Skip to main content
SpringerLink
Log in

Illuminating Both Sides of Line Segments

  • Conference paper
  • First Online:
Discrete and Computational Geometry (JCDCG 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2098))

Included in the following conference series:

Abstract

What is the minimum number of light sources that can collectively illuminate both sides of n disjoint line segments in the plane? We prove that this optimization problem is NP-hard. The worst case analysis shows, however, that ⌊4(n + 1)/5⌋ light sources are always enough and sometimes necessary for all n ≥2.

This problem was motivated by an open problem posed by Czyzowicz et al.: what is the minimal number of light sources that can collectively illuminate any set of n disjoint segments from one side at least.

The author acknowledges support from the Berlin-Zürich European Graduate Program “Combinatorics, Geometrym, and Computation”.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baker, B. S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Math. 41 (1994) 153–180

    MATH  Google Scholar 

  2. Berge, C.: Sur le couplage maximum d’un graphe. C. R. Acad. Sciences, Paris 247 (1958) 258–259

    MATH  MathSciNet  Google Scholar 

  3. Berman, F., Johnson, D., Leighton, T., Shor, P. W., Snyder, L.: Generalized planar matching. J. Algorithms 11 (1990) 153–184.

    Article  MATH  MathSciNet  Google Scholar 

  4. Czyzowicz, J., Rival, I., Urrutia, J.: Galleries, light matchings, and visibility graphs. In: Galleries, light matchings and visibility graphs. Algorithms and data structures (Ottawa, ON, 1989), Lecture Notes in Comput. Sci. 382, Springer-Verlag, Berlin, (1989) 316–324

    Google Scholar 

  5. Czyzowicz, J., Rivera-Campo, E., Urrutia, J.: Illuminating rectangles and triangles in the plane. J. Combin. Theory Ser. B (1993) 1–17

    Google Scholar 

  6. Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., Zaks, J.: Tight bounds for the rectangular art gallery problem, in: Graph-theoretic concepts in computer science (Fischbachau, 1991), Lecture Notes in Comput. Sci. 570, Springer-Verlag, Berlin, (1992) 105–112

    Google Scholar 

  7. Czyzowicz, J., Rivera-Campo, E., Urrutia, J., Zaks, J.: On illuminating line segments in the plane. Discrete Math. 137 (1995) 147–153

    Article  MathSciNet  Google Scholar 

  8. Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximately of some art gallery problems, in: Proceedings 10th Canadian Conference on Computational Geometry (Montréal, 1998)

    Google Scholar 

  9. Fejex, Tóth, L.: Illumination of convex disks. Acta Math. Acad. Sci. Hungar. 29 (1977) 355–360

    Article  MathSciNet  Google Scholar 

  10. Hartvigsen, D., Hell, P.: The k-piece packing problem. Manuscript (2000)

    Google Scholar 

  11. Hoffmann, F.: On the rectilinear Art Gallery Problem. in: Proc ICALP, Lecture Notes in Comput. Sci. 90, Springer-Verlag (1990) 717–728

    Google Scholar 

  12. Kaneko, A.: A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two. Submitted

    Google Scholar 

  13. Kano, M., Katona, G. Y., Király, Z.: On path-factors and subfactors of graphs without paths of length one. Submitted

    Google Scholar 

  14. Karp, R. M.: Reducibility among combinatorial problems, in: Complexity of Computer Computations (R. Miller and J. Thatcher, eds.). Plenum Press, New York (1972) 85–103

    Google Scholar 

  15. Kirkpatrick, D. G., Hell, P.: On the completeness of a generalized matching problem, in: Proc. 10th Ann. ACM Symp. on Theory of Computing (San Diego, Calif., 1978), ACM, New York (1978) 240–245

    Google Scholar 

  16. Lee, D. T., Lin, A. K.: Computational complexity of art gallery problems. IEEE Trans. Inform. Theory 32 (1986) 276–282

    Article  MATH  MathSciNet  Google Scholar 

  17. Nishizeki, T., Baybars, I.: Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math. 28 (1979) 255–267

    Article  MATH  MathSciNet  Google Scholar 

  18. O’Rourke, J.: Open problems in the combinatorics of visibility and illumination, in: Advances in Discrete and Computational Geometry (B. Chazelle, J. E. Goodman, and R. Pollack, eds.) (Contemporary Mathematics), AMS, Providence (1998) 237–243

    Google Scholar 

  19. O’Rourke, J.: Art gallery theorems and algorithms. The International Series of Monographs on Computer Science, Oxford University Press, New York (1987)

    Google Scholar 

  20. Tóth, Cs. D.: Illumination in the presence of opaque line segments in the plane. Submitted

    Google Scholar 

  21. Tóth, Cs. D.: Illuminating disjoint line segments in the plane. Submitted

    Google Scholar 

  22. Urrutia, J.: Art Gallery and Illumination Problems. in: Handbook on Computational Geometry (J. R. Sack, J. Urrutia eds.), Elsevier Science Publishers, Amsterdam (2000) 973–1027

    Chapter  Google Scholar 

  23. Zaks, J.: A note on illuminating line segments in the plane. Manuscript (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ETH Zürich, Institut fü Theoretische Informatik, CH-8092, Zürich, Switzerland

    Csaba D. Tóth

Authors
  1. Csaba D. Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuya-ku, Tokyo, 151-0063, Japan

    Jin Akiyama

  2. Department of Computer and Information Sciences, Ibaraki University, Hitachi, 316-8511, Japan

    Mikio Kano

  3. Department of Mathematics, Tokai University, 3-20-1 Orido Shimizu-shi, Shizuoka, 424-8610, Japan

    Masatsugu Urabe

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Tóth, C.D. (2001). Illuminating Both Sides of Line Segments. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_35

Download citation

  • DOI: https://doi.org/10.1007/3-540-47738-1_35

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42306-5

  • Online ISBN: 978-3-540-47738-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation