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DOI: 10.4230/LIPIcs.ICALP.2016.53

Optimization Algorithms for Faster Computational Geometry

Authors: Zeyuan Allen-Zhu, Zhenyu Liao, and Yang Yuan

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

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.

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Zeyuan Allen-Zhu, Zhenyu Liao, and Yang Yuan. Optimization Algorithms for Faster Computational Geometry. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 53:1-53:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{allenzhu_et_al:LIPIcs.ICALP.2016.53,
  author =	{Allen-Zhu, Zeyuan and Liao, Zhenyu and Yuan, Yang},
  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 =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.53},
  URN =		{urn:nbn:de:0030-drops-63325},
  doi =		{10.4230/LIPIcs.ICALP.2016.53},
  annote =	{Keywords: maximum inscribed balls, minimum enclosing balls, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2016.44

Simultaneous Nearest Neighbor Search

Authors: Piotr Indyk, Robert Kleinberg, Sepideh Mahabadi, and Yang Yuan

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract

Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given k query points Q=q_1,...,q_k, and a compatibility graph G with vertices in Q, and the goal is to return data points p_1,...,p_k that minimize (i) the weighted sum of the distances from q_i to p_i and (ii) the weighted sum, over all edges (i,j) in the compatibility graph G, of the distances between p_i and p_j. The problem has several applications in computer vision and databases, where one wants to return a set of *consistent* answers to multiple related queries. Furthermore, it generalizes several well-studied computational problems, including Nearest Neighbor Search, Aggregate Nearest Neighbor Search and the 0-extension problem. In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point q_i we find its (approximate) nearest neighbor point p'_i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q_1,...,q_k and p'_1,...,p'_k; this can be done efficiently given that k is much smaller than n. Even though p'_1,...,p'_k might not constitute the optimal answers to queries q_1,...,q_k, we show that, for the unweighted case, the resulting algorithm satisfies a O(log k/log log k)-approximation guarantee. Furthermore, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice, e.g., 2D grids, 3D grids or planar graphs. Finally, we validate our theoretical results by preliminary experiments. In particular, we show that the empirical approximation factor provided by the above approach is very close to 1.

Cite as

Piotr Indyk, Robert Kleinberg, Sepideh Mahabadi, and Yang Yuan. Simultaneous Nearest Neighbor Search. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{indyk_et_al:LIPIcs.SoCG.2016.44,
  author =	{Indyk, Piotr and Kleinberg, Robert and Mahabadi, Sepideh and Yuan, Yang},
  title =	{{Simultaneous Nearest Neighbor Search}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{44:1--44:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.44},
  URN =		{urn:nbn:de:0030-drops-59360},
  doi =		{10.4230/LIPIcs.SoCG.2016.44},
  annote =	{Keywords: Approximate Nearest Neighbor, Metric Labeling, 0-extension, Simultaneous Nearest Neighbor, Group Nearest Neighbor}
}

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Optimization Algorithms for Faster Computational Geometry

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Simultaneous Nearest Neighbor Search

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