Unified Compression-Based Acceleration of Edit-Distance Computation

• Published in: Algorithmica Vol. 65; no. 2; pp. 339 - 353
• Main Authors: Hermelin, Danny, Landau, Gad M., Landau, Shir, Weimann, Oren
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.02.2013; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Artificial intelligence; Coding, codes; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Information, signal and communications theory; Learning and adaptive systems; Mathematics of Computing; Pattern recognition; Signal and communications theory; Signal processing; Telecommunications and information theory; Theoretical computing; Theory of Computation; Edit-distance; Straight-line-programming; Data compression; Combinatorial pattern matching; Dictionaries; Worst case method; String matching; Knowledge representation; Logical programming; RLE encoding; Circumscription; Upper bound; Character string; Answer set programming; Dynamic programming; Rational function; Pattern matching; Distance
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-011-9590-6

The edit distance problem is a classical fundamental problem in computer science in general, and in combinatorial pattern matching in particular. The standard dynamic programming solution for this problem computes the edit-distance between a pair of strings of total length O ( N ) in O ( N 2 ) time. To this date, this quadratic upper-bound has never been substantially improved for general strings. However, there are known techniques for breaking this bound in case the strings are known to compress well under a particular compression scheme. The basic idea is to first compress the strings, and then to compute the edit distance between the compressed strings. As it turns out, practically all known o ( N 2 ) edit-distance algorithms work, in some sense, under the same paradigm described above. It is therefore natural to ask whether there is a single edit-distance algorithm that works for strings which are compressed under any compression scheme. A rephrasing of this question is to ask whether a single algorithm can exploit the compressibility properties of strings under any compression method, even if each string is compressed using a different compression. In this paper we set out to answer this question by using straight line programs . These provide a generic platform for representing many popular compression schemes including the LZ-family, Run-Length Encoding, Byte-Pair Encoding, and dictionary methods. For two strings of total length N having straight-line program representations of total size n , we present an algorithm running in O ( nN lg( N / n )) time for computing the edit-distance of these two strings under any rational scoring function, and an O ( n 2/3 N 4/3 ) time algorithm for arbitrary scoring functions. Our new result, while providing a speed up for compressible strings, does not surpass the quadratic time bound even in the worst case scenario.