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++ 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 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# 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 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 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.. ?> |
<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> |
21
Complexity Analysis:
- Time Complexity: O(1)
- Auxiliary Space: O(1)