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Approximating Tverberg points in linear time for any fixed dimension

Authors:

Wolfgang Mulzer,

Daniel WernerAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 303 - 310

https://doi.org/10.1145/2261250.2261294

Published: 17 June 2012 Publication History

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Abstract

Let P be a d-dimensional n-point set. A Tverberg partition of P is a partition of P into r sets P1, ..., Pr such that the convex hulls ch(P1), ..., ch(Pr) have non-empty intersection. A point in the intersection of the convex hulls is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest.

We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)³ in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy.

References

[1]

T. M. Chan. An optimal randomized algorithm for maximum Tukey depth. In Proc. 15th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 430--436, 2004.

Digital Library

Google Scholar

[2]

B. Chazelle. The discrepancy method: randomness and complexity. Cambridge University Press, Cambridge, 2000.

Digital Library

Google Scholar

[3]

K. L. Clarkson, D. Eppstein, G. L. Miller, C. Sturtivant, and S.-H. Teng. Approximating center points with iterative Radon points. Internat. J. Comput. Geom. Appl., 6(3):357--377, 1996.

Crossref

Google Scholar

[4]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to algorithms. MIT Press, Cambridge, MA, third edition, 2009.

Digital Library

Google Scholar

[5]

L. Danzer, B. Grünbaum, and V. Klee. Helly's theorem and its relatives. In Proc. Sympos. Pure Math., Vol. VII, pages 101--180. Amer. Math. Soc., Providence, R.I., 1963.

Crossref

Google Scholar

[6]

H. Edelsbrunner. Algorithms in combinatorial geometry. Springer-Verlag, Berlin, 1987.

Digital Library

Google Scholar

[7]

S. Jadhav and A. Mukhopadhyay. Computing a centerpoint of a finite planar set of points in linear time. Discrete Comput. Geom., 12(3):291--312, 1994.

Digital Library

Google Scholar

[8]

C. Knauer, H. R. Tiwary, and D. Werner. On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011), volume 9, pages 649--660. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2011.

Google Scholar

[9]

J. Matousek. Lectures on Discrete Geometry. Springer, 2002.

Digital Library

Google Scholar

[10]

G. L. Miller and D. R. Sheehy. Approximate centerpoints with proofs. Comput. Geom. Theory Appl., 43(8):647--654, 2010.

Digital Library

Google Scholar

[11]

C. H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498 -- 532, 1994.

Digital Library

Google Scholar

[12]

R. Rado. A theorem on general measure. J. London Math. Soc., 21:291--300, 1946.

Crossref

Google Scholar

[13]

S.-H. Teng. Points, spheres, and separators: a unified geometric approach to graph partitioning. PhD thesis, School of Computer Science, Carnegie Mellon University, 1992.

Digital Library

Google Scholar

[14]

H. Tverberg. A generalization of Radon's theorem. J. London Math. Soc., 41:123--128, 1966.

Crossref

Google Scholar

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Index Terms

Approximating Tverberg points in linear time for any fixed dimension
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 June 2012

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SoCG '12

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Odeyomi OUde BRoy K(2023)Online Decentralized Multi-Agents Meta-Learning With Byzantine ResiliencyIEEE Access10.1109/ACCESS.2023.329167711(68286-68300)Online publication date: 2023
https://doi.org/10.1109/ACCESS.2023.3291677
Mendes HHerlihy MVaidya NGarg V(2015)Multidimensional agreement in Byzantine systemsDistributed Computing10.1007/s00446-014-0240-528:6(423-441)Online publication date: 1-Dec-2015
https://dl.acm.org/doi/10.1007/s00446-014-0240-5
Vaidya NGarg VFatourou PTaubenfeld G(2013)Byzantine vector consensus in complete graphsProceedings of the 2013 ACM symposium on Principles of distributed computing10.1145/2484239.2484256(65-73)Online publication date: 22-Jul-2013
https://dl.acm.org/doi/10.1145/2484239.2484256
Mulzer WStein Y(2013)Algorithms for Tolerated Tverberg PartitionsAlgorithms and Computation10.1007/978-3-642-45030-3_28(295-305)Online publication date: 2013
https://doi.org/10.1007/978-3-642-45030-3_28

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