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Minimum spanning tree cost of given Graphs

  • Difficulty Level : Expert
  • Last Updated : 16 Apr, 2021

Given an undirected graph of V nodes (V > 2) named V1, V2, V3, …, Vn. Two nodes Vi and Vj are connected to each other if and only if 0 < | i – j | ≤ 2. Each edge between any vertex pair (Vi, Vj) is assigned a weight i + j. The task is to find the cost of the minimum spanning tree of such graph with V nodes.
Examples: 
 

Input: V = 4 
 

Output: 13
Input: V = 5 
Output: 21 
 

 



Approach: Starting with a graph with minimum nodes (i.e. 3 nodes), the cost of the minimum spanning tree will be 7. Now for every node i starting from the fourth node which can be added to this graph, ith node can only be connected to (i – 1)th and (i – 2)th node and the minimum spanning tree will only include the node with the minimum weight so the newly added edge will have the weight i + (i – 2)
 

So addition of fourth node will increase the overall weight as 7 + (4 + 2) = 13 
Similarly adding fifth node, weight = 13 + (5 + 3) = 21 
… 
For nth node, weight = weight + (n + (n – 2))
 

This can be generalized as weight = V2 – V + 1 where V is the total nodes in the graph.
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function that returns the minimum cost
// of the spanning tree for the required graph
int getMinCost(int Vertices)
{
    int cost = 0;
 
    // Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1;
 
    return cost;
}
 
// Driver code
int main()
{
    int V = 5;
    cout << getMinCost(V);
 
    return 0;
}

Java




// Java implementation of the approach
class GfG
{
 
// Function that returns the minimum cost
// of the spanning tree for the required graph
static int getMinCost(int Vertices)
{
    int cost = 0;
 
    // Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1;
 
    return cost;
}
 
// Driver code
public static void main(String[] args)
{
    int V = 5;
    System.out.println(getMinCost(V));
}
}
 
// This code is contributed by
// Prerna Saini.

C#




// C# implementation of the above approach
using System;
 
class GfG
{
 
    // Function that returns the minimum cost
    // of the spanning tree for the required graph
    static int getMinCost(int Vertices)
    {
        int cost = 0;
     
        // Calculating cost of MST
        cost = (Vertices * Vertices) - Vertices + 1;
     
        return cost;
    }
     
    // Driver code
    public static void Main()
    {
        int V = 5;
        Console.WriteLine(getMinCost(V));
    }
}
 
// This code is contributed by Ryuga

Python3




# python3 implementation of the approach
  
# Function that returns the minimum cost
# of the spanning tree for the required graph
def getMinCost( Vertices):
    cost = 0
  
    # Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1
  
    return cost
  
# Driver code
if __name__ == "__main__":
 
    V = 5
    print (getMinCost(V))

PHP




<?php
// PHP implementation of the approach
// Function that returns the minimum cost
// of the spanning tree for the required graph
function getMinCost($Vertices)
{
    $cost = 0;
 
    // Calculating cost of MST
    $cost = ($Vertices * $Vertices) - $Vertices + 1;
 
    return $cost;
}
 
// Driver code
$V = 5;
echo getMinCost($V);
 
#This Code is contributed by ajit..
?>

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function that returns the minimum cost
// of the spanning tree for the required graph
function getMinCost(Vertices)
{
    var cost = 0;
 
    // Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1;
 
    return cost;
}
 
// Driver code
var V = 5;
document.write( getMinCost(V));
 
// This code is contributed by rrrtnx.
</script>
Output: 
21

 

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