Prim’s Algorithm (Simple Implementation for Adjacency Matrix Representation)

Last Updated : 31 Aug, 2022

We have discussed Prim’s algorithm and its implementation for adjacency matrix representation of graphs.
As discussed in the previous post, in Prim’s algorithm, two sets are maintained, one set contains list of vertices already included in MST, other set contains vertices not yet included. In every iteration, we consider the minimum weight edge among the edges that connect the two sets.

The implementation discussed in previous post uses two arrays to find minimum weight edge that connects the two sets. Here we use one inMST[V]. The value of MST[i] is going to be true if vertex i is included in the MST. In every pass, we consider only those edges such that one vertex of the edge is included in MST and other is not. After we pick an edge, we mark both vertices as included in MST.

Implementation:

C++

// A simple C++ implementation to find minimum 
// spanning tree for adjacency representation.
#include <bits/stdc++.h>
using namespace std;
#define V 5
 
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
bool isValidEdge(int u, int v, vector<bool> inMST)
{
   if (u == v)
       return false;
   if (inMST[u] == false && inMST[v] == false)
        return false;
   else if (inMST[u] == true && inMST[v] == true)
        return false;
   return true;
}
 
void primMST(int cost[][V])
{  
    vector<bool> inMST(V, false);
 
    // Include first vertex in MST
    inMST[0] = true;
 
    // Keep adding edges while number of included
    // edges does not become V-1.
    int edge_count = 0, mincost = 0;
    while (edge_count < V - 1) {
 
        // Find minimum weight valid edge.  
        int min = INT_MAX, a = -1, b = -1;
        for (int i = 0; i < V; i++) {
            for (int j = 0; j < V; j++) {               
                if (cost[i][j] < min) {
                    if (isValidEdge(i, j, inMST)) {
                        min = cost[i][j];
                        a = i;
                        b = j;
                    }
                }
            }
        }
        if (a != -1 && b != -1) {
            printf("Edge %d:(%d, %d) cost: %d \n", 
                         edge_count++, a, b, min);
            mincost = mincost + min;
            inMST[b] = inMST[a] = true;
        }
    }
    printf("\n Minimum cost= %d \n", mincost);
}
 
// driver program to test above function
int main()
{
    /* Let us create the following graph
          2    3
      (0)--(1)--(2)
       |   / \   |
      6| 8/   \5 |7
       | /     \ |
      (3)-------(4)
            9          */
    int cost[][V] = {
        { INT_MAX, 2, INT_MAX, 6, INT_MAX },
        { 2, INT_MAX, 3, 8, 5 },
        { INT_MAX, 3, INT_MAX, INT_MAX, 7 },
        { 6, 8, INT_MAX, INT_MAX, 9 },
        { INT_MAX, 5, 7, 9, INT_MAX },
    };
 
    // Print the solution
    primMST(cost);
 
    return 0;
}

Java

// A simple Java implementation to find minimum 
// spanning tree for adjacency representation.
import java.util.*;
 
class GFG 
{
 
static int V = 5;
static int INT_MAX = Integer.MAX_VALUE;
 
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
static boolean isValidEdge(int u, int v,
                           boolean[] inMST)
{
    if (u == v)
        return false;
    if (inMST[u] == false && inMST[v] == false)
        return false;
    else if (inMST[u] == true && inMST[v] == true)
        return false;
    return true;
}
 
static void primMST(int cost[][])
{ 
    boolean []inMST = new boolean[V];
 
    // Include first vertex in MST
    inMST[0] = true;
 
    // Keep adding edges while number of included
    // edges does not become V-1.
    int edge_count = 0, mincost = 0;
    while (edge_count < V - 1)
    {
 
