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Top Tree Compression of Tries

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Abstract

We present a compressed representation of tries based on top tree compression [ICALP 2013] that works on a standard, comparison-based, pointer machine model of computation and supports efficient prefix search queries. Namely, we show how to preprocess a set of strings of total length n over an alphabet of size \(\sigma\) into a compressed data structure of worst-case optimal size \(O(n/\log _\sigma n)\) that given a pattern string P of length m determines if P is a prefix of one of the strings in time \(O(\min (m\log \sigma ,m + \log n))\). We show that this query time is in fact optimal regardless of the size of the data structure. Existing solutions either use \(\Omega (n)\) space or rely on word RAM techniques, such as tabulation, hashing, address arithmetic, or word-level parallelism, and hence do not work on a pointer machine. Our result is the first solution on a pointer machine that achieves worst-case o(n) space. Along the way, we develop several interesting data structures that work on a pointer machine and are of independent interest. These include an optimal data structures for random access to a grammar-compressed string and an optimal data structure for a variant of the level ancestor problem.

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Notes

  1. Here we use edge labels instead of nodes label. The two definitions are equivalent and edge labels are more natural for tries.

  2. Here \(\alpha _k(n)\) for any constant k denotes the inverse of the \(k^{th}\) row of Ackermann’s function, defined as \(\alpha _k(n)=1+\alpha _k(\alpha _{k-1}(n))\) so that \(\alpha _1(n)= n/2\), \(\alpha _2(n)=\log n\), \(\alpha _3(n)=\log ^* n\), and so on.

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Authors and Affiliations

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

    Philip Bille & Inge Li Gørtz

  2. University of Wrocław, Wrocław, Poland

    Paweł Gawrychowski

  3. University of Haifa, Haifa, Israel

    Gad M. Landau & Oren Weimann

  4. NYU Tandon School of Engineering, New York University, New York, USA

    Gad M. Landau

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  1. Philip Bille
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  2. Paweł Gawrychowski
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  4. Gad M. Landau
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Correspondence to Philip Bille.

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Philip Bille and Inge Li Gørtz: Supported by the Danish Research Council (DFF—4005-00267, DFF—1323-00178). Gad M. Landau: Supported by the Israel Science Foundation Grants 1475/18, and No. 2018141 from the United States-Israel Binational Science Foundation. Oren Weimann: Supported by the Israel Science Foundation Grant 592/17.

An extended abstract appeared at ISAAC 2019 [17].

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Bille, P., Gawrychowski, P., Gørtz, I.L. et al. Top Tree Compression of Tries. Algorithmica 83, 3602–3628 (2021). https://doi.org/10.1007/s00453-021-00869-w

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