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On Space Efficient Two Dimensional Range Minimum Data Structures

  • Gerth Stølting Brodal
  • Pooya Davoodi
  • S. Srinivasa Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6347)

Abstract

The two dimensional range minimum query problem is to preprocess a static two dimensional m by n array A of size N = m · 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(clog2 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· min {m,logn}) bits with O(1) query time, assuming m ≤ n. This leaves a gap to the lower bound of Ω(Nlogm) bits for this version of the problem.

Keywords

Binary Tree Lookup Table Minimum Element Query Time Additional Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gerth Stølting Brodal
    • 1
  • Pooya Davoodi
    • 1
  • S. Srinivasa Rao
    • 2
  1. 1.MADALGO, Department of Computer ScienceAarhus UniversityÅrhus NDenmark
  2. 2.School of Computer Science and EngineeringSeoul National UniversityS. Korea

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