Minimum cost to merge numbers from 1 to N
Given an integer N, the task is to find the minimum cost to merge all the numbers from 1 to N where the cost of merging two set of numbers A and B is equal to the product of the product of the numbers in the respective sets.
Examples:
Input: N = 4
Output: 32Merging {1} and {2} costs 1 * 2 = 2
Merging {1, 2} and {3} costs 2 * 3 = 6
Merge{1, 2, 3} and {4} costs 6 * 4 = 24
Hence, the minimal cost is 2 + 6 + 24 = 32Input: N = 2
Output: 2
Approach:

The first approach that comes in our mind is sorting. We take first two smallest elements and add them, then continue adding to the rest of the elements in the sorted array. But it fails when the current running sum exceeds the next smallest value in the array coming next.
Take N = 5, If we take the sorting approach, then Merge {1} and {2}  1 * 2 = 2 Merge {1, 2} and {3}  2 * 3 = 6 Merge{1, 2, 3} and {4}  6 * 4 = 24 Merge{1, 2, 3, 4} and {5}  24 * 5 = 120 Total sum = 152
But optimal way is, Merge {1} and {2}  1 * 2 = 2 Merge {1, 2} and {3}  2 * 3 = 6 Merge {4} and {5}  4 * 5 = 20 Merge {1, 2, 3} and {4, 5}  6 * 20 = 120 Total sum = 148 This is the minimal answer.
 So, the correct approach to solve this problem is the Minheap based approach. Initially, we push all the numbers from 1 to N into the MinHeap.
 At every iteration, we extract the minimum and the second minimum element from the MinHeap and insert their product back into it. This ensures that the addition cost generated will be minimum.
 We keep on repeating the above step until there is only one element remaining in the MinHeap. The calculated sum till that instant gives us the required answer.
Below is the implementation of the above approach:
C++
// C++ program to find the Minimum // cost to merge numbers // from 1 to N. #include <bits/stdc++.h> using namespace std; // Function returns the // minimum cost int GetMinCost( int N) { // Min Heap representation priority_queue< int , vector< int >, greater< int > > pq; // Add all elements to heap for ( int i = 1; i <= N; i++) { pq.push(i); } int cost = 0; while (pq.size() > 1) { // First minimum int mini = pq.top(); pq.pop(); // Second minimum int secondmini = pq.top(); pq.pop(); // Multiply them int current = mini * secondmini; // Add to the cost cost += current; // Push the product into the // heap again pq.push(current); } // Return the optimal cost return cost; } // Driver code int main() { int N = 5; cout << GetMinCost(N); } 
Java
// Java program for the above approach import java.util.*; class GFG { // Function returns the // minimum cost static int GetMinCost( int N) { // Min Heap representation PriorityQueue<Integer> pq; pq = new PriorityQueue<>(); // Add all elements to heap for ( int i = 1 ; i <= N; i++) { pq.add(i); } int cost = 0 ; while (pq.size() > 1 ) { // First minimum int mini = pq.remove(); // Second minimum int secondmini = pq.remove(); // Multiply them int current = mini * secondmini; // Add to the cost cost += current; // Push the product into the // heap again pq.add(current); } // Return the optimal cost return cost; } // Driver Code public static void main(String args[]) { int N = 5 ; // Function call System.out.println(GetMinCost(N)); } } // This code is contributed by rutvik_56 
Python3
# python3 program to find the Minimum # cost to merge numbers # from 1 to N. # Function returns the # minimum cost def GetMinCost(N): # Min Heap representation pq = [] # Add all elements to heap for i in range ( 1 , N + 1 , 1 ): pq.append(i) pq.sort(reverse = False ) cost = 0 while ( len (pq) > 1 ): # First minimum mini = pq[ 0 ] pq.remove(pq[ 0 ]) # Second minimum secondmini = pq[ 0 ] pq.remove(pq[ 0 ]) # Multiply them current = mini * secondmini # Add to the cost cost + = current # Push the product into the # heap again pq.append(current) pq.sort(reverse = False ) # Return the optimal cost return cost # Driver code if __name__ = = '__main__' : N = 5 print (GetMinCost(N)) # This code is contributed by Bhupendra_Singh 
148
Time Complexity: O(NlogN)
Auxiliary Space: O(N)
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