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DOI: 10.4230/LIPIcs.SoCG.2018.59

Further Consequences of the Colorful Helly Hypothesis

Authors: Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

Let F be a family of convex sets in R^d, which are colored with d+1 colors. We say that F satisfies the Colorful Helly Property if every rainbow selection of d+1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family F there is a color class F_i subset F, for 1 <= i <= d+1, whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d >= 2 there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in F can be crossed by g(d) lines.

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Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin. Further Consequences of the Colorful Helly Hypothesis. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{martinezsandoval_et_al:LIPIcs.SoCG.2018.59,
  author =	{Mart{\'\i}nez-Sandoval, Leonardo and Rold\'{a}n-Pensado, Edgardo and Rubin, Natan},
  title =	{{Further Consequences of the Colorful Helly Hypothesis}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{59:1--59:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.59},
  URN =		{urn:nbn:de:0030-drops-87726},
  doi =		{10.4230/LIPIcs.SoCG.2018.59},
  annote =	{Keywords: geometric transversals, convex sets, colorful Helly-type theorems, line transversals, weak epsilon-nets, transversal numbers}
}

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DOI: 10.4230/LIPIcs.ESA.2017.4

Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats

Authors: Pankaj K. Agarwal, Natan Rubin, and Micha Sharir

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract

We consider the Approximate Nearest Neighbor (ANN) problem where the input set consists of n k-flats in the Euclidean Rd, for any fixed parameters k<d, and where, for each query point q, we want to return an input flat whose distance from q is at most (1 + epsilon) times the shortest such distance, where epsilon > 0 is another prespecified parameter. We present an algorithm that achieves this task with n^{k+1}(log(n)/epsilon)^O(1) storage and preprocessing (where the constant of proportionality in the big-O notation depends on d), and can answer a query in O(polylog(n)) time (where the power of the logarithm depends on d and k). In particular, we need only near-quadratic storage to answer ANN queries amidst a set of n lines in any fixed-dimensional Euclidean space. As a by-product, our approach also yields an algorithm, with similar performance bounds, for answering exact nearest neighbor queries amidst k-flats with respect to any polyhedral distance function. Our results are more general, in that they also provide a tradeoff between storage and query time.

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Pankaj K. Agarwal, Natan Rubin, and Micha Sharir. Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agarwal_et_al:LIPIcs.ESA.2017.4,
  author =	{Agarwal, Pankaj K. and Rubin, Natan and Sharir, Micha},
  title =	{{Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.4},
  URN =		{urn:nbn:de:0030-drops-78182},
  doi =		{10.4230/LIPIcs.ESA.2017.4},
  annote =	{Keywords: Approximate nearest neighbor search, k-flats, Polyhedral distance functions, Linear programming queries}
}

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Documents authored by Rubin, Natan

Further Consequences of the Colorful Helly Hypothesis

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Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats

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