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Improved Submatrix Maximum Queries in Monge Matrices

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Automata, Languages, and Programming (ICALP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8572))

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Abstract

We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n×n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA‘12] required O(n logn) space and O(log2 n) query time. We also give an alternative data structure with constant query-time and O(n 1 + ε) construction time and space for any fixed ε < 1. For n×n partial Monge matrices we obtain a data structure with O(n) space and O(logn ·α(n)) query time. The data structure of Kaplan et al. required O(n logn ·α(n)) space and O(log2 n) query time. Our improvements are enabled by a technique for exploiting the structure of the upper envelope of Monge matrices to efficiently report column maxima in skewed rectangular Monge matrices. We hope this technique will be useful in obtaining faster search algorithms in Monge partial matrices. In addition, we give a linear upper bound on the number of breakpoints in the upper envelope of a Monge partial matrix. This shows that the inverse Ackermann α(n) factor in the analysis of the data structure of Kaplan et. al is superfluous.

A full version of this paper can be found as Arxiv preprint [18].

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Author information

Authors and Affiliations

  1. MPII, Germany

    Paweł Gawrychowski

  2. IDC Herzliya, Israel

    Shay Mozes

  3. University of Haifa, Israel

    Oren Weimann

Authors
  1. Paweł Gawrychowski
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  2. Shay Mozes
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  3. Oren Weimann
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstrasse 3, 85748, München, Germany

    Javier Esparza

  2. LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Pierre Fraigniaud

  3. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Thore Husfeldt

  4. Department Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Elias Koutsoupias

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Gawrychowski, P., Mozes, S., Weimann, O. (2014). Improved Submatrix Maximum Queries in Monge Matrices. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_44

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  • DOI: https://doi.org/10.1007/978-3-662-43948-7_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43947-0

  • Online ISBN: 978-3-662-43948-7

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