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 rm is the number of words of
length m in the dictionary, and ϕ(r)=maxmln(smrm)/m, the
complexity rate for the string characters to be read by this algorithm is at
most κUBϕ(r) for some constant
κUB. 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
κLBϕ(r). This proves the optimality of the
algorithm, improving and correcting previous claims.Comment: 25 pages, 4 figure