Repetition complexity of words

Abstract

With ideas from data compression and combinatorics on words, we introduce a complexity measure for words, called repetition complexity, which quantifies the amount of repetition in a word. The repetition complexity of ω, R(ω), is defined as the smallest amount of space needed to store ω when reduced by repeatedly applying the following procedure: n consecutive occurrences uu . . . u of the same subword u of ω are stored as (u, n). The repetition complexity has interesting relations with well-known complexity measures, such as subword complexity, sub, and Lempel-Ziv complexity, lz. We have always R(ω) ≥ lz(ω) and could even be that the former is linear while the latter is only logarithmic; e.g., this happens for prefixes of certain infinite words obtained by iterated morphisms. An infinite word α being ultimately periodic is equivalent to: (i) sub(prefn(α)) = O(n), (ii) lz(prefn(α)) = O(1), and (iii) r(prefn(α)) = lgn + O(1). De Bruijn words, well known for their high subword complexity are shown to have almost highest repetition complexity; the precise complexity remains open. R(ω) can be computed in time O(n3(log n)2) and it is open, and probably very difficult, to find very fast algorithms.