# Hyperplane separability and convexity of probabilistic point sets

## Abstract

We describe an O(nd) 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 [6]. 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 [6]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.

## Publication Title

Leibniz International Proceedings in Informatics, LIPIcs

## Recommended Citation

Fink, M., Hershberger, J., Kumar, N., & Suri, S.
(2016). Hyperplane separability and convexity of probabilistic point sets.* Leibniz International Proceedings in Informatics, LIPIcs**, 51*, 38.1-38.16.
https://doi.org/10.4230/LIPIcs.SoCG.2016.38