Average complexity of exact and approximate multiple string matching

We show that the average number of characters examined to search for r random patterns of length m in a text of length n over a uniformly distributed alphabet of size a cannot be less than Omega(n log(sigma)(rm)/m). When we permit up to k insertions, deletions, and/or substitutions of characters in the occurrences of the patterns, the lower bound becomes Omega(n(k + log(sigma)(rm))/m). This generalizes previous single-pattern lower bounds of Yao (for exact matching) and of Chang and Marr (for approximate matching), and proves the optimality of several existing multipattern search algorithms. (C) 2004 Elsevier B.V. All rights reserved.