Optimal Algorithms for Geometric Centers and Depth

TL;DR

This paper introduces a randomized technique for solving implicit linear programming problems and applies it to develop near-optimal algorithms for computing geometric centers like the centerpoint and Tukey median in various dimensions.

Contribution

The paper presents a novel randomized approach for implicit linear programs and provides efficient algorithms for geometric center computations in high-dimensional spaces.

Findings

Algorithms run in $O(n \, \log n)$ expected time for 2D cases.

Expected time for higher dimensions is within one logarithmic factor of $O(n^{d-1})$.

Algorithms are near-optimal for fundamental geometric center problems.

Abstract

$\renewcommand{\Re}{\mathbb{R}}$ We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be exploited in order to obtain efficient linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set $P$ of size $n$ in $\Re^d$, we develop algorithms for computing geometric centers of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For $d=2$, the new algorithms run in $O(n\log n)$ expected time, which is optimal, and for higher constant $d>2$, the expected time bound is within one logarithmic factor of $O(n^{d-1})$, which is also likely near optimal for some of the problems.