On the Exact Complexity of String Matching (Extended Abstract)

of Results We investigate the maximal number of character comparisons made by a linear-time string matching algorithm, given a text of length n and a pattern of length m over a general alphabet. We denote it by c(n,m) or approximate it by (1 + C)n, where C is a universal constant. We add the subscript “on-line” when we restrict attention to on-line algorithms and the superscript “1” when we consider algorithms that find only one occurrence of the pattern in the text. It is well known that n 5 c(n, m) 5 2n m + 1 or 0 5 C 5 1. The upper bound was established 20 years ago [16, 181 and no progress has been made for 19 years. The only lower bound has been the obvious one, c(n,m) 2 n. We improve these bounds and determine Con-line exactly. The bounds below contain the period size z of the pattern.