C = if i=0 or j=0 =c + 1 if I,j > 0 and xi = yj =max ( c, c ) if I,j > 0 and xi ≠ yjĥ4 Example X = A B C B D A B Y = B D C A B A Let c be the length of an LCS of X and Y. If xm ≠ yn, then zk ≠ yn implies that Z is an LCS of X and Yn_1. If xm ≠ yn, then zk ≠ xm implies that Z is an LCS of Xm_1 and Y. If xm = yn, then zk = xm = yn and Zk_1 is an LCS of Xm_1 and Yn_1. Let X = and Y = be sequences, and let Z= be any LCS of X and Y. S1 = ACCGGTCGAGTGCGCGGAAGCCGGCCCGAA S2 = GTCGTTCGGAATGCCGTTGCTCTGTAAA S3= GTCGTCGGAAGCCGGCCGAAĤ8 LC Subsequence A subsequence of a given sequence is just the given sequence with zero or more elements left out Given two sequences X and Y, a sequence Z is a common subsequence of X and Y, if Z is a subsequence of both X and YĤ9 Theorem – Optimal substructure of an LCS

S1 = ACCGGTCGAGTGCGCGGAAGCCGGCCCGAA S2 = GTCGTTCGGAATGCCGTTGCTCTGTAAA Let m be the minimum number of scalar multiplications needed to compute the matrix Ai. A ,An Minimize the number of scalar multiplication Matrix multiplication is associative A1 – 100x100, A2-100x5, A3- 5x50 A1.(A2.A3) need multiplications (A1.A2).A3 need multiplicationsĪssume that matrix Ai has dimensions pi-1 x pi for i =1, 2, 3, … n. Greedy algorithms do not always yield optimal solutions Examples: Minimum spanning tree(Kruskal and Prim), shortest path algorithm (Dijkstra)ģ Activity Selection Problem: Schedule several competing activities that requires exclusive use of a common resource Goal: Select a maximum-size set of mutually compatible activities.Ĥ Activity Selection we have a set S = while w of n matrices Compute the product A1. locally optimal choice leading to a globally optimal solution.

1 Algorithm Design Techniques Greedy Approach vs Dynamic ProgrammingĢ Greedy Algorithms A greedy algorithm always makes the choice that looks best at the moment.

Source code algoritma greedy