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Dynamic Programming

Dynamic Programming. Csc 487/687 Computing for Bioinformatics. Subsequence. A subsequence of a given sequence is just the given sequence with some elements (possible none) left out.

walter-gallegos
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Dynamic Programming

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  1. Dynamic Programming Csc 487/687 Computing for Bioinformatics

  2. Subsequence • A subsequence of a given sequence is just the given sequence with some elements (possible none) left out. • Given a sequence X = <x1,x2,…,xm>, another sequence Z = <z1,z2,…,zk> is a subsequence of X if there exists a strictly increasing sequence <i1,i2,…,ik> of indices of X such that for all j=1,2,…,k, we have xij = zj.

  3. Common subsequence • Given two sequences X and Y, we say that a sequence Z is a common subsequence of X and Y if Z is a subsequence of both X and Y. • Longest-common-subsequence problem(LCS) • We are given two sequences X = <x1,x2,…,xm> and Y = <y1,y2,…,yn> and wish to find a maximum length common subsequence of X and Y.

  4. How to solve it? • Brute force search?

  5. What is dynamic programming? • Divide-and-conquer algorithms? • Not independent? Subproblems share subsubproblems. • Just once and then saves its answer in a table, thereby avoiding the work of recomputing the answer every time

  6. The development of a dynamic programming algorithm • Characterize the structure of an optimal solution. • Recursively define the value of an optimal solution • Compute the value of an optimal solution in bottom-up fashion • Construct an optimal solution from computed information

  7. Optimal-substructure property • Given a sequence X = <x1,x2,…,xm>, we define the ith prefix of X, for i=0,1,…,m, as Xi = <x1,x2,…,xi> . • Optimal substructure of an LCS Let X = <x1,x2,…,xm>, Y = <y1,y2,…,yn> be sequences, and let Z = <z1,z2,…,zk> be any LCS of X and Y • If xm = yn, then zk =xm = yn and Zk-1 is an LCS of Xm-1 and Yn-1 • If xm ≠ yn, then zk ≠xm implies Zis an LCS of Xm-1 and Y. • If xm ≠ yn, then zk ≠yn implies Zis an LCS of Xand Yn-1.

  8. The cost of optimal solution • C [i, j] is the length of an LCS of the sequence Xi and Yj. if either i=0 or j=0, one of the sequences has length of 0, so the LCS has length 0. the optimal substructure of the LCS problem gives the recursive formula 0 if i=0 or j=0 c[i,j]= c[i-1, j-1]+1 if i,j>0 and xi=yj, max(c[i, j-1],c[i-1,j]) if i,j>0 and xi ≠yj,

  9. Example..

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