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Cell Probe Lower Bounds and Approximations for Range Mode

  • Mark Greve
  • Allan Grønlund Jørgensen
  • Kasper Dalgaard Larsen
  • Jakob Truelsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

The mode of a multiset of labels, is a label that occurs at least as often as any other label. The input to the range mode problem is an array A of size n. A range query [i,j] must return the mode of the subarray A[i],A[i + 1],...,A[j]. We prove that any data structure that uses S memory cells of w bits needs \(\Omega(\frac{{\rm log} n}{\log (Sw/n)})\) time to answer a range mode query. Secondly, we consider the related range k-frequency problem. The input to this problem is an array A of size n, and a query [i,j] must return whether there exists a label that occurs precisely k times in the subarray A[i],A[i + 1],...,A[j]. We show that for any constant k > 1, this problem is equivalent to 2D orthogonal rectangle stabbing, and that for k = 1 this is no harder than four-sided 3D orthogonal range emptiness. Finally, we consider approximate range mode queries. A c-approximate range mode query must return a label that occurs at least 1/c times that of the mode. We describe a linear space data structure that supports 3-approximate range mode queries in constant time, and a data structure that uses \(O(\frac{n}{\varepsilon})\) space and supports (1 + ε)-approximation queries in \(O({\rm log} {\frac {1}{\varepsilon}})\) time.

Keywords

Data Structure Range Query Query Time Range Mode Answer Query 
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

  • Mark Greve
    • 1
  • Allan Grønlund Jørgensen
    • 1
  • Kasper Dalgaard Larsen
    • 1
  • Jakob Truelsen
    • 1
  1. 1.MADALGO, Department of Computer ScienceAarhus UniversityDenmark

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