Work

Optimal Algorithms for Geometric Centers and Depth

Year: 2019

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.1912.01639

Authors: Chan, Timothy M., Har-Peled, Sariel, Jones, Mitchell

Institution: University of Illinois Urbana-Champaign

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: AF: Small: Towards Sturdier Geometric Algorithms, AF: Small: Towards better geometric algorithms: Summarizing, partitioning and shrinking data, AF: Small: Fundamental Problems in Geometric Data Structures

OpenAlex ID: W4287999767

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