Extending the Four Russian Algorithm to Compute the Edit Script in Linear Space

Computing the Edit Distance between two strings is one of the most fundamental problems in computer science. Algorithms based on edit distance are used extensively in the alignment of biological sequences. Any improvement either in time or space in solving this problem will be highly desirable. The standard dynamic programming based algorithm to compute the edit distance of two strings S1 = [a1, a2, a2 . . . an] and S2 = [b1, b2, b3 . . . bn] takes O(n2) time. This algorithm computes the value of the edit distance using O(n2) space. Within these resource bounds, it also computes the actual edit script (i.e., a sequence of Inserts, deletes, and changes that transfroms S1 to S2). Often the edit script is more important for several problems (such as sequence alignment) than the value of the edit distance. The first major improvement in the asymptotic runtime for computing the value of the edit distance was acheived in [KF70]. This algorithm is widely known as the Four Russian Algorithm and it improves the running time by a factor of O(log n) (with a run time of O(n2/ log n)) to compute just the value of the edit distance. It does not address the problem of computing the actual edit script, which is of wider interest rather than just the value. Hirschberg [Hir77] has given an algorithm that computes the actual script in O(n2) time. In paper [RA03] linear space parallel algorithms for the sequence alignment problem were given, however they assume that O(n2) is the optimal asymptotic complexity of the sequential algorithm. In this paper we present algorithms that compute both the edit script and value in O( n 2 logn) time using O(n) space.