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On Hardness of Several String Indexing Problems

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Book cover Combinatorial Pattern Matching (CPM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8486))

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Abstract

Let \({\cal D} =\{d_1,d_2,...,d_D\}\) be a collection of D string documents of n characters in total. The two-pattern matching problems ask to index \({\cal D}\) for answering the following queries efficiently.

  • report/count the unique documents containing P 1 and P 2.

  • report/count the unique documents containing P 1 , but not P 2.

Here P 1 and P 2 represent input patterns of length p 1 and p 2 respectively. Linear space data structures with \(O(p_1+p_2+\sqrt{nk}\log^{O(1)} n)\) query cost are already known for the reporting version, where k represents the output size. For the counting version (i.e., report the value k), a simple linear-space index with \(O(p_1+p_2+ \sqrt{n})\) query cost can be constructed in O(n 3/2) time. However, it is still not known if these are the best possible bounds for these problems. In this paper, we show a strong connection between these string indexing problems and the boolean matrix multiplication problem. Based on this, we argue that these results cannot be improved significantly using purely combinatorial techniques. We also provide an improved upper bound for a related problem known as two-dimensional substring indexing.

Work supported in part by the Danish National Research Foundation grant DNRF84 through Center for Massive Data Algorithmics (MADALGO).

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Author information

Authors and Affiliations

  1. Aarhus University, Denmark

    Kasper Green Larsen & Jesper Sindahl Nielsen

  2. University of Waterloo, Canada

    J. Ian Munro & Sharma V. Thankachan

Authors
  1. Kasper Green Larsen
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  2. J. Ian Munro
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  3. Jesper Sindahl Nielsen
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  4. Sharma V. Thankachan
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Editor information

Editors and Affiliations

  1. St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, 191023, St. Petersburg, Russia

    Alexander S. Kulikov

  2. National Research University Higher School of Economics, Bldg 3, Bolshoy Trekhsvyatitelskiy pereulok, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. University of California at San Diego, 9500 Gilman Drive, EBU3b 4236, 92093-0404, La Jolla, CA, USA

    Pavel Pevzner

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Larsen, K.G., Munro, J.I., Nielsen, J.S., Thankachan, S.V. (2014). On Hardness of Several String Indexing Problems. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds) Combinatorial Pattern Matching. CPM 2014. Lecture Notes in Computer Science, vol 8486. Springer, Cham. https://doi.org/10.1007/978-3-319-07566-2_25

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  • DOI: https://doi.org/10.1007/978-3-319-07566-2_25

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07565-5

  • Online ISBN: 978-3-319-07566-2

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