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Range Minimum Query Indexes in Higher Dimensions

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Combinatorial Pattern Matching (CPM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9133))

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Abstract

Range minimum queries (RMQs) are essential in many algorithmic procedures. The problem is to prepare a data structure on an array to allow for fast subsequent queries that find the minimum within a range in the array. We study the problem of designing indexing RMQ data structures which only require sub-linear space and access to the input array while querying. The RMQ problem in one-dimensional arrays is well understood with known indexing data structures achieving optimal space and query time. The two-dimensional indexing RMQ data structures have received the attention of researchers recently. There are also several solutions for the RMQ problem in higher dimensions. Yuan and Atallah [SODA’10] designed a brilliant data structure of size \(O(N)\) which supports RMQs in a multi-dimensional array of size \(N\) in constant time for a constant number of dimensions. In this paper we consider the problem of designing indexing data structures for RMQs in higher dimensions. We design a data structure of size \(O(N)\) bits that supports RMQs in constant time for a constant number of dimensions. We also show how to obtain trade-offs between the space of indexing data structures and their query time.

J. Iacono—Research supported by NSF grant CCF-1018370 and BSF grant 2010437.

G.M. Landau—Research partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 571/14, Grant No. 2008217 from the United States- Israel Binational Science Foundation (BSF) and DFG.

M. Lewenstein—Research supported by BSF grant 2010437, a Google Research Award and GIF grant 1147/2011.

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Authors and Affiliations

  1. Polytechnic Institute of New York University, New York, USA

    Pooya Davoodi, John Iacono & Gad M. Landau

  2. Haifa University, Haifa, Israel

    Gad M. Landau

  3. Bar-Ilan University, Ramat Gan, Israel

    Moshe Lewenstein

Authors
  1. Pooya Davoodi
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  2. John Iacono
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  3. Gad M. Landau
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  4. Moshe Lewenstein
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Corresponding author

Correspondence to Moshe Lewenstein .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Verona, Verona, Italy

    Ferdinando Cicalese

  2. Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Ely Porat

  3. Department of Computer Science, University of Salerno, Fisciano, Italy

    Ugo Vaccaro

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Davoodi, P., Iacono, J., Landau, G.M., Lewenstein, M. (2015). Range Minimum Query Indexes in Higher Dimensions. In: Cicalese, F., Porat, E., Vaccaro, U. (eds) Combinatorial Pattern Matching. CPM 2015. Lecture Notes in Computer Science(), vol 9133. Springer, Cham. https://doi.org/10.1007/978-3-319-19929-0_13

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  • DOI: https://doi.org/10.1007/978-3-319-19929-0_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19928-3

  • Online ISBN: 978-3-319-19929-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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