Longest Common Subsequence from Fragmentsvia Sparse Dynamic

Sparse Dynamic Programming has emerged as an essential tool for the design of eecient algorithms for optimization problems coming from such diverse areas as Computer Science, Computational Biology and Speech Recognition 7, 11, 15]. We provide a new Sparse Dynamic Programming technique that extends the Hunt-Szymanski 2, 9, 8] paradigm for the computation of the Longest Common Subsequence (LCS) and apply it to solve the LCS from Fragments problem: given a pair of strings X and Y (of length n and m, resp.) and a set M of matching substrings of X and Y , nd the longest common subse-quence based only on the symbol correspondences induced by the sub-strings. This problem arises in an application to analysis of software systems. Our algorithm solves the problem in O(jMj log jM j) time using balanced trees, or O(jMj log log min(jMj; nm=jMj)) time using John-son's version of Flat Trees 10]. These bounds apply for two cost measures. The algorithm can also be adapted to nding the usual LCS in O((m + n) log jj + jM j log jM j) using balanced trees or O((m + n) log jj + jM j log log min(jMj; nm=jMj)) using Johnson's Flat Trees, where M is the set of maximal matches between substrings of X and Y and is the alphabet.