License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.53
URN: urn:nbn:de:0030-drops-63325
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6332/
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Allen-Zhu, Zeyuan ; Liao, Zhenyu ; Yuan, Yang

Optimization Algorithms for Faster Computational Geometry

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LIPIcs-ICALP-2016-53.pdf (0.4 MB)

Abstract

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by m hyperplanes, and the minimum enclosing ball (MinEB) of a set of n points, both in d-dimensional space. We improve the running time of iterative algorithms on

MaxIB from ~O(m*d*alpha^3/epsilon^3) to ~O(m*d + m*sqrt(d)*alpha/epsilon), a speed-up up to ~O(sqrt(d)*alpha^2/epsilon^2), and

MinEB from ~O(n*d/sqrt(epsilon)) to ~O(n*d + n*sqrt(d)/sqrt(epsilon)), a speed-up up to ~O(sqrt(d)).

Our improvements are based on a novel saddle-point optimization framework. We propose a new algorithm L1L2SPSolver for solving a class of regularized saddle-point problems, and apply a randomized Hadamard space rotation which is a technique borrowed from compressive sensing. Interestingly, the motivation of using Hadamard rotation solely comes from our optimization view but not the original geometry problem: indeed, it is not immediately clear why MaxIB or MinEB, as a geometric problem, should be easier to solve if we rotate the space by a unitary matrix. We hope that our optimization perspective sheds lights on solving other geometric problems as well.

BibTeX - Entry

@InProceedings{allenzhu_et_al:LIPIcs:2016:6332,
  author =	{Zeyuan Allen-Zhu and Zhenyu Liao and Yang Yuan},
  title =	{{Optimization Algorithms for Faster Computational Geometry}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{53:1--53:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6332},
  URN =		{urn:nbn:de:0030-drops-63325},
  doi =		{10.4230/LIPIcs.ICALP.2016.53},
  annote =	{Keywords: maximum inscribed balls, minimum enclosing balls, approximation algorithms}
}

Keywords: maximum inscribed balls, minimum enclosing balls, approximation algorithms
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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