Relating the Average-Case Costs of the Brute-Force and Knuth-Morris-Pratt String Matching Algorithm

Abstract

Among the algorithms which are able to check in 0(n+m) steps, if a string PATTERN = p₁p₂...p_m occurs in a string TEXT = t₁t₂...t_n, the one developed by Knuth, Morris and Pratt (KMP, for short) plays kind of a fundamental role. Its linear worst-case time complexity contrasts sharply with the 0(n.m) upper bound for the overhead of the brute-force strategy which naively probes each position i, for 1≤i≤n−m+1, in TEXT for a complete match

$$ {{p}_{1}}{{p}_{2}}...{{p}_{m}}={{t}_{i}}{{t}_{i+1}}...{{t}_{i+m-1}} $$

However, the average-case performance of the KMP algorithm has been suspected not to be drastically better than that of the naive method [9]:

“The Knuth-Morris-Pratt algorithm is not likely to be significantly faster than the brute-force method in most actual applications, ...”

The main objective of this paper is to elaborate on this observation and to present a detailed and accurate average-case analysis of both the brute-force and the KMP algorithm. The analysis exploits results from Markov chain theory. This approach is believed to be practically sound, since string matching can be modeled conveniently by finite-state devices.