Question

Given an array of real-valued numbers, A[1..n,1..n], I wish to find the sub-array

B = A[i..j,s..t]

with 1 <= i <= j <= n, and 1 <= s <= t <= n

such that the sum of the numbers in B is maximized. Is it possible to solve this using dynamic programming? I spoke to one of the OR professors at Aarhus University, and he didn’t know how to do it, and said that he had difficulty seeing how it could have the optimal substructure quality.

But IS it possible? If yes, how? If not, why?

I already know of an algorithm that runs in O(n^3) time, by reducing it to n(n+1)/2 subproblems of complexity O(n), but that seems like it’s a bit slow. I know that an optimal algorithm would run in Omega(n) time, but I’m hoping that dynamic programming could be used for making it run in O(n^2) time.

Original question summarized

I added this section because I felt some people has misinterpreted the point of my question. The original question was:

1. Is it possible to use dynamic programming to solve the above problem in O(n^2) time? If yes, how? If no, why not?

Additional questions:

I added a new questions here. More might be added later:

1. In order to use dynamic programming, I need to make use of solutions to easily solved subproblems (Or else the point is moot). The structure of the problem is such, that if we take a subarray B = A[1..m,1..m] of A[1..n,1..n] where m < n, then the optimal solution for array B is at most as good as in A, trivially, since the same solution is feasible in A. To use dynamic programming, it is therefore reasonable to ask: What is the relationship between the optimal subarray of A[1..i,1..i] and the optimal subarray of A[1..i+1,1..i+1]?

Answer 1

From The Algorithmist, if you have an n by n array, the best you can do is O(n^3):

• First, calculate the vertical prefix sum for all columns (an O(n^2) algorithm).
• Second, assume that the maximum sub-array will be between row a and row b, inclusive. There are only O(n^2) a, b pairs such that a < b. Try each of them.
• Since we already have the vertical prefix sum for all columns, the sum of elements in arr[a..b][c] for column c can be computed in O(1) time. This allows us to imagine each column sum as if it is a single element of a one-dimensional array across all columns (one dimensional array with one row and n columns).
• There’s an O(n) algorithm to compute the maximum sub-array for a one-dimensional array, known as Kadane’s Algorithm.
• Applying the Kadane’s algorithm inside each a and b combination gives the total complexity of O(n3).

Given that you have an n by 2 array, you can get it down to O(n^2). The key, as above, is to use Kadane’s algorithm