On the separability of stochastic geometric objects, with applications

Jie Xue, Yuan Li, Ravi Janardan

Research output: Contribution to journalArticle

1 Scopus citations

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=SR∪SB be a given set of stochastic bichromatic points, and define n=min⁡{|SR|,|SB|} and N=max⁡{|SR|,|SB|}. We show that the separable-probability (SP) of S can be computed in O(nNd−1) time for d≥3 and O(min⁡{nNlog⁡N,N2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nNd) time for d≥2. In addition, we give an Ω(nNd−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(nNd) and O(nNd+1) time, respectively. Finally, we present some applications of our algorithms to stochastic convex hull-related problems.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalComputational Geometry: Theory and Applications
Volume74
DOIs
StatePublished - Oct 2018

Keywords

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

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