Let A be an array of integers (positive or negative). Assume that the array is of size n. The subarray A[i..j] is the part of the array that starts at index i and ends at index j, where 0 i j n-1. Let s_ij equal the sum of the integers in A[i..j].

We wish to solve the following problem:

Find the maximum value for s_ij over all subarrays in array A, where 0 i j n-1.

The three algorithms given below solve this problem. NOTE: If all of the values in the array are negative, then the maximum value for s_ij is 0 by default.

Example: If the array contains the values { -1, 12, -3, 14, -4, 3 }, then the maximum sum over all subarrays is 23 (for the subarray {12, -3, 14}). If the array contains the values { 2, -3, 5, -1, 0, 7}, then the maximum sum over all subarrays is 11 (for the subarray {5, -1, 0, 7}).

1.

Start with a maximum sum of 0. Compute the sum of each 1-element subarray, then compute the sum of each 2-element subarray, then compute the sum of each 3-element subarray, etc. For each sum you compute, if it is larger than the maximum sum you've seen, then it becomes the maximum sum.

2.

Start with a maximum sum of 0. Compute the sum of each subarray that starts at index 0, then compute the sum of each subarray that starts at index 1, then compute the sum of each subarray that starts at index 2, etc. For each sum you compute, if it is larger than the maximum sum you've seen, then it becomes the maximum sum.

EFFICIENCY NOTE: Once you compute the sum of the subarray from A[i] to A[j], the sum of the subarray from A[i] to A[j+1] is just the previous sum you computed plus A[j+1]. Don't add up all of the previous values all over again.

3.

Start with a maximum sum of 0. Compute the sum of all subarrays starting from index 0. Use the efficiency note from Algorithm 2. If the sum you compute is negative, start computing the sum of all subarrays starting from the next index in the array. Repeat this special rule if necessary. For each sum you compute, if it is larger than the maximum sum you've seen, then it becomes the maximum sum.

For example, as you compute the sum of all subarrays starting from index 0, if you find the sum of the integers in A[0..5] is negative, then compute the sum of all subarrays starting from index 6 next. If the sum of the integers from A[6..14] is negative, then compute the sum of all subarrays starting from index 15, Etc.

You are to write a Java program that determines the amount of work each of these algorithms does to compute its answer for arrays of various sizes. Using this data, you are to determine the runtime complexity of each algorithm.