On Maximal Suffices and Constant-Space Linear-Time Versions of KMP Algorithm

Abstract

We investigate several simple ways of transforming Knuth- Morris-Pratt algorithm (KMP) into a constant-space and still linear time string-matching algorithm. We also identify a class of very special patterns for which the transformation is particularly simple and show usefulness of the class. Constant-space linear-time string-matching algorithms are usually very sophisticated. Most of them consist of two phases: (very technical) preprocessing phase and searching phase. An exception is onephase Crochemore’s algorithm [2]. It is an on-line version of KMP algorithm with “on-the-fly” computation of pattern shifts (as approximate periods). We explore further Crochemore’s approach, and construct alternative algorithms which are differently structured. In Crochemore’s algorithm the approximate-period function is restarted from inside, which means that several internal variables of this function are changing globally, also Crochemore’s algorithm strongly depends on the concrete implementation of approximate-periods computation. We present a simple modification of KMP algorithm which works in O(1)-space, O(n)-time for any function which computes periods or approximate periods in O(1)- space, and linear time. The approximate-period function can be treated as a black box. We show also that lexicographically self-maximal patterns are especially well suited for Crochemore-style string matching. A new O(1) space string-matching algorithm, MaxSuffix-Matching, is proposed in the paper, which gives yet another example of applicability of maximal suffices.