Let XX[1: nn] be a real array. A prefix of X is a subarray XX[1: kk] where 1 ≤ kk ≤ nn, and a suffix of X is a subarray XX[kk: nn]. The minimum-prefix-product problem is the problem of taking XX[1:nn] as input and returning the k such that XX[1] × XX[2] × … × XX[kk] is the smallest possible. The minimum-suffix-product problem is to find k such that XX[kk] × XX[kk + 1] × … × XX[nn] is the smallest possible. The minimum-subarray-product problem is the problem of taking XX[1: nn] as input and returning two integers k and r, (1 ≤ kk ≤ rr ≤ nn), such that XX[kk] × XX[kk + 1] × … × XX[rr] is the smallest possible. a. Write a divide-and-conquer algorithm for solving the minimum-prefix-product problem, and another divide-and-conquer algorithm for solving the minimum-suffix-product problem.
b. Write a divide-and-conquer algorithm for solving the minimum-subarray-product problem.
c. Analyze the time complexity of your algorithms.