Discrete & Computational Geometry

, Volume 40, Issue 3, pp 377–394 | Cite as

Tight Bounds for Connecting Sites Across Barriers

  • David Krumme
  • Eynat Rafalin
  • Diane L. SouvaineEmail author
  • Csaba D. Tóth


Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines, it is known that there is a spanning tree such that every barrier is crossed by \(O(\sqrt{m}\,)\) tree edges, and this bound is asymptotically optimal. Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Lower bound constructions are known with 3 crossings per barrier and 2n total cost.

We obtain tight bounds on the minimum cost of spanning trees in the special case where the barriers are interior disjoint line segments that form a convex subdivision of the plane and there is a point in every cell of the subdivision. In particular, we show that there is a spanning tree such that every barrier crosses at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are the best possible.


Crossing number Spanning trees 


  1. 1.
    Asano, T., de Berg, M., Cheong, O., Guibas, L.J., Snoeyink, J., Tamaki, H.: Spanning trees crossing few barriers. Discrete Comput. Geom. 30(4), 591–606 (2003) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Chazelle, B., Welzl, E.: Quasi-optimal range searching in spaces of finite VC-dimension. Discrete Comput. Geom. 4(5), 467–489 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Edelsbrunner, H., Guibas, L., Stolfi, J.: Optimal point location in a monotone subdivision. SIAM J. Comput. 15(2), 317–340 (1983) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Guibas, L.J., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Trans. Graph. 4(2), 74–123 (1985) zbMATHCrossRefGoogle Scholar
  5. 5.
    Hoffmann, M., Tóth, Cs.D.: Connecting points in the presence of obstacles in the plane. In: Proc. 14th Canad. Conf. on Comput. Geom., pp. 63–67, 2002 Google Scholar
  6. 6.
    Hoffmann, M., Tóth, Cs.D.: Spanning trees across axis-parallel segments. In: Proc. 18th Canadian Conf. on Comput. Geom., pp. 101–104, 2006 Google Scholar
  7. 7.
    Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM J. Comput. 12(1), 28–35 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Krumme, D., Perkins, G., Rafalin, E., Souvaine, D.L.: Upper and lower bounds for connecting sites across barriers. TUFTS-CS Technical Report 2003-6, Tufts University, Medford, MA (2003) Google Scholar
  9. 9.
    Matoušek, J.: Spanning trees with low crossing number. RAIRO Inform. Théor. Appl. 25(2), 103–123 (1991) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Matoušek, J.: Efficient partition trees. Discrete Comput. Geom. 8, 315–334 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sarnak, N., Tarjan, R.: Planar point location using persistent search trees. Commun. ACM 29(7), 669–679 (1986) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Snoeyink, J.: Open problem presented at the 9th Canadian Conference on Computational Geometry, 1997 Google Scholar
  13. 13.
    Snoeyink, J., van Kreveld, M.: Linear-time reconstruction of Delaunay triangulations with applications. In: Proc. 5th European Sympos. on Algorithms. Lecture Notes in Comp. Sci., vol. 1284, pp. 459–471. Springer, Berlin (1997) Google Scholar
  14. 14.
    Welzl, E.: Partition trees for triangle counting and other range searching problems. In: Proc. 4th Sympos. on Comput. Geom., pp. 23–33. ACM Press, New York (1988) Google Scholar
  15. 15.
    Welzl, E.: On spanning trees with low crossing numbers. In: Data Structures And Efficient Algorithms. Lecture Notes in Comp. Sci., vol. 594, pp. 233–249. Springer, Berlin (1992) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • David Krumme
    • 1
  • Eynat Rafalin
    • 2
  • Diane L. Souvaine
    • 1
    Email author
  • Csaba D. Tóth
    • 3
  1. 1.Department of Computer ScienceTufts UniversityMedfordUSA
  2. 2.Google Inc.Mountain ViewUSA
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada

Personalised recommendations