On space efficient two dimensional range minimum data structures

  • Authors:

  • Gerth Stølting Brodal;Pooya Davoodi;S. Srinivasa Rao

  • Affiliations:

  • MADALGO, Department of Computer Science, Aarhus University, IT Parken, Århus N, Denmark;MADALGO, Department of Computer Science, Aarhus University, IT Parken, Århus N, Denmark;School of Computer Science and Engineering, Seoul National University, S. Korea

  • Venue:
  • ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
  • Year:
  • 2010

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Abstract

The two dimensional range minimum query problem is to preprocess a static two dimensional m by n array A of size N = m ċ n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using O(N/c) bits additional space requires Ω(c) query time, for any c where 1 ≤ c ≤ N. This lower bound holds for any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits additional space which can be preprocessed in O(N) time and achieves O(c log2 c) query time. For c = O(1), this is the first O(1) query time algorithm using optimal O(N) bits additional space. For the case where queries can not probe A, we give a data structure of size O(N ċ min{m, log n}) bits with O(1) query time, assuming m ≤ n. This leaves a gap to the lower bound of Ω(N log m) bits for this version of the problem.