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Journey to the Center of the Point Set

Published: 31 December 2020 Publication History

Abstract

Let P be a set of n points in R d. For a parameter α ∈ (0,1), an α-centerpoint of P is a point p ∈ R d such that all closed halfspaces containing P also contain at least α n points of P. We revisit an algorithm of Clarkson et al. [1996] that computes (roughly) a 1/(4d2)-centerpoint in Õ(d9) randomized time, where Õ hides polylogarithmic terms. We present an improved algorithm that can compute centerpoints with quality arbitrarily close to 1/d2 and runs in randomized time Õ(d7). While the improvements are (arguably) mild, it is the first refinement of the algorithm by Clarkson et al. [1996] in over 20 years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

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  • (2024)Computing Approximate Centerpoints in Polynomial Time2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00104(1654-1668)Online publication date: 27-Oct-2024

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cover image ACM Transactions on Algorithms
ACM Transactions on Algorithms  Volume 17, Issue 1
January 2021
335 pages
ISSN:1549-6325
EISSN:1549-6333
DOI:10.1145/3446616
  • Editor:
  • Edith Cohen
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 31 December 2020
Accepted: 01 October 2020
Revised: 01 August 2020
Received: 01 September 2019
Published in TALG Volume 17, Issue 1

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Author Tags

  1. Computational geometry
  2. centerpoints
  3. random walks

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  • (2024)Computing Approximate Centerpoints in Polynomial Time2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00104(1654-1668)Online publication date: 27-Oct-2024

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