Timothy M. Chan's Publications: Statistical depth

On approximate range counting and depth

Peyman Afshani)

Improving previous methods by Aronov and Har-Peled (SODA'05) and Kaplan and Sharir (SODA'06), we present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log (n/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.

An optimal randomized algorithm for maximum Tukey depth

We present the first optimal algorithm to compute the maximum Tukey depth (also known as location or halfspace depth) for a non-degenerate point set in the plane. The algorithm is randomized and requires O(n log n) expected time for n data points. In a higher fixed dimension d >= 3, the expected time bound is O(n^{d-1}), which is probably optimal as well. The result is obtained using an interesting variant of the author's randomized optimization technique, capable of solving "implicit" linear-programming-type problems; some other applications of this technique are briefly mentioned.

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Timothy Chan (Last updated Aug 2023)