Algorithmica (2017) 79:340–367
Convex Hulls Under Uncertainty
Pankaj K. Agarwal
· Sariel Har-Peled
· Hakan Yıldız
· Wuzhou Zhang
Received: 26 March 2015 / Accepted: 29 July 2016 / Published online: 15 August 2016
© Springer Science+Business Media New York 2016
Abstract We study the convex-hull problem in a probabilistic setting, motivated by
the need to handle data uncertainty inherent in many applications, including sensor
databases, location-based services and computer vision. In our framework, the uncer-
tainty of each input point is described by a probability distribution over a ﬁnite number
of possible locations including a null location to account for non-existence of the point.
Our results include both exact and approximation algorithms for computing the proba-
bility of a query point lying inside the convex hull of the input, time–space tradeoffs for
the membership queries, a connection between Tukey depth and membership queries,
as well as a new notion of β-hull that may be a useful representation of uncertain hulls.
Keywords Convex hull · Membership probability · Tukey depth · Uncertainty
A preliminary version of this paper appeared in the proceedings of 22nd European Symposium of
P. Agarwal and W. Zhang were supported by NSF under Grants CCF-09-40671, CCF-10-12254, and
CCF-11-61359, by ARO Grants W911NF-07-1-0376 and W911NF-08-1-0452, and by an ERDC Contract
W9132V-11-C-0003. S. Har-Peled was supported by NSF Grants CCF-09-15984 and CCF-12-17462.
S. Suri and H. Yıldız were supported by NSF Grants CCF-1161495 and CNS-1035917.
Duke University, Durham, NC, USA
University of Illinois at Urbana-Champaign, Champaign, IL, USA
University of California, Santa Barbara, Santa Barbara, CA, USA
Microsoft Corporation, 1 Microsoft Way, Redmond, WA 98052, USA
Apple Inc., 1 Inﬁnite Loop, Cupertino, CA 95014, USA