        // Find minimum weight valid edge. 
        int min = INT_MAX, a = -1, b = -1;
        for (int i = 0; i < V; i++) 
        {
            for (int j = 0; j < V; j++) 
            {             
                if (cost[i][j] < min) 
                {
                    if (isValidEdge(i, j, inMST)) 
                    {
                        min = cost[i][j];
                        a = i;
                        b = j;
                    }
                }
            }
        }
         
        if (a != -1 && b != -1) 
        {
            System.out.printf("Edge %d:(%d, %d) cost: %d \n", 
                                    edge_count++, a, b, min);
            mincost = mincost + min;
            inMST[b] = inMST[a] = true;
        }
    }
    System.out.printf("\n Minimum cost = %d \n", mincost);
}
 
// Driver Code
public static void main(String[] args)
{
    /* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | /     \ |
    (3)-------(4)
            9         */
    int cost[][] = {{ INT_MAX, 2, INT_MAX, 6, INT_MAX },
                    { 2, INT_MAX, 3, 8, 5 },
                    { INT_MAX, 3, INT_MAX, INT_MAX, 7 },
                    { 6, 8, INT_MAX, INT_MAX, 9 },
                    { INT_MAX, 5, 7, 9, INT_MAX }};
 
    // Print the solution
    primMST(cost);
}
} 
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation to find minimum 
# spanning tree for adjacency representation.
from sys import maxsize
INT_MAX = maxsize
V = 5
 
# Returns true if edge u-v is a valid edge to be 
# include in MST. An edge is valid if one end is 
# already included in MST and other is not in MST.
def isValidEdge(u, v, inMST):
    if u == v:
        return False
    if inMST[u] == False and inMST[v] == False:
        return False
    elif inMST[u] == True and inMST[v] == True:
        return False
    return True
 
def primMST(cost):
    inMST = [False] * V
 
    # Include first vertex in MST
    inMST[0] = True
 
    # Keep adding edges while number of included 
    # edges does not become V-1.
    edge_count = 0
    mincost = 0
    while edge_count < V - 1:
 
        # Find minimum weight valid edge.
        minn = INT_MAX
        a = -1
        b = -1
        for i in range(V):
            for j in range(V):
                if cost[i][j] < minn:
                    if isValidEdge(i, j, inMST):
                        minn = cost[i][j]
                        a = i
                        b = j
 
        if a != -1 and b != -1:
            print("Edge %d: (%d, %d) cost: %d" %
                 (edge_count, a, b, minn))
            edge_count += 1
            mincost += minn
            inMST[b] = inMST[a] = True
 
    print("Minimum cost = %d" % mincost)
 
# Driver Code
if __name__ == "__main__":
    ''' Let us create the following graph 
        2 3 
    (0)--(1)--(2) 
    | / \ | 
    6| 8/ \5 |7 
    | /     \ | 
    (3)-------(4) 
            9         '''
 
    cost = [[INT_MAX, 2, INT_MAX, 6, INT_MAX], 
            [2, INT_MAX, 3, 8, 5],
            [INT_MAX, 3, INT_MAX, INT_MAX, 7], 
            [6, 8, INT_MAX, INT_MAX, 9],
            [INT_MAX, 5, 7, 9, INT_MAX]]
 
    # Print the solution
    primMST(cost)
 
# This code is contributed by
# sanjeev2552

C#

// A simple C# implementation to find minimum 
// spanning tree for adjacency representation.
using System;
 
class GFG 
{
 
static int V = 5;
static int INT_MAX = int.MaxValue;
 
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
static bool isValidEdge(int u, int v,
                        bool[] inMST)
{
    if (u == v)
        return false;
    if (inMST[u] == false && inMST[v] == false)
        return false;
    else if (inMST[u] == true && 
             inMST[v] == true)
        return false;
    return true;
}
 
static void primMST(int [,]cost)
{ 
    bool []inMST = new bool[V];
 
    // Include first vertex in MST
    inMST[0] = true;
 
    // Keep adding edges while number of 
    // included edges does not become V-1.
    int edge_count = 0, mincost = 0;
    while (edge_count < V - 1)
    {
 
