lec14-handout.pdf - Announcements CMPSCI 311 Introduction...

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CMPSCI 311: Introduction to Algorithms Lecture 14: Dynamic Programming 3 Akshay Krishnamurthy University of Massachusetts Last Compiled: March 27, 2018 Announcements Quiz due tonight Homework 4 out Hopefully Hw 3 graded by end of week Discussion on friday as usual Recap Three dynamic programming problems Weighted interval scheduling Subproblems are prefixes Subset sum Subproblems are prefixes and remaining budget RNA Folding Subproblems are intervals Today Sequence Alignment Shortest paths with negative weights All-pairs shortest paths Sequence Alignment Biologists use genetic similarity to determine evolutionary relationships. But how do we say if two gene sequences are similar or not? We align them. Also used in spell-checkers and search engines. Sequence Alignment Example. TAIL vs TALE For two strings X = x 1 x 2 . . . x m , Y = y 1 y 2 . . . y n , an alignment M is a matching between { 1 , . . . , m } and { 1 , . . . , n } . M is valid if Matching. Each element appears in at most one pair in M . No crossings. If ( i, j ) , ( k, ) M , then i < k and j < . Cost of M : Gap penalty. For each unmatched character, you pay δ . Alignment cost. For a match ( i, j ) , you pay C ( x i , y j ) . cost ( M ) = δ ( m + n - 2 | M | ) + ( i,j ) M C ( x i , y j ) .
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Sequence Alignment Problem. Given strings X, Y gap-penalty δ and cost matrix C , find valid alignment of minimal cost.
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