#

// C# program for the above algorithm
using System;
class GFG
{
  public static int V = 5;
 
  // Function to find index of max-weight
  // vertex from set of unvisited vertices
  static int findMaxVertex(bool[] visited,
                           int[] weights)
  {
 
    // Stores the index of max-weight vertex
    // from set of unvisited vertices
    int index = -1;
 
    // Stores the maximum weight from
    // the set of unvisited vertices
    int maxW = int.MinValue;
 
    // Iterate over all possible
    // nodes of a graph
    for (int i = 0; i < V; i++)
    {
 
      // If the current node is unvisited
      // and weight of current vertex is
      // greater than maxW
      if (visited[i] == false && weights[i] > maxW)
      {
 
        // Update maxW
        maxW = weights[i];
 
        // Update index
        index = i;
      }
    }
    return index;
  }
 
  // Utility function to find the maximum
  // spanning tree of graph
  static void printMaximumSpanningTree(int[, ] graph,
                                       int[] parent)
  {
 
    // Stores total weight of
    // maximum spanning tree
    // of a graph
    int MST = 0;
 
    // Iterate over all possible nodes
    // of a graph
    for (int i = 1; i < V; i++)
    {
 
      // Update MST
      MST += graph[i, parent[i]];
    }
 
    Console.WriteLine(
      "Weight of the maximum Spanning-tree " + MST);
 
    Console.WriteLine();
    Console.WriteLine("Edges \tWeight");
 
    // Print the Edges and weight of
    // maximum spanning tree of a graph
    for (int i = 1; i < V; i++) {
      Console.WriteLine(parent[i] + " - " + i + " \t"
                        + graph[i, parent[i]]);
    }
  }
 
  // Function to find the maximum spanning tree
  static void maximumSpanningTree(int[, ] graph)
  {
 
    // visited[i]:Check if vertex i
    // is visited or not
    bool[] visited = new bool[V];
 
    // weights[i]: Stores maximum weight of
    // graph to connect an edge with i
    int[] weights = new int[V];
 
    // parent[i]: Stores the parent node
    // of vertex i
    int[] parent = new int[V];
 
    // Initialize weights as -INFINITE,
    // and visited of a node as false
    for (int i = 0; i < V; i++) {
      visited[i] = false;
      weights[i] = int.MinValue;
    }
 
    // Include 1st vertex in
    // maximum spanning tree
    weights[0] = int.MaxValue;
    parent[0] = -1;
 
    // Search for other (V-1) vertices
    // and build a tree
    for (int i = 0; i < V - 1; i++) {
 
      // Stores index of max-weight vertex
      // from a set of unvisited vertex
      int maxVertexIndex
        = findMaxVertex(visited, weights);
 
      // Mark that vertex as visited
      visited[maxVertexIndex] = true;
 
      // Update adjacent vertices of
      // the current visited vertex
      for (int j = 0; j < V; j++) {
 
        // If there is an edge between j
        // and current visited vertex and
        // also j is unvisited vertex
        if (graph[j, maxVertexIndex] != 0
            && visited[j] == false) {
 
          // If graph[v][x] is
          // greater than weight[v]
          if (graph[j, maxVertexIndex]
              > weights[j]) {
 
            // Update weights[j]
            weights[j]
              = graph[j, maxVertexIndex];
 
            // Update parent[j]
            parent[j] = maxVertexIndex;
          }
        }
      }
    }
 
    // Print maximum spanning tree
    printMaximumSpanningTree(graph, parent);
  }
 
  // Driver Code
  static public void Main()
  {
 
    // Given graph
    int[, ] graph = { { 0, 2, 0, 6, 0 },
                     { 2, 0, 3, 8, 5 },
                     { 0, 3, 0, 0, 7 },
                     { 6, 8, 0, 0, 9 },
                     { 0, 5, 7, 9, 0 } };
 
