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On the Approximation Ratio of LZ-End to LZ77

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String Processing and Information Retrieval (SPIRE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12944))

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Abstract

A family of Lempel-Ziv factorizations is a well-studied string structure. The LZ-End factorization is a member of the family that achieved faster extraction of any substrings (Kreft & Navarro, TCS 2013). One of the interests for LZ-End factorizations is the possible difference between the size of LZ-End and LZ77 factorizations. They also showed families of strings where the approximation ratio of the number of LZ-End phrases to the number of LZ77 phrases asymptotically approaches 2. However, the alphabet size of these strings is unbounded. In this paper, we analyze the LZ-End factorization of the period-doubling sequence. We also show that the approximation ratio for the period-doubling sequence asymptotically approaches 2 for the binary alphabet.

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Notes

  1. 1.

    This version of LZ77 is often called non-overlapping LZ77 or LZ77 without self-references, since each phrase \(p_i\) never overlaps with any of its sources.

  2. 2.

    This definition of LZ77 is different from the original one [33] (see [21] for more information).

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP20J11983 (TM), JP20J21147 (MF), JP18K18002 (YN), JP21K17705 (YN), JP18H04098 (MT), JP20H05964 (MT), and by JST PRESTO Grant Number JPMJPR1922 (SI).

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Authors and Affiliations

  1. Department of Information Science and Technology, Kyushu University, Fukuoka, Japan

    Takumi Ideue

  2. Department of Informatics, Kyushu University, Fukuoka, Japan

    Takuya Mieno, Mitsuru Funakoshi, Yuto Nakashima, Shunsuke Inenaga & Masayuki Takeda

  3. Japan Society for the Promotion of Science, Tokyo, Japan

    Takuya Mieno & Mitsuru Funakoshi

  4. PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan

    Shunsuke Inenaga

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  1. Takumi Ideue
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Correspondence to Takumi Ideue .

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Editors and Affiliations

  1. Université de Rouen Normandie, Mont-St-Aignan, France

    Thierry Lecroq

  2. CNRS, CRIStAL, Villeneuve d'Ascq, France

    Hélène Touzet

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Ideue, T., Mieno, T., Funakoshi, M., Nakashima, Y., Inenaga, S., Takeda, M. (2021). On the Approximation Ratio of LZ-End to LZ77. In: Lecroq, T., Touzet, H. (eds) String Processing and Information Retrieval. SPIRE 2021. Lecture Notes in Computer Science(), vol 12944. Springer, Cham. https://doi.org/10.1007/978-3-030-86692-1_10

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