Squares, cubes, and time-space efficient string searching

Abstract

We address several technical problems related to the time-space optimal string-matching algorithm of Galil and Seiferas (called the GS algorithm). This algorithm contains a parameterk on which the complexity depends and that originally satisfiesk ≥ 4. We show thatk=3 is the least integer for which the GS algorithm works. This value of the parameterk also minimizes the time of the search phase of the string-searching algorithm. With the parameterk=2 we consider a simpler version of the algorithm working in linear time and logarithmic space. This algorithm is based on the following fact: any word of lengthn starts by less than log_Φ n squares of primitive prefixes. Fibonacci words have a logarithmic number of square prefixes. Hence, the combinatorics of prefix squares and cubes is essential for string-matching with small memory.

We give a time-space optimal sequential computation of the period of a word based on the GS algorithm. The latter corrects the algorithm given in [GS2] for the computation of periods. We present an optimal parallel algorithm for pattern preprocessing. This paper also provides a cleaner version and a simpler analysis of the GS algorithm.