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Approximate Range Mode and Range Median Queries

  • Prosenjit Bose
  • Evangelos Kranakis
  • Pat Morin
  • Yihui Tang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3404)

Abstract

We consider data structures and algorithms for preprocessing a labelled list of length n so that, for any given indices i and j we can answer queries of the form: What is the mode or median label in the sequence of labels between indices i and j. Our results are on approximate versions of this problem. Using \(O(\frac{n}{1-\alpha})\) space, our data structure can find in \(O({\rm log}{\rm log}_\frac{1}{\alpha} n)\) time an element whose number of occurrences is at least α times that of the mode, for some user-specified parameter 0 < α< 1. Data structures are proposed to achieve constant query time for α=1/2,1/3 and 1/4, using storage space of O(n log n), O(n log log n) and O(n), respectively. Finally, if the elements are comparable, we construct an \(O(\frac{n}{1-\alpha})\) space data structure that answers approximate range median queries. Specifically, given indices i and j, in O(1) time, an element whose rank is at least \(\alpha \times \lfloor|j-i+1|/2\rfloor\) and at most \((2-\alpha)\times\lfloor|j-i+1|/2\rfloor\) is returned for 0 < α< 1.

Keywords

Storage Space Lookup Table Query Range Query Time Approximation Factor 
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 2005

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Evangelos Kranakis
    • 1
  • Pat Morin
    • 1
  • Yihui Tang
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawa, OntarioCanada

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