## Abstract

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint and multipoint uncertainty models. Let S=S_{R}∪S_{B} be a given set of stochastic bichromatic points, and define n=min{|S_{R}|,|S_{B}|} and N=max{|S_{R}|,|S_{B}|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d−1}) time for d≥3 and O(min{nNlogN,N^{2}}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^{d}) time for d≥2. In addition, we give an Ω(nN^{d−1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^{d}) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex hull-related problems.

Original language | English (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Computational Geometry: Theory and Applications |

Volume | 74 |

DOIs | |

State | Published - Oct 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier B.V.

## Keywords

- Convex hull
- Expected separation-margin
- Linear separability
- Separable-probability
- Stochastic objects