Optimal algorithms to embed trees in a point set

  • Prosenjit Bose
  • Michael McAllister
  • Jack Snoeyink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

We present optimal Θ(n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted-tree embeddings and degree-constrained embeddings. In the rooted-tree embedding problem we are given a rooted-tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the root at point p. In the degree-constrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n}-2 and are asked to embed a tree in P using straight lines that respects the degrees assigned to each point of P. In both problems, the points of P must be in general position and the embeddings have no crossing edges.

References

  1. 1.
    P. Bose, G. Di Battista, W. Lenhart, and G. Liotta. Proximity constraints and representable trees. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 340–351. Springer-Verlag, 1995.Google Scholar
  2. 2.
    P. Bose, W. Lenhart, and G. Liotta. Characterizing proximity trees. Report TR-SOCS-93.9, School of Comp. Sci., McGill Univ., Montreal, Quebec, Canada, 1993.Google Scholar
  3. 3.
    B. Chazelle. On the convex layers of a planar set. IEEE Transactions on Information Theory, IT-31:509–517, 1985.CrossRefGoogle Scholar
  4. 4.
    P. Crescenzi and A. Piperno. Optimal-area upward drawings of AVL trees. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 307–317. Springer-Verlag, 1995.Google Scholar
  5. 5.
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Comput. Geom. Theory Appl., 4:235–282, 1994.Google Scholar
  6. 6.
    P. Eades and S. Whitesides. The realization problem for Euclidean minimum spanning trees is NP-hard. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 49–56, 1994.Google Scholar
  7. 7.
    J. Hershberger and S. Suri. Applications of a semi-dynamic convex hull algorithm. BIT, 32:249–267, 1992.CrossRefGoogle Scholar
  8. 8.
    Y. Ikebe, M. Perles, A. Tamura, and S. Tokunaga. The rooted tree embedding problem into points in the plane. Discrete & Computational Geometry, 11:51–63, 1994.Google Scholar
  9. 9.
    G. Kant, G. Liotta, R. Tamassia, and I. Tollis. Area requirement of visibility representations of trees. In Proc. 5th Canad. Conf. Comput. Geom., pages 192–197, Waterloo, Canada, 1993.Google Scholar
  10. 10.
    D. E. Knuth. Fundamental Algorithms, volume 1 of The Art of Computer Programming. Addison-Wesley, second edition, 1973.Google Scholar
  11. 11.
    J. Manning and M. J. Atallah. Fast detection and display of symmetry in trees. Congressus Numerantium, 64:159–169, 1988.Google Scholar
  12. 12.
    A. Melkman. On-line construction of the convex hull of a simple polyline. Information Processing Letters, 25:11–12, 1987.CrossRefGoogle Scholar
  13. 13.
    C. Monma and S. Suri. Transitions in geometric minimum spanning trees. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 239–249, 1991.Google Scholar
  14. 14.
    A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, 1992.Google Scholar
  15. 15.
    J. O'Rourke. Computational Geometry in C. Cambridge University Press, 1994.Google Scholar
  16. 16.
    M. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166–204, 1981.CrossRefGoogle Scholar
  17. 17.
    J. Pach and J. Törőcsik. Layout of rooted trees. In W. T. Trotter, editor, Planar Graphs, volume 9 of DIMACS Series, pages 131–137. American Mathematical Society, 1993.Google Scholar
  18. 18.
    W. Paul and J. Simon. Decision trees and random access machines. Logic and Algorithmics, Monograph 30, L'Enseignement Mathématique, 1987.Google Scholar
  19. 19.
    F. P. Preparata and M. I. Shamos. Computational Geometry: an Introduction. Springer-Verlag, New York, NY, 1985.Google Scholar
  20. 20.
    A. Tamura and Y. Tamura. Degree constrained tree embedding into points in the plane. Information Processing Letters, 44:211–1214, 1992.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Michael McAllister
    • 1
  • Jack Snoeyink
    • 1
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

Personalised recommendations