Skip to main content
SpringerLink
Log in

Stabbing Pairwise Intersecting Disks by Four Points

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

In their seminal work, Danzer (Stud. Sci. Math. Hungar. 21(1–2), 111–134 (1986)) and Stachó (Mat. Lapok 32(1–3), 19–47 (1981–84)) established that every set D of pairwise intersecting disks in the plane can be stabbed by four points. Recently, Har-Peled et al. (Discrete Math. 344(7), # 112403 (2021)) presented a relatively simple linear-time algorithm for finding five points that stab D. We present an alternative (somewhat less involved) proof to the assertion that four points are sufficient to stab D. Moreover, our proof is constructive and provides a simple linear-time algorithm for finding the stabbing points. As a warmup, we present a nearly-trivial linear-time algorithm with an elementary proof for finding five points that stab D.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Notes

  1. Stachó, on the other hand, starts with the maximum inscribed disk d defined by three arbitrary disks in D that do not have a common point. Notice that d does not necessarily intersect all disks in D.

References

  1. Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Danzer, L.: Zur Lösung des Gallaischen Problems über Kreisscheiben in der Euklidischen Ebene. Stud. Sci. Math. Hungar. 21(1–2), 111–134 (1986)

    MATH  Google Scholar 

  3. Grünbaum, B.: On intersections of similar sets. Portugal. Math. 18, 155–164 (1959)

    MathSciNet  MATH  Google Scholar 

  4. Hadwiger, H., Debrunner, H.: Ausgewählte Einzelprobleme der kombinatorischen Geometrie in der Ebene. Enseign. Math. 1, 56–89 (1955)

    MathSciNet  MATH  Google Scholar 

  5. Har-Peled, S., Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P., Sharir, M., Willert, M.: Stabbing pairwise intersecting disks by five points. Discrete Math. 344(7), # 112403 (2021)

  6. Helly, E.: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math.-Ver. 32, 175–176 (1923)

    MATH  Google Scholar 

  7. Helly, E.: Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatsh. Math. Phys. 37, 281–302 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  8. Löffler, M., van Kreveld, M.J.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. 43(4), 419–433 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4–5), 498–516 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Radon, J.: Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. 83(1–2), 113–115 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stachó, L.: Über ein Problem für Kreisscheibenfamilien. Acta Sci. Math. (Szeged) 26, 273–282 (1965)

    MathSciNet  MATH  Google Scholar 

  12. Stachó, L.: A solution of Gallai’s problem on pinning down circles. Mat. Lapok 32(1–3), 19–47 (1981–84)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Paz Carmi & Matthew J. Katz

  2. School of Computer Science, Carleton University, Ottawa, Canada

    Pat Morin

Authors
  1. Paz Carmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew J. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pat Morin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paz Carmi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carmi, P., Katz, M.J. & Morin, P. Stabbing Pairwise Intersecting Disks by Four Points. Discrete Comput Geom 70, 1751–1784 (2023). https://doi.org/10.1007/s00454-023-00567-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-023-00567-0

Keywords

Mathematics Subject Classification

Search

Navigation