More dynamic programming Longest common subsequence

1. More dynamic programming Longest common subsequence

2. Definition of Subsequences on board. • Proof of optimal substructure on board. • Recursive Formula derivation on board. • Example from CLRS.

3. Lambda is the null string. We fill the first row and column with zeros because the LCS of a null string with anything is zero. 0 1 2 3 4 5 6 j λ C A B A yj i B D λ 0 1 A 2 B C 3 4 B 5 D A 6 7 B xi

4. Begin filling in row major order. We place a pointer to tell where the value was obtained from to help with traceback. 0 1 2 3 4 5 6 j λ C A B A yj i B D λ 0 1 A 2 B C 3 4 B 5 D A 6 7 B xi

5. Continue filling in row major order. 0 1 2 3 4 5 6 j λ C A B A yj i B D λ 0 1 A 2 B C 3 4 B 5 D A 6 7 B xi

6. Final Result (from CLRS page 395)

7. The following procedure (from CLRS page 396) will print the LCS. Initial call is to (b, X, 7, 6) • This will print BCBA. PrintLCS(7,6) (6,6) print A (5,5) (4,5) print B (3,4) (3,3) print C (2,2) (2,1) print B (1,0) See previous slide. It prints on return from call with