Stabbing pairwise intersecting disks by five points

Sariel Har-Peled; Haim Kaplan; Wolfgang Mulzer; Liam Roditty; Paul Seiferth; Micha Sharir; Max Willert

doi:10.4230/LIPIcs.ISAAC.2018.50

Stabbing pairwise intersecting disks by five points

Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, Max Willert

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

Original language	English
Title of host publication	29th International Symposium on Algorithms and Computation, ISAAC 2018
Editors	Chung-Shou Liao, Wen-Lian Hsu, Der-Tsai Lee
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	50:1-50:12
ISBN (Electronic)	9783959770941
DOIs	https://doi.org/10.4230/LIPIcs.ISAAC.2018.50
State	Published - 1 Dec 2018
Event	29th International Symposium on Algorithms and Computation, ISAAC 2018 - Jiaoxi, Yilan, Taiwan, Province of China Duration: 16 Dec 2018 → 19 Dec 2018

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	123
ISSN (Print)	1868-8969

Conference

Conference	29th International Symposium on Algorithms and Computation, ISAAC 2018
Country/Territory	Taiwan, Province of China
City	Jiaoxi, Yilan
Period	16/12/18 → 19/12/18

Bibliographical note

Publisher Copyright:
© Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert;

Funding

Partially supported by a NSF AF awards CCF-1421231, and CCF-1217462. 2 Partially supported by DFG grant MU/3501/1 and ERC STG 757609. 3 Partially supported by DFG grant MU/3501/1. 4 Partially supported by ISF Grant 892/13, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Funding Work on this paper was supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF). 1 Partially supported by a NSF AF awards CCF-1421231, and CCF-1217462. 2 Partially supported by DFG grant MU/3501/1 and ERC STG 757609. 3 Partially supported by DFG grant MU/3501/1. 4 Partially supported by ISF Grant 892/13, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work on this paper was supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF).

Funders	Funder number
Blavatnik Research Fund in Computer Science
German–Israeli Science Foundation
Hermann Minkowski-MINERVA Center for Geometry
National Science Foundation	CCF-1421231, CCF-1217462
Horizon 2020 Framework Programme
H2020 European Research Council	MU/3501/1
Engineering Research Centers	STG 757609
Deutsche Forschungsgemeinschaft
Israel Science Foundation	892/13
Tel Aviv University
Israeli Centers for Research Excellence	4/11
National Science Foundation

Keywords

Disk graph
LP-type problem
Piercing set

Access to Document

10.4230/LIPIcs.ISAAC.2018.50

Cite this

Har-Peled, S., Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P., Sharir, M., & Willert, M. (2018). Stabbing pairwise intersecting disks by five points. In C.-S. Liao, W.-L. Hsu, & D.-T. Lee (Eds.), 29th International Symposium on Algorithms and Computation, ISAAC 2018 (pp. 50:1-50:12). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 123). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2018.50

Har-Peled, Sariel ; Kaplan, Haim ; Mulzer, Wolfgang et al. / Stabbing pairwise intersecting disks by five points. 29th International Symposium on Algorithms and Computation, ISAAC 2018. editor / Chung-Shou Liao ; Wen-Lian Hsu ; Der-Tsai Lee. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. pp. 50:1-50:12 (Leibniz International Proceedings in Informatics, LIPIcs).

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abstract = "Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.",

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Har-Peled, S, Kaplan, H, Mulzer, W, Roditty, L, Seiferth, P, Sharir, M & Willert, M 2018, Stabbing pairwise intersecting disks by five points. in C-S Liao, W-L Hsu & D-T Lee (eds), 29th International Symposium on Algorithms and Computation, ISAAC 2018. Leibniz International Proceedings in Informatics, LIPIcs, vol. 123, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 50:1-50:12, 29th International Symposium on Algorithms and Computation, ISAAC 2018, Jiaoxi, Yilan, Taiwan, Province of China, 16/12/18. https://doi.org/10.4230/LIPIcs.ISAAC.2018.50

Stabbing pairwise intersecting disks by five points. / Har-Peled, Sariel; Kaplan, Haim; Mulzer, Wolfgang et al.
29th International Symposium on Algorithms and Computation, ISAAC 2018. ed. / Chung-Shou Liao; Wen-Lian Hsu; Der-Tsai Lee. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. p. 50:1-50:12 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 123).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.

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Har-Peled S, Kaplan H, Mulzer W, Roditty L, Seiferth P, Sharir M et al. Stabbing pairwise intersecting disks by five points. In Liao CS, Hsu WL, Lee DT, editors, 29th International Symposium on Algorithms and Computation, ISAAC 2018. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2018. p. 50:1-50:12. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.ISAAC.2018.50

Stabbing pairwise intersecting disks by five points

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Keywords

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