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Full-Fledged Real-Time Indexing for Constant Size Alphabets

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Abstract

In this paper we describe a data structure that supports pattern matching queries on a dynamically arriving text over an alphabet of constant size. Each new symbol can be prepended to T in O(1) worst-case time. At any moment, we can report all occurrences of a pattern P in the current text in \(O(|P|+k)\) time, where |P| is the length of P and k is the number of occurrences. This resolves, under assumption of constant size alphabet, a long-standing open problem of existence of a real-time indexing method for string matching (see Amir and Nor in Real-time indexing over fixed finite alphabets, pp. 1086–1095, 2008).

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Notes

  1. Henceforth, \(\log ^{(3)}n=\log \log \log n\).

  2. For simplicity we assume that \(\log ^{(3)}n\) and \(\log \log n\) are integers and \(\log ^{(3)}n \) divides \(\log \log n\). If this is not the case, we can find \(d'\) and d that satisfy these requirements such that \(\log \log n\le d\le 2\log \log n\) and \(\log ^{(3)}n\le d'\le 2\log ^{(3)}n\).

  3. In fact, the query time is even slightly better.

  4. In fact, it would suffice to store \(3d-1\) most recently read symbols in compact form.

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Acknowledgments

GK has been supported by the Labex Bézout program funded by the French government. This work was done during the visit of YN to the Laboratoire d’Informatique Gaspard Monge, supported by Université Paris-Est Marne-la-Vallée and CNRS. We thank the anonymous reviewers for helpful comments.

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Authors and Affiliations

  1. Laboratoire d’Informatique Gaspard Monge, Université Paris-Est & CNRS, Marne-la-Vallée, France

    Gregory Kucherov

  2. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Yakov Nekrich

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  1. Gregory Kucherov
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  2. Yakov Nekrich
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Correspondence to Yakov Nekrich.

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Kucherov, G., Nekrich, Y. Full-Fledged Real-Time Indexing for Constant Size Alphabets. Algorithmica 79, 387–400 (2017). https://doi.org/10.1007/s00453-016-0199-7

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