Efficient parallel algorithms for solving set recurrence equations and applications

The authors 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 algorithm uses O(M\´(n/(log n √loglog n))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 O(M"(n/log^1.5n)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 a CREW PRAM. The authors show that linear context-free languages can be recognized in O(log²n) time using o(M(n)) processors on an EREW PRAM. They give applications to other problems such as string shuffling, recognition of languages accepted by two-head nondeterministic finite automata, and transductions defined by two-tape nondeterministic finite transducers