Finding the longest non-negative sub array

Question

I\'m looking for an more efficient alternative to brute force for finding the longest sub array of an array with a non-negative sum. The numbers range from -5 to 5 in this a

Answer 1

HINTS

You can do this in linear time by:

1. First compute the cumulative sum of your array A
2. Then compute an array B giving the greatest value in A to the right of index i
3. And compute an array C giving the smallest value in A to the left of index i

Both B and C will be non-increasing arrays.

Then for each start position in array C, compute the greatest end position in array B such that B[end]>C[start]. This can be done in linear time by:

1. start with start=0
2. increment end until B[end+1]<=C[start]
3. increment start and return to step 2

The greatest value of end-start corresponds to the longest subarray.

Explanation

The main idea is that once you have a cumulative sum of your array you can compute the value of a subarray by doing a subtraction.

For example, the array:

4 2 -5 3 0 -2 

has cumulative values:

A = [4 6 1 4 4 2]

So to find the sum of the second, third, fourth entries (indices 1,2,3 with values 2,-5,3) we can compute:

A[3]-A[0] = 4 - 4 = 0

and so your problem now reduces to finding pairs of values in array A that are furthest apart and also have A[end]>A[start].