Longest Common Subsequence (LCS)

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Longest Common Subsequence (LCS) 張智星 (Roger Jang) jang@mirlab.org http://mirlab.org/jang 多媒體資訊檢索實驗室台灣大學資訊工程系

Longest Common Subsequence Given a string, we can delete some elements to form a subsequence: s1=uvwxyz  s2=uwyz (after deleting v and x) s2 is a subsequence of s1. Longest common subsequence (LCS) The similarity of two string can be define as the length of the LCS between them. Example: abcdefg and xzackdfwgh have acdfg as a longest common subsequence For word suggestion!

Brute-Force Approach to LCS A brute-force solution Enumerate all subsequences of X Test which ones are also subsequences of Y Pick the longest one. Analysis: If X is of length n, then it has 2n subsequences This is an exponential-time algorithm!

DP for LCS: 3-step Formula Quiz!

DP for LCS: Table Filling (1/2) Note that all DP problems can be visualized as path finding…

DP for LCS: Table Filling (2/2) Observations LCS=‘properi’ or ‘propert’ (which is obtained by keeping multiple back-tracking paths) A match occurs when the node has a 45-degree back-tracking path

DP for LCS: Quiz for Table Filling Hints Create a (m+1)*(n+1) matrix for table filling Fill row 0 and column 0 with 0 first to establish the base cases of boundary conditions Fill all the other elements in a layer-by-layer manner.

Quiz Solution To create this plot Download Machine Learning Toolbox Run lcs('about', 'aeiopu', 1) under MATLAB