Document

# Stabbing Pairwise Intersecting Disks by Five Points

## File

LIPIcs.ISAAC.2018.50.pdf
• Filesize: 0.72 MB
• 12 pages

## Cite As

Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert. Stabbing Pairwise Intersecting Disks by Five Points. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 50:1-50:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.50

## 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Combinatorics
##### Keywords
• Disk graph
• piercing set
• LP-type problem

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Noga Alon and Daniel J. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Adv. Math., 96(1):103-112, 1992.
2. Timothy M. Chan. An optimal randomized algorithm for maximum Tukey depth. In Proc. 15th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 430-436, 2004.
3. Bernard Chazelle. The Discrepancy Method - Randomness and Complexity. Cambridge University Press, Cambridge, 2001.
4. Bernard Chazelle and Jiřı Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms, 21(3):579-597, 1996.
5. Ludwig Danzer. Zur Lösung des Gallaischen Problems über Kreisscheiben in der Euklidischen Ebene. Studia Sci. Math. Hungar., 21(1-2):111-134, 1986.
6. Adrian Dumitrescu and Minghui Jiang. Piercing translates and homothets of a convex body. Algorithmica, 61(1):94-115, 2011.
7. Branko Grünbaum. On intersections of similar sets. Portugal. Math., 18:155-164, 1959.
8. Hugo Hadwiger and Hans Debrunner. Ausgewählte Einzelprobleme der kombinatorischen Geometrie in der Ebene. Enseignement Math. (2), 1:56-89, 1955.
9. Eduard Helly. Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175-176, 1923.
10. Eduard Helly. Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatshefte für Mathematik, 37(1):281-302, 1930.
11. Johann Radon. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Mathematische Annalen, 83(1):113-115, 1921.
12. Raimund Seidel. Small-Dimensional Linear Programming and Convex Hulls Made Easy. Discrete Comput. Geom., 6:423-434, 1991.
13. Micha Sharir and Emo Welzl. A combinatorial bound for linear programming and related problems. Proc. 9th Sympos. Theoret. Aspects Comput. Sci. (STACS), pages 567-579, 1992.
14. Lajos Stachó. Über ein Problem für Kreisscheibenfamilien. Acta Sci. Math. (Szeged), 26:273-282, 1965.
15. Lajos Stachó. A solution of Gallai’s problem on pinning down circles. Mat. Lapok, 32(1-3):19-47, 1981/84.
X

Feedback for Dagstuhl Publishing