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Maximum Spanning Tree using Prim’s Algorithm
  • Difficulty Level : Expert
  • Last Updated : 15 Feb, 2021

Given undirected weighted graph G, the task is to find the Maximum Spanning Tree of the Graph using Prim’s Algorithm

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.

Examples:

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.



Maximum Spanning Tree:

Given an undirected weighted graph, a maximum spanning tree is a spanning tree having maximum weight. It can be easily computed using Prim’s algorithm. The goal here is to find the spanning tree with the maximum weight out of all possible spanning trees.

Prim’s Algorithm:

Prim’s algorithm is a greedy algorithm, which works on the idea that a spanning tree must have all its vertices connected. The algorithm works by building the tree one vertex at a time, from an arbitrary starting vertex, and adding the most expensive possible connection from the tree to another vertex, which will give us the Maximum Spanning Tree (MST).

Follow the steps below to solve the problem:

  • Initialize a visited array of boolean datatype, to keep track of vertices visited so far. Initialize all the values with false.
  • Initialize an array weights[], representing the maximum weight to connect that vertex. Initialize all the values with some minimum value.
  • Initialize an array parent[], to keep track of the maximum spanning tree.
  • Assign some large value, as the weight of the first vertex and parent as -1, so that it is picked first and has no parent.
  • From all the unvisited vertices, pick a vertex v having a maximum weight and mark it as visited.
  • Update the weights of all the unvisited adjacent vertices of v. To update the weights, iterate through all the unvisited neighbors of v. For every adjacent vertex x, if the weight of the edge between v and x is greater than the previous value of v, update the value of v with that weight.

Below is the implementation of the above algorithm:

C++

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// C++ program for the above algorithm
 
#include <bits/stdc++.h>
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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output: 

Weight of the maximum Spanning-tree 30

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

 

Time Complexity: O(V2) where V is the number of nodes in the graph.
Auxiliary Space: O(V2)

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