Tighter lower bounds on the exact complexity of string matching

Abstract

This paper considers the exact number of character comparisons needed to find all occurrences of a pattern of length m in a text of length n using on-line and general algorithms. For on-line algorithms, a lower bound of about $(1 + \frac{9}{4(m + 1)}) \cdot n$ character comparisons is obtained. For general algorithms, a lower bound of about $(1 + \frac{2}{m + 3}) \cdot n$ character comparisons is obtained. These lower bounds complement an on-line upper bound of about $(1 + \frac{8}{3(m + 1)}) \cdot n$ comparisons obtained recently by Cole and Hariharan. The lower bounds are obtained by finding patterns with interesting combinatorial properties. It is also shown that for some patterns off-line algorithms can be more efficient than on-line algorithms.