Tighter upper bounds on the exact complexity of string matching

Abstract

This paper considers how many character comparisons are needed to find all occurrences of a pattern of length m in a text of length n. The main contribution is to show an upper bound of the form of n + O (n/m) character comparisons, following preprocessing. Specifically, we show an upper bound of $n + \frac{8}{3(m+1)}(n-m)$ character comparisons. This bound is achieved by an online algorithm which performs O (n) work in total and requires O (m) space and O (m2) time for preprocessing. The current best lower bound for online algorithms is $n + \frac {16}{7m+27}(n-m)$ character comparisons for $m = 16k + 19$, for any integer $k\geq 1$ and for general algorithms is $n+\frac{2}{m+3}(n-m)$ character comparisons, for $m = 2k+1$, for any integer $k\geq 1$.