Angle-Restricted Steiner Arborescences for Flow Map Layout

  • Kevin Buchin
  • Bettina Speckmann
  • Kevin Verbeek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

Abstract

We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source.

We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. Computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid “positive monotone” obstacles.

Keywords

Steiner Tree Logarithmic Spiral Angle Restriction Radial Order Steiner Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    CSISS - Spatial Tools: Tobler’s Flow Mapper, http://www.csiss.org/clearinghouse/FlowMapper
  2. 2.
    Aichholzer, O., Aurenhammer, F., Icking, C., Klein, R., Langetepe, E., Rote, G.: Generalized self-approaching curves. Discr. Appl. Mathem. 109(1-2), 3–24 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Awerbuch, B., Baratz, A., Peleg, D.: Cost-sensitive analysis of communication protocols. In: Proc. 9th ACM Symposium on Principles of Distributed Computing, pp. 177–187. ACM (1990)Google Scholar
  4. 4.
    Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient-constrained minimum networks. I. Fundamentals. Journal of Global Optimization 21, 139–155 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brazil, M., Thomas, D.A.: Network optimization for the design of underground mines. Networks 49, 40–50 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Buchin, K., Speckmann, B., Verbeek, K.: Flow map layout via spiral trees. IEEE Transactions on Visualization and Computer Graphics (to appear, 2011) (Proceedings Visualization / Information Visualization 2011)Google Scholar
  7. 7.
    Córdova, J., Lee, Y.: A heuristic algorithm for the rectilinear Steiner arborescence problem. Technical report, Engineering Optimization (1994)Google Scholar
  8. 8.
    Dent, B.D.: Cartography: Thematic Map Design, 5th edn. McGraw-Hill, New York (1999)Google Scholar
  9. 9.
    Krozel, J., Lee, C., Mitchell, J.: Turn-constrained route planning for avoiding hazardous weather. Air Traffic Control Quarterly 14(2), 159–182 (2006)CrossRefGoogle Scholar
  10. 10.
    Lu, B., Ruan, L.: Polynomial time approximation scheme for the rectilinear Steiner arborescence problem. J. Comb. Optimization 4(3), 357–363 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mitchell, J.: L 1 shortest paths among polygonal obstacles in the plane. Algorithmica 8, 55–88 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Phan, D., Xiao, L., Yeh, R., Hanrahan, P., Winograd, T.: Flow map layout. In: Proc. IEEE Symposium on Information Visualization, pp. 219–224 (2005)Google Scholar
  13. 13.
    Ramnath, S.: New approximations for the rectilinear Steiner arborescence problem. IEEE Trans. Computer-Aided Design Integ. Circuits Sys. 22(7), 859–869 (2003)CrossRefGoogle Scholar
  14. 14.
    Rao, S., Sadayappan, P., Hwang, F., Shor, P.: The rectilinear Steiner arborescence problem. Algorithmica 7, 277–288 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Shi, W., Su, C.: The rectilinear Steiner arborescence problem is NP-complete. In: Proc. 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 780–787 (2000)Google Scholar
  16. 16.
    Shi, W., Su, C.: The rectilinear Steiner arborescence problem is NP-complete. SIAM Journal on Computing 35(3), 729–740 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Slocum, T.A., McMaster, R.B., Kessler, F.C., Howard, H.H.: Thematic Cartography and Geovisualization, 3rd edn. Pearson, New Jersey (2010)Google Scholar
  18. 18.
    Tobler, W.: Experiments in migration mapping by computer. The American Cartographer 14(2), 155–163 (1987)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kevin Buchin
    • 1
  • Bettina Speckmann
    • 1
  • Kevin Verbeek
    • 1
  1. 1.Dep. of Mathematics and Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations