Given two sequences, the shared subsequence is a sequence that is a subsequence of both sequences.
A maximum-size shared subsequence is a shared subsequence of maximum size.
For example, the maximum-size shared subsequence of (3,2,8,2,3,9,4,3,9) and (1,3,2,3,7,9) is (3,2,3,9).
In this homework we will develop a dynamic programming algorithm to find a maximum-size shared subsequence of two subsequences (the two subsequences will be given as input).
1A: (10, 3,2,8,2,3,9,4,3,9) and (10, 1,3,2,3,7,9)
1B: (1, 3,2,8,2,3,9,4,3,9) and (10, 1,3,2,3,7,9)
2. Prove the following claims:
Given two sequences A[1..n] = (A[1], A[2],..., A[n]) and B[1..m] = (B[1], B[2], ...., B[m]), define MCS(A, n, B, m) to be the maximum size of any subsequence shared by A[1..n] and B[1..m].
2A: If A[n] = B[m], then MCS(A, n, B, m) ≥ 1 + MCS(A, n-1, B, m-1).
2B: In all cases, MCS(A, n, B, m) ≤ 1 + MCS(A, n-1, B, m-1).
2C: If A[n] = B[m] then MCS(A, n, B, m) = 1 + MCS(A, n-1, B, m-1). (You may use the facts you proved in 2A and 2B.)
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2D: In all cases, MCS(A, n, B, m) ≥ max(MCS(A, n-1, B, m), MCS(A, n, B, m-1).
2E: If A[n] ≠ B[m], then MCS(A, n, B, m) ≤ max(MCS(A, n-1, B, m), MCS(A, n, B, m-1).
2F: If A[n] ≠ B[m], then MCS(A, n, B, m) = max(MCS(A, n-1, B, m), MCS(A, n, B, m-1). (You may use the facts you proved in 2D and 2E.)
3. The facts proved in 2C and 2F lead to the following recursive algorithm to compute MCS(A, n, B, m):
MCS(A, n, B, m) if (n == 0 or m ==0) return 0; if (A[n] == B[m]) return 1+MCS(A, n-1, B, m-1); return max(MCS(A, n, B, m-1), MCS(A, n-1, B, m));
3A: Give the best big-Ω lower bound you can on worst-case running time of MCS(A, n, B, m) as a function of n and m. Explain your reasoning.
3B: Give the best big-O upper bound you can on worst-case running time of MCS(A, n, B, m) as a function of n and m. Explain your reasoning.
4A: Precisely describe a faster algorithm (running in time O(n m)) for computing MCS(A, n, B, m). Describe the algorithm with the lowest big-O worst-case running time that you can.
4B: Explain why your algorithm is correct.
4C: Give the best big-O upper bound you can on the worst-case running time of your algorithm, in terms of n and m. Explain your reasoning.
5. Implement your algorithm and use it to find the maximum size of any subsequence shared by the following two sequences:
int A[100] = {48, 29, 25, 7, 21, 32, 32, 13, 38, 16, 13, 29, 8, 28, 0, 21, 11, 27, 17, 44, 28, 10, 49, 23, 20, 33, 35, 40, 4, 15, 40, 34, 23, 40, 3, 39, 26, 45, 16, 23, 22, 39, 25, 32, 2, 34, 3, 46, 16, 19, 4, 25, 36, 14, 37, 30, 34, 49, 5, 9, 32, 19, 19, 6, 33, 9, 28, 32, 1, 29, 41, 42, 11, 12, 31, 13, 33, 5, 31, 6, 35, 10, 27, 36, 45, 48, 38, 5, 27, 21, 34, 23, 11, 20, 22, 25, 11, 44, 3, 32};
int B[100] = {33, 31, 9, 41, 49, 35, 12, 3, 43, 2, 47, 43, 11, 29, 11, 24, 4, 15, 28, 48, 3, 28, 9, 20, 10, 0, 1, 26, 35, 37, 48, 26, 32, 8, 14, 48, 9, 45, 16, 27, 13, 21, 6, 28, 36, 1, 16, 4, 41, 33, 49, 36, 20, 44, 46, 26, 36, 42, 22, 29, 29, 24, 30, 3, 20, 42, 3, 36, 14, 1, 44, 26, 35, 9, 47, 32, 43, 47, 29, 45, 36, 20, 0, 48, 10, 18, 40, 20, 41, 42, 11, 5, 30, 32, 46, 20, 38, 9, 19, 24};
You may use C++, Python, or Perl to implement your algorithm. Include a print-out of your algorithm and the length of the shared subsequence that it finds.