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On approximate range counting and depth. (English) Zbl 1180.68124
Summary: We improve the previous results by B. Aronov and S. Har-Peled [SIAM J. Comput. 38, No. 3, 899–921 (2008; Zbl 1180.68278)] and H. Kaplan and M. Sharir [“Randomized incremental constructions of three-dimensional convex hulls and planar Voronoi diagrams, and approximate range counting”, in: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’06, 484–493 (2006)] 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.

##### MSC:
 68P05 Data structures 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68W20 Randomized algorithms 68W25 Approximation algorithms
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##### References:
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