Dr. Watson has been kidnaped! Sherlock Holmes was contacted by the kidnapper for ransom. Moments later he received a message from Dr. Watson's phone. The message contained three random strings. Sherlock being Sherlock, was able to deduce immediately that Dr. Watson was trying to give a hint about his location. He figured out that the longest common subsequence between the 3 words is the location. But since it was too easy for him, he got bored and asked you to find out what the actual location is. Your task is to find the longest common subsequence from the 3 given strings before it is too late. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements. For instance, given a sequence "drew"; "d", "w", "de", "drw", "drew" are all examples of valid subsequences (there are also others), while "er", "wdre" are not. Design a dynamic programming algorithm which takes three input sequences X, Y, and Z, of lengths m, n, and p, respectively, and returns their longest common subsequence. For full marks your algorithm should run in (mnp) time. Note that W is a common subsequence of X, Y, and Z if and only if W is a subsequence of X, W is a subsequence of Y, and W is a subsequence of Z. (a) Describe the set of subproblems that your dynamic programming algorithm will consider. Your solution should look something like "For every [...], we define OPT([...]) to be [...]."
(b) Give a recurrence expressing the solution to each subproblem in terms of the solution to smaller subproblems. Provide a few sentences justifying why your recurrence is correct.
(c) Describe in pseudocode an algorithm that determines the longest common subse- quence of X, Y, and Z. Your implementation may be either "bottom-up" or "top-down."
(d) Analyze the running time and space usage of your algorithm.