    // Function call
    maximumSpanningTree(graph);
  }
}
 
// This code is contributed by Dharanendra L V

Javascript

 
// Javascript program for the above algorithm
 
var V = 5;
 
// Function to find index of max-weight
// vertex from set of unvisited vertices
function findMaxVertex(visited, weights)
{
 
    // Stores the index of max-weight vertex
    // from set of unvisited vertices
    var index = -1;
 
    // Stores the maximum weight from
    // the set of unvisited vertices
    var maxW = -1000000000;
 
    // Iterate over all possible
    // nodes of a graph
    for (var i = 0; i < V; i++) {
 
        // If the current node is unvisited
        // and weight of current vertex is
        // greater than maxW
        if (visited[i] == false
            && weights[i] > maxW) {
 
            // Update maxW
            maxW = weights[i];
 
            // Update index
            index = i;
        }
    }
    return index;
}
 
// Utility function to find the maximum
// spanning tree of graph
function printMaximumSpanningTree(graph, parent)
{
 
    // Stores total weight of
    // maximum spanning tree
    // of a graph
    var MST = 0;
 
    // Iterate over all possible nodes
    // of a graph
    for (var i = 1; i < V; i++) {
 
        // Update MST
        MST += graph[i][parent[i]];
    }
 
    document.write( "Weight of the maximum Spanning-tree "
         + MST + '<br>'
         + '<br>');
 
    document.write( "Edges \tWeight<br>");
 
    // Print the Edges and weight of
    // maximum spanning tree of a graph
    for (var i = 1; i < V; i++) {
        document.write( parent[i] + " - " + i + "      "
             + graph[i][parent[i]] + " <br>");
    }
}
 
// Function to find the maximum spanning tree
function maximumSpanningTree(graph)
{
 
    // visited[i]:Check if vertex i
    // is visited or not
    var visited = Array(V).fill(false);
 
    // weights[i]: Stores maximum weight of
    // graph to connect an edge with i
    var weights = Array(V).fill(-1000000000);
 
    // parent[i]: Stores the parent node
    // of vertex i
    var parent = Array(V).fill(0);
 
 
    // Include 1st vertex in
    // maximum spanning tree
    weights[0] = 1000000000;
    parent[0] = -1;
 
    // Search for other (V-1) vertices
    // and build a tree
    for (var i = 0; i < V - 1; i++) {
 
        // Stores index of max-weight vertex
        // from a set of unvisited vertex
        var maxVertexIndex
            = findMaxVertex(visited, weights);
 
        // Mark that vertex as visited
        visited[maxVertexIndex] = true;
 
        // Update adjacent vertices of
        // the current visited vertex
        for (var j = 0; j < V; j++) {
 
            // If there is an edge between j
            // and current visited vertex and
            // also j is unvisited vertex
            if (graph[j][maxVertexIndex] != 0
                && visited[j] == false) {
 
                // If graph[v][x] is
                // greater than weight[v]
                if (graph[j][maxVertexIndex] > weights[j]) {
 
                    // Update weights[j]
                    weights[j] = graph[j][maxVertexIndex];
 
                    // Update parent[j]
                    parent[j] = maxVertexIndex;
                }
            }
        }
    }
 
    // Print maximum spanning tree
    printMaximumSpanningTree(graph, parent);
}
 
// Driver Code
// Given graph
var graph = [ [ 0, 2, 0, 6, 0 ],
                    [ 2, 0, 3, 8, 5 ],
                    [ 0, 3, 0, 0, 7 ],
                    [ 6, 8, 0, 0, 9 ],
                    [ 0, 5, 7, 9, 0 ] ];
// Function call
maximumSpanningTree(graph);
 
// This code is contributed by rutvik_56.

输出：

Weight of the maximum Spanning-tree 30

Edges     Weight
3 - 1     8 
4 - 2     7 
0 - 3     6 
3 - 4     9

时间复杂度： O(V ² )，其中 V 是图中的节点数。
辅助空间： O(V ² )

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