Maximize Array sum by replacing middle elements with min of Subarray corners

Given an array A[] of length N. Then your task is to output the maximum sum that can be achieved by using the given operation at any number of times (possibly zero):

• Select a subarray, whose length is at least 3.
• Replace all the middle elements (Except the first and last) with a minimum of elements at both ends.

Examples:

Input: N = 3, A[] = {8, 1, 7}
Output: 22
Explanation: Let us take the subarray A[1, 3] = {8, 1, 7}. Middle element(s) are: {1}. Now minimum of A[1] and A[3] is = min(A[1], A[3]) = min(8, 7) = 7. Then, we replaced all the middle elements equal to min according to the operation. So, updated A[] = {8, 7, 7}. The sum of updated A[] = 22. Which is the maximum possible using a given operation. Therefore, output is 22.

Input: N = 2, A[] = {5, 2}
Output: 7
Explanation: No subarray of length at least 3 can be chosen, Thus can't apply any operation. So the maximum possible sum is 7.

Approach: Implement the idea below to solve the problem

The problem is based on the observations and can be solved using the Prefix and Suffix arrays. It must be noticed that for any index i, we can not make the element A[i] to be anything more than the minimum of the greatest element to its left and the greatest element to its right.

Steps were taken to solve the problem:

• Create two arrays let say Prefix[] and Suffix[]
• Set prefix[0] = A[0]
• Run a loop for i = 1 to i < N and follow the below-mentioned steps under the scope of the loop
  • prefix[i] = max(prefix[i - 1], A[i])
• Set suffix[N - 1] = A[N - 1]
• Run a loop for i = N - 2 to I >=0 and follow the below-mentioned steps under the scope of the loop
  • suffix[i] = max(suffix[i + 1], A[i])
• Create a variable let say Sum to store the max possible sum.
• Run a loop for i = 1 to i < N - 1 and follow below mentioned steps under the scope of loop
  • Max1 = Prefix[i]
  • Max2 = Suffix[i]
  • sum += min(Max1, Max2)
• Sum += A[0] + A[N - 1]
• Output the value store in Sum.

Code to implement the approach:

7

Time Complexity: O(N)
Auxiliary Space: O(2*N), As Prefix and Suffix arrays of length N are used.