        // Find minimum weight valid edge. 
        int min = INT_MAX, a = -1, b = -1;
        for (int i = 0; i < V; i++) 
        {
            for (int j = 0; j < V; j++) 
            {         
                if (cost[i, j] < min) 
                {
                    if (isValidEdge(i, j, inMST)) 
                    {
                        min = cost[i, j];
                        a = i;
                        b = j;
                    }
                }
            }
        }
         
        if (a != -1 && b != -1) 
        {
            Console.Write("Edge {0}:({1}, {2}) cost: {3} \n", 
                                    edge_count++, a, b, min);
            mincost = mincost + min;
            inMST[b] = inMST[a] = true;
        }
    }
    Console.Write("\n Minimum cost = {0} \n", mincost);
}
 
// Driver Code
public static void Main(String[] args)
{
    /* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | / \ |
    (3)-------(4)
            9     */
    int [,]cost = {{ INT_MAX, 2, INT_MAX, 6, INT_MAX },
                   { 2, INT_MAX, 3, 8, 5 },
                   { INT_MAX, 3, INT_MAX, INT_MAX, 7 },
                   { 6, 8, INT_MAX, INT_MAX, 9 },
                   { INT_MAX, 5, 7, 9, INT_MAX }};
 
    // Print the solution
    primMST(cost);
}
} 
 
// This code is contributed by PrinciRaj1992

Javascript

<script>
 
// JavaScript implementation to find minimum 
// spanning tree for adjacency representation.
 
let V = 5;
let let_MAX = Number.MAX_VALUE;
   
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
function isValidEdge(u, v, inMST)
{
    if (u == v)
        return false;
    if (inMST[u] == false && inMST[v] == false)
        return false;
    else if (inMST[u] == true && inMST[v] == true)
        return false;
    return true;
}
   
function primMST(cost)
{ 
    let inMST = Array(V).fill(false);
   
    // Include first vertex in MST
    inMST[0] = true;
   
    // Keep adding edges while number of included
    // edges does not become V-1.
    let edge_count = 0, mincost = 0;
    while (edge_count < V - 1)
    {
   
        // Find minimum weight valid edge. 
        let min = let_MAX, a = -1, b = -1;
        for (let i = 0; i < V; i++) 
        {
            for (let j = 0; j < V; j++) 
            {             
                if (cost[i][j] < min) 
                {
                    if (isValidEdge(i, j, inMST)) 
                    {
                        min = cost[i][j];
                        a = i;
                        b = j;
                    }
                }
            }
        }
           
        if (a != -1 && b != -1) 
        {
            document.write("Edge " + 
            edge_count++ + ": (" +  a +"," + b + 
            ") " + "cost: " + min + "<br/>");
            mincost = mincost + min;
            inMST[b] = inMST[a] = true;
        }
    }
     document.write("<br>");
     document.write("  ");
     document.write("Minimum cost = " + mincost);
} 
 
// driver code
 
      /* Let us create the following graph
        2 3
    (0)--(1)--(2)
    | / \ |
    6| 8/ \5 |7
    | /     \ |
    (3)-------(4)
            9         */
    let cost = [[ let_MAX, 2, let_MAX, 6, let_MAX ],
                    [ 2, let_MAX, 3, 8, 5 ],
                    [ let_MAX, 3, let_MAX, let_MAX, 7 ],
                    [ 6, 8, let_MAX, let_MAX, 9 ],
                    [ let_MAX, 5, 7, 9, let_MAX ]];
   
    // Print the solution
    primMST(cost);
   
</script>

Output

Edge 0:(0, 1) cost: 2 
Edge 1:(1, 2) cost: 3 
Edge 2:(1, 4) cost: 5 
Edge 3:(0, 3) cost: 6 

 Minimum cost= 16

Complexity Analysis:

Time Complexity: O(V³)
Note that time complexity of previous approach that uses adjacency matrix is O(V²) and time complexity of the adjacency list representation implementation is O((E+V)LogV).
Auxiliary Space: O(E + V)

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Practice Tags :

Prim’s Algorithm (Simple Implementation for Adjacency Matrix Representation)

C++

Java

Python3

C#

Javascript

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