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New Advances in Rightmost Lempel-Ziv

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String Processing and Information Retrieval (SPIRE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14240))

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Abstract

The Lempel-Ziv (LZ) 77 factorization of a string is a widely-used algorithmic tool that plays a central role in compression and indexing. For a length-n string over a linearly-sortable alphabet, e.g., \(\varSigma = \{1, \dots , \sigma \}\) with \({\sigma = n^{\mathcal O(1)}}\), it can be computed in \(\mathcal O(n)\) time. It is unknown whether this time can be achieved for the rightmost LZ parsing, where each referencing phrase points to its rightmost previous occurrence. The currently best solution takes \({\mathcal O(n (1 + {\log \sigma }/{\sqrt{\log n})})}\) time (Belazzougui & Puglisi SODA2016). We show that this problem is much easier to solve for the LZ-End factorization (Kreft & Navarro DCC2010), where the rightmost factorization can be obtained in \(\mathcal O(n)\) time for the greedy parsing (with phrases of maximal length), and in \(\mathcal O(n + z \sqrt{\log z})\) time for any LZ-End parsing of z phrases. We also make advances towards a linear time solution for the general case. We show how to solve multiple non-trivial subsets of the phrases of any LZ-like parsing in \(\mathcal O(n)\) time. As a prime example, we can find the rightmost occurrence of all phrases of length \(\varOmega (\log ^{6.66} n / \log ^2 \sigma )\) in \(\mathcal O(n / \log _\sigma n)\) time and space.

Supported by Danish Research Council grant DFF-8021-002498.

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Authors and Affiliations

  1. Technical University of Dortmund, Dortmund, Germany

    Jonas Ellert & Johannes Fischer

  2. Technical University of Denmark, DTU Compute, Lyngby, Denmark

    Max Rishøj Pedersen

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  1. Jonas Ellert
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  2. Johannes Fischer
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Correspondence to Jonas Ellert .

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  1. ISTI-CNR, Pisa, Italy

    Franco Maria Nardini

  2. University of Pisa, Pisa, Italy

    Nadia Pisanti

  3. University of Pisa, Pisa, Italy

    Rossano Venturini

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Ellert, J., Fischer, J., Pedersen, M.R. (2023). New Advances in Rightmost Lempel-Ziv. In: Nardini, F.M., Pisanti, N., Venturini, R. (eds) String Processing and Information Retrieval. SPIRE 2023. Lecture Notes in Computer Science, vol 14240. Springer, Cham. https://doi.org/10.1007/978-3-031-43980-3_15

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  • DOI: https://doi.org/10.1007/978-3-031-43980-3_15

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