Efficient Representation and Parallel Computation of String-Substring Longest Common Subsequences

Given two strings a, b of length m, n respectively, the string-substring longest common subsequence (SS-LCS) problem consists in computing the length of the longest common subsequence of a and every substring of b. An explicit representation of the output lengths is of size Θ(n). We show that the output can be represented implicitly by a set of n two-dimensional integer points, where individual output lengths are obtained by dominance counting queries. This leads to a data structure of size O(n), which allows to query an individual output length in time O( logn log logn ), using a recent result by JaJa, Mortensen and Shi. The currently best sequential SS-LCS algorithm by Alves et al. can be adapted to produce the output in the above geometric representation. We also develop a new parallel SS-LCS algorithm that runs on a p-processor coarse-grained computer in O( p ) local computation, O(n log p) communication, O(log p) barrier synchronisations, and O(n) memory per processor, producing the output in the above geometric representation. Compared to previously known results, our approach presents a substantial improvement in algorithm functionality, output representation efficiency, communication efficiency and/or memory efficiency.