Path Simplification for Metro Map Layout

  • Damian Merrick
  • Joachim Gudmundsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

Abstract

We investigate the problem of creating simplified representations of polygonal paths. Specifically, we look at a path simplification problem in which line segments of a simplification are required to conform with a restricted set of directions \({\cal C}\). An algorithm is given to compute such simplified paths in \(\O(|{\cal C}|^3 n^2)\) time, where n is the number of vertices in the original path. This result is extended to produce an algorithm for graphs induced by multiple intersecting paths. The algorithm is applied to construct schematised representations of real world railway networks, in the style of metro maps.

References

  1. 1.
    Alt, H., Godau, M.: Measuring the resemblance of polygonal curves. In: Proc. 8th Annual Symposium on Computational Geometry, pp. 102–109 (1992)Google Scholar
  2. 2.
    Agarwal, P.K., Varadarajan, K.R.: Efficient Algorithms for Approximating Polygonal Chains. Discrete & Computational Geometry 23(2), 273–291 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bose, J., Cheong, O., Cabello, S., Gudmundsson, J., van Kreveld, M., Speckmann, B.: Area-preserving approximations of polygonal paths. J. Discrete Alg. (2006)Google Scholar
  4. 4.
    Barequet, G., Chen, D.Z., Daescu, O., Goodrich, M.T., Snoeyink, J.: Efficiently approximating polygonal paths in three and higher dimensions. Algorithmica 33(2), 150–167 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cabello, S., de Berg, M., van Kreveld, M.: Schematization of networks. Computational Geometry and Applications 30, 223–238 (2005)MATHCrossRefGoogle Scholar
  6. 6.
    Chan, W.S., Chin, F.: Approximation of polygonal curves with minimum number of line segments or minimum error. IJCGA 6, 59–77 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chen, D.Z., Daescu, O.: Space-efficient algorithms for approximating polygonal curves in two-dimensional space. IJCGA 13, 95–111 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, D.Z., Daescu, O., Hershberger, J., Kogge, P.M., Mi, N., Snoeyink, J.: Polygonal path simplification with angle constraints. CGTA 32(3), 173–187 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    CityRail network map. Web page: http://www.cityrail.info/networkmaps/mainmap.jsp (Accessed 6th Sept 2006)
  10. 10.
    Douglas, D., Peucker, T.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. The Canadian Cartographer 10(2), 112–122 (1973)Google Scholar
  11. 11.
    Eu, D., Toussaint, G.T.: On Approximating Polygonal Curves in Two and Three Dimensions. CVGIP: Graphical Model and Image Processing 56(3), 231–246 (1994)Google Scholar
  12. 12.
    Friedrich, C.: jjGraph. Personal communicationGoogle Scholar
  13. 13.
    Goodrich, M.T.: Efficient piecewise-linear function approximation using the uniform metric. Discrete & Computational Geometry 14, 445–462 (1995)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gudmundsson, J., van Kreveld, M., Merrick, D.: Schematisation of Tree Drawings. Submitted to Graph Drawing (June 2006)Google Scholar
  15. 15.
    Gudmundsson, J., Narasimhan, G., Smid, M.H.M.: Distance-Preserving Approximations of Polygonal Paths. To appear in CGTA (2006)Google Scholar
  16. 16.
    Guibas, L.J., Hershberger, J., Mitchell, J.S.B., Snoeyink, J.: Approximating Polygons and Subdivisions with Minimum Link Paths. IJCGA 3(4), 383–415 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hershberger, J., Snoeyink, J.: Cartographic line simplification and polygon CSG formulæ in O(n log* n) time. Theory & Applications 11(3-4), 175–185 (1998)MATHMathSciNetGoogle Scholar
  18. 18.
    Hong, S.-H., Merrick, D., do Nascimento, H.A.D.: The metro map layout problem. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 482–491. Springer, Heidelberg (2005)Google Scholar
  19. 19.
    Imai, H., Iri, M.: Computational-geometric methods for polygonal approximations of a curve. Comp. Vision, Graphics and Image Processing 36, 31–41 (1986)CrossRefGoogle Scholar
  20. 20.
    Imai, H., Iri, M.: An optimal algorithm for approximating a piecewise linear function. Journal of Information Processing 9(3), 159–162 (1986)MATHMathSciNetGoogle Scholar
  21. 21.
    Imai, H., Iri, M.: Polygonal approximations of a curve-formulations and algorithms. In: Toussaint, G.T. (ed.) Computational Morphology, North-Holland, Amsterdam (1988)Google Scholar
  22. 22.
    London Underground network map. Web page: http://www.tfl.gov.uk/tube/maps/ (Accessed 6th Sept 2006)
  23. 23.
    Melkman, A., O’Rourke, J.: On polygonal chain approximation. In: Toussaint, G.T. (ed.) Computational Morphology, North-Holland, Amsterdam (1988)Google Scholar
  24. 24.
    Merrick, D., Gudmundsson, J.: Increasing the readability of graph drawings with centrality-based scaling. In: Proc. APVIS 2006, pp. 67–76 (2006)Google Scholar
  25. 25.
    Merrick, D., Gudmundsson, J.: \(\mathcal C\)-Directed Path Simplification for Metro Map Layout. http://www.dmist.net/metromap.pdf (Accessed 6th Sept 2006)
  26. 26.
    Neyer, G.: Line simplification with restricted orientations. In: Proc. 6th International Workshop on Algorithms and Data Structures, pp. 13–24 (1999)Google Scholar
  27. 27.
    Nöllenburg, M., Wolff, A.: A mixed-integer program for drawing high-quality metro maps. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Stott, J., Rodgers, P.: Metro map layout using multicriteria optimization. In: Proc. 8th Interational Conference on Information Visualisation, pp. 355–362 (2004)Google Scholar
  29. 29.
    Toussaint, G.T.: On the Complexity of Approximating Polygonal Curves in the Plane. In: Proc. of the International Symposium on Robotics and Automation (IASTED), pp. 311–318 (1985)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Damian Merrick
    • 1
    • 2
  • Joachim Gudmundsson
    • 2
  1. 1.School of Information Technologies, University of SydneyAustralia
  2. 2.National ICT Australia, SydneyAustralia

Personalised recommendations