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Repetition Complexity of Words

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Computing and Combinatorics (COCOON 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2387))

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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 w, r(w), is defined as the smallest amount of space needed to store w when reduced by repeatedly applying the following procedure: n consecutive occurrences uu... u of the same subword u of w 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(w) ≥ lz(w) 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)) = \( \mathcal{O} \) (n), (ii) lz(prefn(α)) = \( \mathcal{O} \) (1), and (iii) r(prefn(α)) = lgn + \( \mathcal{O} \) (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(w) can be computed in time \( \mathcal{O} \) (n 3(logn)2) and it is open, and probably very difficult, to find very fast algorithms.

Research partially supported by NSERC grant R3143A01.

Research partially supported by NSERC grant OGP0041630.

Research partially supported by NSERC grant OGP0046373.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Western Ontario, N6A 5B7, London, Ontario, Canada

    Lucian Ilie, Sheng Yu & Kaizhong Zhang

Authors
  1. Lucian Ilie
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  2. Sheng Yu
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  3. Kaizhong Zhang
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, California, 93106, USA

    Oscar H. Ibarra

  2. Department of Mathematics, National University of Singapore, Singapore, Singapore, 117543

    Louxin Zhang

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© 2002 Springer-Verlag Berlin Heidelberg

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Ilie, L., Yu, S., Zhang, K. (2002). Repetition Complexity of Words. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_35

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  • DOI: https://doi.org/10.1007/3-540-45655-4_35

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43996-7

  • Online ISBN: 978-3-540-45655-1

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