Minimum cost to connect weighted nodes represented as array

• Difficulty Level : Easy
• Last Updated : 30 Apr, 2021

Given an array of N elements(nodes), where every element is weight of that node.Connecting two nodes will take a cost of product of their weights.You have to connect every node with every other node(directly or indirectly).Output the minimum cost required.
Examples:

Input : a[] = {6, 2, 1, 5}
Output :  13
Explanation :
Here, we connect the nodes as follows:
connect a and a, cost = 6*1 = 6,
connect a and a, cost = 1*2 = 2,
connect a and a, cost = 1*5 = 5.
every node is reachable from every other node:
Total cost = 6+2+5 = 13.

Input  : a[] = {5, 10}
Output : 50
Explanation : connections:
connect a and a, cost = 5*10 = 50,
Minimum cost = 50.

We need to make some observations that we have to make a connected graph with N-1 edges. As the output will be sum of products of two numbers, we have to minimize the product of every term in that sum equation. How can we do it? Clearly, choose the minimum element in the array and connect it with every other. In this way we can reach every other node from a particular node.
Let, minimum element = a[i], (let it’s index be 0)
Minimum Cost
So, answer is product of minimum element and sum of all the elements except minimum element.

C++

 // cpp code for Minimum Cost Required to connect weighted nodes#include using namespace std;int minimum_cost(int a[], int n){    int mn = INT_MAX;    int sum = 0;    for (int i = 0; i < n; i++) {         // To find the minimum element        mn = min(a[i], mn);         // sum of all the elements        sum += a[i];    }     return mn * (sum - mn);} // Driver codeint main(){    int a[] = { 4, 3, 2, 5 };    int n = sizeof(a) / sizeof(a);    cout << minimum_cost(a, n) << endl;    return 0;}

Java

 // Java code for Minimum Cost Required to// connect weighted nodes import java.io.*; class GFG {         static int minimum_cost(int a[], int n)    {                 int mn = Integer.MAX_VALUE;        int sum = 0;                 for (int i = 0; i < n; i++) {             // To find the minimum element            mn = Math.min(a[i], mn);             // sum of all the elements            sum += a[i];        }         return mn * (sum - mn);    }     // Driver code    public static void main(String[] args)    {        int a[] = { 4, 3, 2, 5 };        int n = a.length;                 System.out.println(minimum_cost(a, n));    }} // This code is contributed by vt_m.

Python 3

 # Python 3 code for Minimum Cost# Required to connect weighted nodesimport sys def minimum_cost(a, n):     mn = sys.maxsize    sum = 0    for i in range(n):         # To find the minimum element        mn = min(a[i], mn)         # sum of all the elements        sum += a[i]     return mn * (sum - mn) # Driver codeif __name__ == "__main__":         a = [ 4, 3, 2, 5 ]    n = len(a)    print(minimum_cost(a, n)) # This code is contributed# by ChitraNayal

C#

 // C# code for Minimum Cost Required// to connect weighted nodesusing System; class GFG {         // Function to calculate minimum cost    static int minimum_cost(int []a, int n)    {                 int mn = int.MaxValue;        int sum = 0;                 for (int i = 0; i < n; i++)        {             // To find the minimum element            mn = Math.Min(a[i], mn);             // sum of all the elements            sum += a[i];        }         return mn * (sum - mn);    }     // Driver code    public static void Main()    {        int []a = {4, 3, 2, 5};        int n = a.Length;                 Console.WriteLine(minimum_cost(a, n));    }} // This code is contributed by vt_m.



Javascript



Output:

24

Time complexity: O(n).
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