Find the maximum sum of Plus shape pattern in a 2-D array

Given a 2-D array of size N*M where, . The task is to find the maximum value achievable by a + shaped pattern. The elements of the array can be negative.
The plus(+) shape pattern is formed by taking any element with co-ordinate (x, y) as a center and then expanding it in all four directions(if possible). 

A plus(+) shape has atleast five elements which are { (x-1, y), (x, y-1), (x, y), (x+1, y), (x, y+1) } i.e. the arms should have length>1 but not necessarily need to have same length.

Examples: 

Input: N = 3, M = 4
       1 1 1 1
      -6 1 1 -4
       1 1 1 1
Output: 0
Here, (x, y)=(2, 3) center of pattern(+).
Other four arms are, left arm = (2, 2), right arm = (2, 4), 
up arm = (1, 3), down arm = (2, 3).
Hence sum of all elements are ( 1 + 1 + (-4) + 1 + 1 ) = 0.

Input: N = 5, M = 3
       1 2 3
      -6 1 -4
       1 1 1
       7 8 9
       6 3 2
Output: 31

Approach: This problem is an application of the standard Largest Sum Contiguous Subarray.

We quickly pre-compute the maximum contiguous sub-sequence (subarray) sum for each row and column, in 4 directions, namely, Up, Down, Left and Right. This can be done using the standard Maximum contiguous sub-sequence sum of a 1-D array.

We make four 2-D array’s 1 for each direction. 

1. up[i][j]– Maximum sum contiguous sub-sequence of elements in upward direction, from rows 1, 2, 3, …, i More formally, it represents the maximum sum obtained by adding a contiguous sub-sequence of elements from list of arr[1][j], arr[2][j], …, arr[i][j]
2. down[i][j] -Maximum sum contiguous sub-sequence of elements in downward direction, from rows i, i+1, i+2,,…, N More formally, it represents the maximum sum obtained by adding a contiguous sub-sequence of elements from list of arr[i][j], arr[i+1][j], …, arr[N][j]
3. left[i][j]– Maximum sum contiguous sub-sequence of elements in left direction, from columns 1, 2, 3, …, j More formally, it represents the maximum sum obtained by adding a contiguous sub-sequence of elements from list of arr[i][1], arr[i][2], …, arr[i][j]
4. right[i][j]– Maximum sum contiguous sub-sequence of elements in right direction, from columns j, j+1, j+2, …, M More formally, it represents the maximum sum obtained by adding a contiguous sub-sequence of elements from list of arr[i][j], arr[i][j+1], …, arr[i][M]

All that’s left is, to check each cell as a possible center of the + and use pre-computed data to find the value achieved by + shape in O(1). 

Below is the implementation of the above approach:

0

Time Complexity: O(N*M) for given N rows and M columns

Auxiliary Space: O(N*M)