TY - GEN
T1 - On the separability of stochastic geometric objects, with applications
AU - Xue, Jie
AU - Li, Yuan
AU - Janardan, Ravi
PY - 2016/6/1
Y1 - 2016/6/1
N2 - In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/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{nN log N, 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.
AB - In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/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{nN log N, 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.
KW - Convex hull
KW - Expected separation-margin
KW - Linear separability
KW - Separable-probability
KW - Stochastic objects
UR - http://www.scopus.com/inward/record.url?scp=84976907246&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84976907246&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2016.62
DO - 10.4230/LIPIcs.SoCG.2016.62
M3 - Conference contribution
AN - SCOPUS:84976907246
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 62.1-62.16
BT - 32nd International Symposium on Computational Geometry, SoCG 2016
A2 - Fekete, Sandor
A2 - Lubiw, Anna
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 32nd International Symposium on Computational Geometry, SoCG 2016
Y2 - 14 June 2016 through 17 June 2016
ER -