Minimum pair merge operations required to make Array non-increasing

Minimum pair merge operations required to make Array non-increasing

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Given an array A[], the task is to find the minimum number of operations required in which two adjacent elements are removed from the array and replaced by their sum, such that the array is converted to a non-increasing array.

Note: An array with a single element is considered as non-increasing.

Examples:

Input:_ A[] = {1, 5, 3, 9, 1}_

Output:_ 2_

Explanation:

Replacing {1, 5} by {6} modifies the array to {6, 3, 9, 1}

Replacing {6, 3} by {9} modifies the array to {9, 9, 1}

Input:_ A[] = {0, 1, 2}_

Output:_ 2_

*Approach: *The idea is to use Dynamic Programming. A memoization table is used to store the minimum count of operations required to make subarrays non-increasing from right to left of the given array. Follow the steps below to solve the problem:

  • Initialize an array dp[] where dp[i] stores the minimum number of operations required to make the subarray {A[i], …, A[N]} non-increasing. Therefore, the target is to compute dp[0].
  • Find a minimal subarray {A[i] .. A[j]} such that sum({A[i] .. A[j]}) > val[j+1], where, val[j + 1] is the merged sum obtained for the subarray {A[j + 1], … A[N]}.
  • Update dp[i] to j – i + dp[j+1] *and *vals[i] to sum({A[i] .. A[j]}).

Below is the implementation of the above approach:

  • C++

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// C++ program to implement

// the above approach

#include <bits/stdc++.h>

**using** **namespace** std;

// Function to find the minimum operations

// to make the array Non-increasing

**int** solve(vector<``**int**``>& a)

{

// Size of the array

**int** n = a.size();

// Dp table initialization

vector<``**int**``> dp(n + 1, 0), val(n + 1, 0);

// dp[i]: Stores minimum number of 

// operations required to make 

// subarray {A[i], ..., A[N]} non-increasing

**for** (``**int** i = n - 1; i >= 0; i--) {

**long** **long** sum = a[i];

**int** j = i;

**while** (j + 1 < n and sum < val[j + 1]) {

// Increment the value of j

j++;

// Add current value to sum

sum += a[j];

}

// Update the dp tables

dp[i] = (j - i) + dp[j + 1];

val[i] = sum;

}

// Return the answer

**return** dp[0];

}

// Driver code

**int** main()

{

vector<``**int**``> arr = { 1, 5, 3, 9, 1 };

cout << solve(arr);

}

Output:

2

Time Complexity:_ O(N2)_

Auxiliary Space:_ O(N)_

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arrays dynamic programming mathematical memoization

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