# Minimum pair merge operations required to make Array non-increasing

• Difficulty Level : Hard
• Last Updated : 07 May, 2021

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 non-increasing.

Examples:

Input: A[] = {1, 5, 3, 9, 1}
Output:
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.
• 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++

 `// C++ program to implement``// the above approach` `#include ``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;``}` `// Driver code``int` `main()``{``    ``vector<``int``> arr = { 1, 5, 3, 9, 1 };``    ``cout << solve(arr);``}`

## Java

 `// Java program to implement``// the above approach``import` `java.util.*;``class` `GFG{` `// Function to find the minimum operations``// to make the array Non-increasing``static` `int` `solve(``int` `[]a)``{` `    ``// Size of the array``    ``int` `n = a.length;` `    ``// Dp table initialization``    ``int` `[]dp = ``new` `int``[n + ``1``];``    ``int` `[]val = ``new` `int``[n + ``1``];` `    ``// 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--)``    ``{``        ``int` `sum = a[i];``        ``int` `j = i;` `        ``while` `(j + ``1` `< n && 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``public` `static` `void` `main(String[] args)``{``    ``int` `[]arr = { ``1``, ``5``, ``3``, ``9``, ``1` `};``    ``System.out.print(solve(arr));``}``}` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 program to implement``# the above approach` `# Function to find the minimum operations``# to make the array Non-increasing``def` `solve(a):` `    ``# Size of the array``    ``n ``=` `len``(a)` `    ``# Dp table initialization``    ``dp ``=` `[``0``] ``*` `(n ``+` `1``)``    ``val ``=` `[``0``] ``*` `(n ``+` `1``)` `    ``# dp[i]: Stores minimum number of``    ``# operations required to make``    ``# subarray {A[i], ..., A[N]} non-increasing``    ``for` `i ``in` `range``(n ``-` `1``, ``-``1``, ``-``1``):``        ``sum` `=` `a[i]``        ``j ``=` `i` `        ``while``(j ``+` `1` `< n ``and` `sum` `< val[j ``+` `1``]):` `            ``# Increment the value of j``            ``j ``+``=` `1` `            ``# 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``arr ``=` `[ ``1``, ``5``, ``3``, ``9``, ``1` `]` `# Function call``print``(solve(arr))` `# This code is contributed by Shivam Singh`

## C#

 `// C# program to implement``// the above approach``using` `System;``class` `GFG{` `// Function to find the minimum operations``// to make the array Non-increasing``static` `int` `solve(``int` `[]a)``{` `    ``// Size of the array``    ``int` `n = a.Length;` `    ``// Dp table initialization``    ``int` `[]dp = ``new` `int``[n + 1];``    ``int` `[]val = ``new` `int``[n + 1];` `    ``// 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--)``    ``{``        ``int` `sum = a[i];``        ``int` `j = i;` `        ``while` `(j + 1 < n && 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;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `[]arr = { 1, 5, 3, 9, 1 };``    ``Console.Write(solve(arr));``}``}` `// This code is contributed by PrinciRaj1992`

## Javascript

 ``
Output:
`2`

Time Complexity: O(N2
Auxiliary Space: O(N)

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