The complexity of the multiple pattern matching problem for random strings

Abstract

We generalise a multiple string pattern matching algorithm, proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size s. If rm is the number of words of length m in the dictionary, and φ(r) = maxm ln(smrm)/m, the complexity rate for the string characters to be read by this algorithm is at most κUB φ(r) for some constant κUB. Then, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with arbitrary dictionaries, and determine it to be at least κLB φ(r). This proves the optimality of the algorithm, improving and correcting previous claims. Furthermore, we establish a tightness result for dictionaries with the same set {rm}: the worst-case, average-case, and best-case complexities (the latter, up to a finite fraction of the dictionaries) are all equal, up to a finite multiplicative constant.