Longest Common Subsequence from Fragments via Sparse Dynamic Programming

Abstract

Sparse Dynamic Programming has emerged as an essential tool for the design of efficient 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, find the longest common subsequence based only on the symbol correspondences induced by the substrings. This problem arises in an application to analysis of software systems. Our algorithm solves the problem in O(|M| log |M|) time using balanced trees, or O(|M| log log min(|M|,nm/|M|)) time using Johnson’s version of Flat Trees [10]. These bounds apply for two cost measures. The algorithm can also be adapted to finding the usual LCS in O((m + n) log |Σ| + |M|log|M|) using balanced trees or O((m + n)log|Σ| + |M|log log min(|M|; nm/|M|)) using Johnson’s Flat Trees, where M is the set of maximal matches between substrings of X and Y and Σ is the alphabet.

Work Supported in part by Grants from the Italian Ministry of Scientific Research and by the Italian National Research Council. Part of this work was done while the author was visiting Bell Laboratories of Lucent Technologies