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.SoCG.2016.38
URN: urn:nbn:de:0030-drops-59305
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5930/
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Fink, Martin ; Hershberger, John ; Kumar, Nirman ; Suri, Subhash

Hyperplane Separability and Convexity of Probabilistic Point Sets

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Abstract

We describe an O(n^d) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d+1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [Agarwal et al., ESA, 2014]. In addition, our algorithms can handle "input degeneracies" in which more than k+1 points may lie on a k-dimensional subspace, thus resolving an open problem in [Agarwal et al., ESA, 2014]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n^2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.

BibTeX - Entry

@InProceedings{fink_et_al:LIPIcs:2016:5930,
  author =	{Martin Fink and John Hershberger and Nirman Kumar and Subhash Suri},
  title =	{{Hyperplane Separability and Convexity of Probabilistic Point Sets}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{38:1--38:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5930},
  URN =		{urn:nbn:de:0030-drops-59305},
  doi =		{10.4230/LIPIcs.SoCG.2016.38},
  annote =	{Keywords: probabilistic separability, uncertain data, 3-SUM hardness, topological sweep, hyperplane separation, multi-dimensional data}
}

Keywords: probabilistic separability, uncertain data, 3-SUM hardness, topological sweep, hyperplane separation, multi-dimensional data
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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