On Approximate Range Counting and Depth

Authors:
Peyman Afshani;Timothy M. Chan
Affiliations:
University of Waterloo, School of Computer Science, N2L 3G1, Waterloo, Ontario, Canada;University of Waterloo, School of Computer Science, N2L 3G1, Waterloo, Ontario, Canada
Venue:
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Year:
2009

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Abstract

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.