Pattern-matching for strings with short descriptions

TLDR: In this article, it was shown that for a pattern P and text T of descriptive sizes m, n, an occurrence of P in T can be found (if there is any) in time polynomial with respect to n.

Abstract: 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 the 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 m, n, 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 arithmetic progressions The second tool is arithmetics: a simple application of Euclid algorithm Hence a textual problem for exponentially long strings is reduced here to simple arithmetics on integers with (only) linearly many bits We present also an NP-complete version of the pattern-matching for shortly described strings