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Path Schematization for Route Sketches

  • Daniel Delling
  • Andreas Gemsa
  • Martin Nöllenburg
  • Thomas Pajor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

Abstract

Motivated from drawing route sketches, we consider the following path schematization problem. We are given a simple embedded polygonal path P = (v 1, ..., v n ) and a set \(\mathcal{C}\) of admissible edge orientations including the coordinate axes. The problem is to redraw P schematically such that all edges are drawn as line segments that are parallel to one of the specified orientations. We also require that the path preserves the orthogonal order and that it remains intersection-free. Finally, we want the drawing to maximize the number of edges having their preferred edge direction and to minimize the path length.

In this paper we first present an efficient two-step approach for schematizing monotone paths. It consists of an O(n 2)-time algorithm to assign edge directions optimally and a subsequent linear program to minimize the path length. In order to schematize non-monotone paths we propose a heuristic that first splits the input into k monotone subpaths and then combines the optimal embeddings of the monotone subpaths into a single, intersection-free embedding of the initial path in O(k 2 + n) time.

Keywords

Prefer Angle Horizontal Edge Path Schematization Edge Orientation Total Path Length 
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.

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References

  1. 1.
    Agrawala, M., Stolte, C.: Rendering effective route maps: Improving usability through generalization. In: Fiume, E. (ed.) SIGGRAPH, pp. 241–249. ACM Press, New York (2001)Google Scholar
  2. 2.
    Brandes, U., Pampel, B.: On the hardness of orthogonal-order preserving graph drawing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 266–277. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  4. 4.
    Delling, D., Gemsa, A., Nöllenburg, M., Pajor, T.: Path Schematization for Route Sketches. Technical Report 2010-02, Karlsruhe Institute of Technology (2010)Google Scholar
  5. 5.
    Dwyer, T., Koren, Y., Marriott, K.: Stress majorization with orthogonal ordering constraints. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 141–152. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: STOC’84, pp. 302–311. ACM, New York (1984)Google Scholar
  7. 7.
    Merrick, D., Gudmundsson, J.: Path simplification for metro map layout. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 258–269. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Misue, K., Eades, P., Lai, W., Sugiyama, K.: Layout adjustment and the mental map. J. Visual Languages and Computing 6(2), 183–210 (1995)CrossRefGoogle Scholar
  9. 9.
    Neyer, G.: Line simplification with restricted orientations. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 13–24. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Ovenden, M.: Metro Maps of the World. Capital Transport Publishing (2003)Google Scholar
  11. 11.
    Tversky, B.: Cognitive maps, cognitive collages, and spatial mental models. In: Campari, I., Frank, A.U. (eds.) COSIT 1993. LNCS, vol. 716, pp. 14–24. Springer, Heidelberg (1993)Google Scholar
  12. 12.
    Tversky, B., Lee, P.U.: Pictorial and verbal tools for conveying routes. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 51–64. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Daniel Delling
    • 1
  • Andreas Gemsa
    • 2
  • Martin Nöllenburg
    • 2
    • 3
  • Thomas Pajor
    • 2
  1. 1.Microsoft Research Silicon ValleyMountain View
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Department of Computer ScienceUniversity of CaliforniaIrvine

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