Pattern-Matching for Strings with Short Descriptions

We consider strings which are succinctly described. The description is in terms of straight-line programs in which the constants are symbols and the only operation is the concatenation. Such descriptions correspond to systems of recurrences or to context-free grammars generating single words. The descriptive size of a string is the length n of a straight-line program (or size of a grammar) which defines this string. Usually the strings of descriptive size n are of exponential length. Fibonacci and Thue-Morse words are examples of such strings. We show that for a pattern P and text T of descriptive sizes n, m, an occurrence of P in T can be found (if there is any) in time polynomial with respect to n. This is nontrivial, since the actual lengths of P and T could be exponential, and none of the known string-matching algorithms is directly applicable. Our first tool is the periodicity lemma, which allows to represent some sets of exponentially many positions in terms of feasibly many a...

• Type: Conference
• Year: 1995
• Where: CPM
• Authors: Marek Karpinski, Wojciech Rytter, Ayumi Shinohara
• Keywords: Combinatorics, CPM 1995, Descriptive Sizes, Exponential Length, Straight-line Program