Discrete & Computational Geometry

, Volume 42, Issue 1, pp 3–21 | Cite as

On Approximate Range Counting and Depth

  • Peyman Afshani
  • Timothy M. Chan


We improve the previous results by Aronov and Har-Peled (SODA’05) and Kaplan and Sharir (SODA’06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in \(O(\log{n\over k})\) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo. In addition, we describe new results for several related problems, including approximate Tukey depth queries in 3D, approximate regression depth queries in 2D, and approximate linear programming with violations in low dimensions.


Range searching Data structures Approximation algorithms Randomized algorithms Statistical depth 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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