An optimal randomized algorithm for maximum Tukey depth

Abstract

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 fiexed dimension d ≥ 3, the expected time bound is O(nd-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.