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Convex Hulls Under Uncertainty

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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 uncertainty 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 probability 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 \(\beta \)-hull that may be a useful representation of uncertain hulls.

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Notes

  1. Although we only consider a discrete distribution for the possible locations of an uncertain point, one can also use a continuous distribution, described by a non-negative function \(\pi \) such that \(\int _{\mathbb {R}^d} \pi (x) \mathrm {d}x \le 1\); we allow the total measure to be less than 1 to account for the possibility of the point not existing.

  2. Some progress has been made on this problem since the original submission of our paper. In [16], Fink et al. present an \(O(n^{d-1})\) time algorithm for computing the membership probability in dimension \(d \ge 3\), which can also handle degenerate inputs.

  3. Collinearities strictly among input points do not need any special care in our algorithm.

  4. If \(B\) consists of a single site \(p_{i}\), then \(C\) is the line segment \({qp_{i}}\). In this case, we consider the boundary of \(C\) to be a cycle formed by two edges: one going from \(q\) to \(p_{i}\), and one going from \(p_{i}\) back to \(q\).

  5. Any degeneracies that are strictly among P and do not involve \(q\) have no side effect on our algorithm, similar to the planar case.

  6. If there are multiple points with the same \(x_d\)-coordinate, we further order them using their indices (the i in \(p_{i}\)) and break the ties between them.

  7. Notice that dimension of the condition \(p_{i} =\lambda ({\mathsf {R}\cup \{q\}})\) stays the same throughout all levels of the recursion. In other words, at all levels, we condition on \(p_{i}\) being the lowest point along \(x_d\)-axis and set \(G_{2}\) to be the set of sites below \(p_{i}\) along \(x_d\)-axis. Dimension reduction happens only for the second part of condition (\(q\in V\)). Also notice that we do not need to condition on the lowest point on any other dimension other than the dth dimension as we go down the recursion.

  8. For ease of presentation, we assume that the arrangement is non-degenerate. It is straightforward to apply our technique on degenerate arrangements by using standard techniques (such as perturbation) to create a non-degenerate arrangement. We note that, even if we perturb the points to create a non-degenerate arrangement, we still use the old coordinates of the points and utilize the degeneracy handling rules of Sect. 2.3 while computing probabilities.

  9. Haussler and Welzl [19] defined \({\varepsilon }\)-net for general range spaces and proved the bound on the size of \({\varepsilon }\)-nets for range spaces with finite VC-dimension but we need their result for this special case.

  10. In [7, Section 2.4], the running time for (B) was stated as \(O(n\mathop {\mathrm {polylog}}(n))\) since the authors did not aim for the most efficient implementation.

  11. The supporting vertex of \(\ell \) on \(\mathcal {U}^*\) is the vertex on which a line parallel to \(\ell \) is tangent to \(\mathcal {U}^*\).

  12. If \(|L_\triangle | < m\), we specify only those values of \(\beta _\triangle ^i\) for which \(L_\triangle \cap P_i^*\ne \emptyset \).

  13. We note that up to three subproblems might be solved for a fixed \(\tau \in \varXi \), which is why we did not define one subproblem per triangle of \(\varXi \).

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Author information

Authors and Affiliations

  1. Duke University, Durham, NC, USA

    Pankaj K. Agarwal

  2. University of Illinois at Urbana-Champaign, Champaign, IL, USA

    Sariel Har-Peled

  3. University of California, Santa Barbara, Santa Barbara, CA, USA

    Subhash Suri

  4. Microsoft Corporation, 1 Microsoft Way, Redmond, WA, 98052, USA

    Hakan Yıldız

  5. Apple Inc., 1 Infinite Loop, Cupertino, CA, 95014, USA

    Wuzhou Zhang

Authors
  1. Pankaj K. Agarwal
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  2. Sariel Har-Peled
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  3. Subhash Suri
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  4. Hakan Yıldız
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  5. Wuzhou Zhang
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Corresponding author

Correspondence to Wuzhou Zhang.

Additional information

A preliminary version of this paper appeared in the proceedings of 22nd European Symposium of Algorithms (ESA’14).

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.

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Agarwal, P.K., Har-Peled, S., Suri, S. et al. Convex Hulls Under Uncertainty. Algorithmica 79, 340–367 (2017). https://doi.org/10.1007/s00453-016-0195-y

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