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Linked Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space

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Abstract

The dynamic trie is a fundamental data structure with applications in many areas of computer science. This paper proposes a new technique for maintaining a dynamic trie T of size at most 2w nodes under the unit-cost RAM model with a fixed word size w. It is based on the idea of partitioning T into a set of linked small tries, each of which can be maintained efficiently. Our method is not only space-efficient, but also allows the longest common prefix between any query pattern P and the strings currently stored in T to be computed in o(|P|) time for small alphabets, and allows any leaf to be inserted into or deleted from T in o(log|T|) time. To demonstrate the usefulness of our new data structure, we apply it to LZ-compression. Significantly, we obtain the first algorithm for generating the lz78 encoding of a given string of length n over an alphabet of size σ in sublinear (o(n)) time and sublinear (o(nlogσ) bits) working space for small alphabets (\(\sigma= 2^{o(\log n \frac{\log\log\log n}{(\log\log n)^{2}})}\)). Moreover, the working space for our new algorithm is asymptotically less than or equal to the space for storing the output compressed text, regardless of the alphabet size.

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Notes

  1. In this paper, log denotes the base-2 logarithm, ln is the natural logarithm, and log σ is the base-σ logarithm. Furthermore, ⌈ and ⌉ symbols have been omitted to increase the readability.

  2. We use the term size of a trie to refer to the number of nodes in the trie, not the number of bits used to represent the trie.

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments.

Author information

Authors and Affiliations

  1. Laboratory of Mathematical Bioinformatics (Akutsu Laboratory), Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan

    Jesper Jansson

  2. National Institute of Informatics (NII), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Kunihiko Sadakane

  3. School of Computing, National University of Singapore, 13 Computing Drive, Singapore, 117417, Singapore

    Wing-Kin Sung

  4. Genome Institute of Singapore, 60 Biopolis Street, Genome, Singapore, 138672, Singapore

    Wing-Kin Sung

Authors
  1. Jesper Jansson
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  2. Kunihiko Sadakane
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  3. Wing-Kin Sung
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Corresponding author

Correspondence to Jesper Jansson.

Additional information

Parts of the results presented in this paper appeared in preliminary form in Proceedings of the 27th Annual International Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2007), volume 4855 of Lecture Notes in Computer Science, Springer-Verlag, pp. 424–435, 2007.

J.J. was funded by the Hakubi Project at Kyoto University and the Japan Society for the Promotion of Science (JSPS).

K.S. was partially supported by the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)”.

W.-K.S. was partially supported by the MOE AcRF Tier 2 funding R-252-000-444-112.

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Jansson, J., Sadakane, K. & Sung, WK. Linked Dynamic Tries with Applications to LZ-Compression in Sublinear Time and Space. Algorithmica 71, 969–988 (2015). https://doi.org/10.1007/s00453-013-9836-6

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  • DOI: https://doi.org/10.1007/s00453-013-9836-6

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