Maximize the sum of products of the degrees between any two vertices of the tree in C++

C++Server Side ProgrammingProgramming

Given the task is to construct a tree with a given integer N such that, the sum of degree(x) * degree(y) for all ordered pairs (x,y) is maximum and x is not equal to y.

Input −N=5

Output −50

Explanation 

   1
    \
     2
      \
       3
         \
          4
            \
             5
Degree of 1st node = 1
Degree of 2nd node = 2
Degree of 3rd node = 2
Degree of 4th node = 2
Degree of 5th node = 1

Product of all degrees for all ordered pairs (x, y) −

1st node = 1*2 + 1*2 + 1*2 + 1*1 = 7
2nd node = 2*1 + 2*2 + 2*2 + 2*1 = 12
3rd node = 2*1 + 2*2 + 2*2 + 2*1 = 12
4th node = 2*1 + 2*2 + 2*2 + 2*1 = 12
5th node = 1*2 + 1*2 + 1*2 + 1*1 = 7
Total sum = 50

Input −N=7

Output −122

Approach used in the below program as follows

  • Sum of degrees of all nodes in a tree is − (2 * N) – 2. Here N=number of nodes in the tree. In order to maximize the sum, the number of leaf nodes have to be minimized.

  • Inside Max() function initialize int sum=0 and create nested loops initializing x=1 and y=1 having conditions x<N and y<N.

  • Inside the nested loops first check if(x==y), if so then add continue; statement

  • Else initialize int degree=2 and using an if statement check if(x==1 || x==n). If so then put degreeX=1. Then initialize int degree=2 and do the same for variable y

  • Finally, before closing the loops, update the sum variable by writing − sum = (degreeX + degreeY)

Example

 Live Demo

#include <bits/stdc++.h>
using namespace std;
int Max(int N){
   int sum = 0;
   for (int x = 1; x <= N; x++){
      for (int y = 1; y <= N; y++){
         if (x == y)
            continue;
         //Initializing degree for node x to 2
         int degreeX = 2;
         //If x is the leaf node or the root node
         if (x == 1 || x == N)
            degreeX = 1;
         //Initializing degree for node y to 2
         int degreeY = 2;
         //If y is the leaf node or the root node
         if (y == 1 || y == N)
            degreeY = 1;
         //Updating sum
         sum += (degreeX * degreeY);
      }
   }
   return sum;
}
//Main function
int main(){
   int N = 5;
   cout << Max(N);
}

Output

If we run the above code we will get the following output −

50
raja
Published on 17-Aug-2020 12:28:29
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