The complexity of the Multiple Pattern Matching Problem for random strings

Abstract

International audienceWe 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(s m rm)/m, the complexity rate for the string characters to be read by this algorithm is at most κ UB φ(r) for some constant κ UB. 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 κ LB 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