Algorithms for center and Tverberg points

Agarwal, Pankaj K.; Sharir, Micha; Welzl, Emo

doi:10.1145/997817.997830

Cited by 14 publications

(24 citation statements)

References 18 publications

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“…In this section, we present an O(n 2 log 4 n)-time algorithm for computing the set of points of Tukey depth at least with respect to S for a given value . We achieve our algorithm by modifying the previously best known algorithm for this problem given by Agarwal, Sharir and Welzl [3]. Thus we first provide a sketch of their algorithm.…”

Section: Computing the Center Region In Rmentioning

confidence: 99%

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Computing the center region and its variants

Ahn

2019

Theoretical Computer Science

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We present an O(n 2 log 4 n)-time algorithm for computing the center region of a set of n points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes O(n 2+ ) time for any > 0. It is known that the combinatorial complexity of the center region is Ω(n 2 ) in the worst case, thus our algorithm is almost tight. We also consider the problem of computing a colored version of the center region in the two-dimensional Euclidean space and present an O(n log 4 n)-time algorithm.

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Section: Computing the Center Region In Rmentioning

confidence: 99%

“…The running time of the procedure in[3] appears as O(m polylog(m + |CH(Γj)|)) time. We provide a tighter bound.…”

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confidence: 99%

Computing the center region and its variants

Ahn

2019

Theoretical Computer Science

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“…We now describe the algorithm for computing U * , which is similar to the one used for computing the convex hull of a level in an arrangement of lines [ASW08,Mat91]. We begin by describing a simpler procedure, which will be used as a subroutine in the overall algorithm.…”

Section: β-Hullmentioning

confidence: 99%

“…The following two procedures can be developed by plugging Lemma 5.3 into the parametric-search technique [ASW08,Mat91].…”

Section: β-Hullmentioning

confidence: 99%

Convex Hulls Under Uncertainty

et al. 2016

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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 site 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 β-hull that may be a useful representation of uncertain hulls.

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“…3 We emphasize that this is a rather different problem than that of finding a (geometric) Tverberg point (which is guaranteed to exist) for a given set P of N = (d + 1)(r − 1) + 1 points in R d , as considered, e.g., in [1,21].…”

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Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Mabillard

Wagner

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into R d and study conditions under which such multiple points can be eliminated.The most classical case is that of embeddings (i.e., maps without double points) of a k-dimensional complex K into R 2k . For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen obstruction). For k ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K → R 2k , which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f . One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign.We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y ∈ R d an r-fold Tverberg point of a map f :The analogue of (non-)embeddability that we study is the problem Tverberg r k→d : Given a k-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f :Tverberg point? Here, we show that for fixed r, k and d of the form d = rm and k = (r − 1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings).Our main tool is an r-fold analogue of the Whitney trick: * Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948).Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org. SoCG'14, June 8-11, 2014, Kyoto, Japan. Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM 978-1-4503-2594-3/14/06 ...$15.00.Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y− of opposite intersection sign, we can eliminate y+ and y− by a local isotopy of f .In a subsequent paper, we plan to develop this further and present a generalization of the classical Haefliger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of k-complexes into R d for a wider range of dimensions) to intersection points of higher multiplicity.

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Algorithms for center and Tverberg points

Cited by 14 publications

References 18 publications

Computing the center region and its variants

Computing the center region and its variants

Convex Hulls Under Uncertainty

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

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