Fink, Martin ;
Hershberger, John ;
Kumar, Nirman ;
Suri, Subhash
Hyperplane Separability and Convexity of Probabilistic Point Sets
Abstract
We describe an O(n^d) time algorithm for computing the exact probability that two ddimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in dspace is the usual point, but with an associated (independent) probability of existence. We also show that the ddimensional 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 kdimensional 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 kSUM problem, which shows in particular that our O(n^2) algorithms for 2dimensional separability and 3dimensional 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 = {38:138:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5930},
URN = {urn:nbn:de:0030drops59305},
doi = {10.4230/LIPIcs.SoCG.2016.38},
annote = {Keywords: probabilistic separability, uncertain data, 3SUM hardness, topological sweep, hyperplane separation, multidimensional data}
}
2016
Keywords: 

probabilistic separability, uncertain data, 3SUM hardness, topological sweep, hyperplane separation, multidimensional data 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016)

Issue date: 

2016 
Date of publication: 

2016 