Periodic Prefixes in Texts

Abstract

We give some new results on the combinatorics of periodic prefixes in texts. This is related to the time-space optimal string-searching algorithm of Galil and Seiferas (called GS algorithm [GS 83]). The algorithm contains a parameter k that says how many times some prefixes repeat at the beginning of the text. The complexity of GS algorithm essentially depends on k and originally the GS algorithm works only for k≥4. We improve on the value of k and we show that k=3 is the least integer for which GS algorithm works. This value of the parameter k also minimizes the time of the search phase of the string-searching algorithm. With parameter k=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 length n starts by less than logon squares of primitive prefixes. Fibonacci words have a logarithmic number of square prefixes. Hence the combinatorics of square prefixes and cube prefixes is essential for string-matching with small memory. We give a time-space optimal sequential computation of the period of a word based on GS algorithm. The latter corrects the algorithm given in [GS 83] for the computation of periods. We also present an optimal parallel algorithm for the pattern preprocessing.

Work by this author is supported by PRC “Mathématiques-Informatique” & NATO Grant CRG 900293.