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Optimal Algorithms for Geometric Centers and Depth $\newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$
Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones
We develop a general randomized technique for solving implicit 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 used to obtain faster 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.
PDF. : arXiv : DBLP (SICOMP) : DBLP (SoCG).