Minimum spanning tree cost of given Graphs
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++
#include <bits/stdc++.h>
using namespace std;
int getMinCost( int Vertices)
{
int cost = 0;
cost = (Vertices * Vertices) - Vertices + 1;
return cost;
}
int main()
{
int V = 5;
cout << getMinCost(V);
return 0;
}
|
Java
class GfG
{
static int getMinCost( int Vertices)
{
int cost = 0 ;
cost = (Vertices * Vertices) - Vertices + 1 ;
return cost;
}
public static void main(String[] args)
{
int V = 5 ;
System.out.println(getMinCost(V));
}
}
|
C#
using System;
class GfG
{
static int getMinCost( int Vertices)
{
int cost = 0;
cost = (Vertices * Vertices) - Vertices + 1;
return cost;
}
public static void Main()
{
int V = 5;
Console.WriteLine(getMinCost(V));
}
}
|
Python3
def getMinCost( Vertices):
cost = 0
cost = (Vertices * Vertices) - Vertices + 1
return cost
if __name__ = = "__main__" :
V = 5
print (getMinCost(V))
|
PHP
<?php
function getMinCost( $Vertices )
{
$cost = 0;
$cost = ( $Vertices * $Vertices ) - $Vertices + 1;
return $cost ;
}
$V = 5;
echo getMinCost( $V );
#This Code is contributed by ajit..
?>
|
Javascript
<script>
function getMinCost(Vertices)
{
var cost = 0;
cost = (Vertices * Vertices) - Vertices + 1;
return cost;
}
var V = 5;
document.write( getMinCost(V));
</script>
|
Complexity Analysis:
- Time Complexity: O(1)
- Auxiliary Space: O(1)
Last Updated :
15 Sep, 2022
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