Hyperplane separability and convexity of probabilistic point sets

Martin Fink, John Hershberger, Nirman Kumar, Subhash Suri

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 $d \geq 2$. A probabilistic point in $d$-space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent.

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 problem by a factor of $n$. 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 2013. Finally, we prove lower bounds for the separability problem via a reduction from the $k$-SUM problem, which show in particular that our $O(n^2)$ algorithms for $2$-dimensional separability and $3$-dimensional convex hull membership are nearly optimal.


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DOI: http://dx.doi.org/10.20382/jocg.v8i2a3

ISSN: 1920-180X