The complexity of the Multiple Pattern Matching Problem for random strings.

We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $phi(r) = max_m ln(s, m, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $kappa_{{}_textrm{UB}}, phi(r)$ for some constant $kappa_{{}_textrm{UB}}$. On the other side, 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 $kappa_{{}_textrm{LB}}, phi(r)$. This proves the optimality of the algorithm, improving and correcting previous claims.