On Efficient Parallel Algorithms for Solving Set Recurrence Equations

Abstract

We present two EREW PRAM algorithms and one CREW PRAM algorithm for solving set recurrence equations of the type commonly used in dynamic programming solutions to many problems in pattern matching, sequence comparison, and language recognition. All three algorithms run in O(log²n) time. The first EREW PRAM algorithm uses O(M(n/log n)log²n) processors, where M(n) is the number of processors needed to multiply two n × n Boolean matrices in O(log n) time on an EREW PRAM. The second EREW PRAM algorithm uses O(M′(n/(log n[formula]))log²n) processors, where M′(n) is the number of processors needed to multiply two n × n matrices over an arbitrary ring in O(log n) time on an EREW PRAM. The CREW PRAM algorithm uses $\text{O(M″(n/log}{}^{1.5}\text{n )log}{}^2\text{n)}$ processors, where M″(n) is the number of processors needed to multiply two n × n matrices over an arbitrary ring in O(log n) time on a CREW PRAM. As an application, we show that linear context-free languages can be recognized in O(log²n) time using o(M(n)) processors on an EREW PRAM. This result answers an open question in Atallah et al. (in "Proceedings, ACM Symp. on Parallel Alg. and Arch. 1989," pp. 421-431).