Trajectory-Based Dynamic Map Labeling

  • Andreas Gemsa
  • Benjamin Niedermann
  • Martin Nöllenburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


In this paper we introduce trajectory-based labeling, a new variant of dynamic map labeling where a movement trajectory for the map viewport is given. We define a general labeling model and study the active range maximization problem in this model. The problem is \(\cal NP\)-complete and \(\mathcal W[1]\)-hard. In the restricted, yet practically relevant case that no more than k labels can be active at any time, we give polynomial-time algorithms. For the general case we present a practical ILP formulation with an experimental evaluation as well as approximation algorithms.


Activity Model Integer Linear Programming Dynamic Programming Algorithm Interval Graph Left Endpoint 
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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  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory & Appl. 11(3-4), 209–218 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Been, K., Daiches, E., Yap, C.: Dynamic map labeling. IEEE Trans. Visualization and Computer Graphics 12(5), 773–780 (2006)CrossRefGoogle Scholar
  3. 3.
    Been, K., Nöllenburg, M., Poon, S.-H., Wolff, A.: Optimizing active ranges for consistent dynamic map labeling. Comput. Geom. Theory & Appl. 43(3), 312–328 (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carlisle, M.C., Lloyd, E.L.: On the k-coloring of intervals. Discr. Appl. Math. 59(3), 225–235 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: ACM-SIAM Symp. Discr. Algorithms (SODA 2009), pp. 892–901 (2009)Google Scholar
  6. 6.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett. 12(3), 133–137 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gemsa, A., Niedermann, B., Nöllenburg, M.: Trajectory-based dynamic map labeling. CoRR, arXiv:1309.3963 (2013)Google Scholar
  8. 8.
    Gemsa, A., Nöllenburg, M., Rutter, I.: Consistent labeling of rotating maps. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 451–462. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Hsiao, J.Y., Tang, C.Y., Chang, R.S.: An efficient algorithm for finding a maximum weight 2-independent set on interval graphs. Inform. Process. Lett. 43(5), 229–235 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Marx, D.: Efficient approximation schemes for geometric problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Niedermann, B.: Consistent labeling of dynamic maps using smooth trajectories. Master’s thesis, Karlsruhe Institute of Technology (June 2012)Google Scholar
  12. 12.
    Sester, M., Brenner, C.: Continuous generalization for visualization on small mobile devices. In: Fisher, P.F. (ed.) Spatial Data Handling (SDH 2004), pp. 355–368. Springer (2004)Google Scholar
  13. 13.
    Wagner, F., Wolff, A.: A practical map labeling algorithm. Comput. Geom. Theory Appl. 7, 387–404 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wagner, F., Wolff, A., Kapoor, V., Strijk, T.: Three rules suffice for good label placement. Algorithmica 30, 334–349 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andreas Gemsa
    • 1
  • Benjamin Niedermann
    • 1
  • Martin Nöllenburg
    • 1
  1. 1.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyGermany

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