# dpkt - Dynamic Programming Optimal substructure property...

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Dynamic Programming Optimal substructure property. Characterizing the solution. Recurrence. Bottom up computation. 1

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Sequence alignment Let X = h x 1 ,x 2 ,...,x m i and Y = h y 1 ,y 2 ,...,y n i be two sequences of characters from a ﬁnite alphabet Σ. Consider the index sets: { 1 , 2 , ··· ,m } and { 1 , 2 , ··· ,n } . A matching is a collection of ordered pairs from these two sets such that for any two pairs ( i,j ) and ( i 0 ,j 0 ) in the matching, we have i 6 = i 0 and j 6 = j 0 . An alignment is a matching in which there are no crossing pairs: for any two pairs ( i,j ) and ( i 0 ,j 0 ) in the matching, if i < i 0 then j < j 0 . With every alignment associate a cost as follows: For each position of X or Y that is not matched there is a gap cost of δ . For each ( i,j ) in the alignment there is a mismatch cost of α X i Y j . Assume that α pp = 0 for all characters from the alphabet Σ. The cost of the alignment is the sum of the gap costs and the mismatch costs. Problem: Given two sequences X and Y ﬁnd an alignment of minimum cost. 2
Sequence alignment: Characterization of a solution Let X 1 ..i = h x 1 , ··· ,x i i and Y 1 ..j = h y 1 , ··· ,y j i for 1 i m and 1 j n . Let opt ( i,j ) denote the minimum cost of an alignment of X 1 ..i with Y 1 ..j . First we need the following fact: Fact: In any alignment of X 1 ..i with Y 1 ..j , if the character x i is not matched with the character y j then either x i or y j has no matching in the alignment. Let

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dpkt - Dynamic Programming Optimal substructure property...

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