Skip to main content

Multi-sided Boundary Labeling


In the Boundary Labeling problem, we are given a set of n points, referred to as sites, inside an axis-parallel rectangle R, and a set of n pairwise disjoint rectangular labels that are attached to R from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend. In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of R (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossing-free leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18


  1. Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  2. Bastert, O., Fekete, S.P.: Geometrische Verdrahtungsprobleme. Technical Report 96–247, Universität zu Köln (1996)

  3. Bekos, M.A., Cornelsen, S., Fink, M., Hong, S., Kaufmann, M., Nöllenburg, M., Rutter, I., Symvonis, A.: Many-to-one boundary labeling with backbones. In: Wismath, S., Wolff, A. (eds.) Proceedings of 21st International Symposium on Graph Drawing (GD’13), Volume 8242 of Lecture Notes in Computer Science, pp. 244–255. Springer, Berlin (2013)

    Google Scholar 

  4. Bekos, M.A., Kaufmann, M., Nöllenburg, M., Symvonis, A.: Boundary labeling with octilinear leaders. Algorithmica 57(3), 436–461 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bekos, M.A., Kaufmann, M., Potika, K., Symvonis, A.: Area-feature boundary labeling. Comput. J. 53(6), 827–841 (2010)

    Article  Google Scholar 

  6. Bekos, M.A., Kaufmann, M., Symvonis, A., Wolff, A.: Boundary labeling: Models and efficient algorithms for rectangular maps. Comput. Geom. Theory Appl. 36(3), 215–236 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  7. Benkert, M., Haverkort, H.J., Kroll, M., Nöllenburg, M.: Algorithms for multi-criteria boundary labeling. J. Graph Algorithms Appl. 13(3), 289–317 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chan, T.M., Hoffmann, H.-F., Kiazyk, S., Lubiw, A.: Minimum length embedding of planar graphs at fixed vertex locations. In: Wismath, S.K., Wolff, A. (eds.) Proceedings of the 21st International Symposium on Graph Drawing (GD’13) volume 8242 of Lecture Notes in Computer Science, pp. 376–387. Springer, Berlin (2013)

    Google Scholar 

  9. Chazelle, B., 36 co-authors.: The computational geometry impact task force report. In: Chazelle, B. Goodman, J.E., Pollack, R., (eds.) Advances in Discrete and Computational Geometry, vol. 223, pp. 407–463. American Mathematical Society, Providence (1999)

  10. Fink, M., Haunert, J.-H., Schulz, A., Spoerhase, J., Wolff, A.: Algorithms for labeling focus regions. IEEE Trans. Visual. Comput. Graphics 18(12), 2583–2592 (2012)

    Article  Google Scholar 

  11. Freeman, H., Marrinan, S., Chitalia, H.: Automated labeling of soil survey maps. In: Proceedings of ASPRS-ACSM Annual Convention, Baltimore, Vol. 1, pp. 51–59 (1996)

  12. Gemsa, A., Haunert, J.-H., Nöllenburg, M.: Boundary-labeling algorithms for panorama images. In: Proceedings of the 19th ACM SIGSPATIAL International Conference on Advanced Geographic Information Systems (ACM-GIS’11), pp. 289–298 (2011)

  13. Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified positions. Am. Math. Mon. 98, 165–166 (1991)

    MathSciNet  Article  Google Scholar 

  14. Hirschberg, D.S.: A linear space algorithm for computing maximal common subsequences. Commun. ACM 18(6), 341–343 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  15. Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-geodesic embedding of planar graphs. In: Eppstein, D., Gansner, E.R. (eds) Proceedings of the 17th International Symposium Graph Drawing (GD’09), Volume 5849 of Lecture Notes in Computer Science, pp. 207–218. Springer, Berlin (2010)

  16. Liebling, T.M., Margot, F., Müller, D., Prodon, A., Stauffer, L.: Disjoint paths in the plane. ORSA J. Comput. 7(1), 84–88 (1995)

    Article  MATH  Google Scholar 

  17. Lin, C.-C.: Crossing-free many-to-one boundary labeling with hyperleaders. In: Proceedings of the IEEE Pacific Visualization Symposium(PacificVis’10), pp. 185–192 (2010)

  18. Lin, C.-C., Kao, H.-J., Yen, H.-C.: Many-to-one boundary labeling. J. Graph Algorithms Appl. 12(3), 319–356 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  19. Morrison, J.L.: Computer technology and cartographic change. In: Taylor, D. (ed.) The Computer in Contemporary Cartography. Johns Hopkins University Press, Baltimore (1980)

    Google Scholar 

  20. Nöllenburg, M., Polishchuk, V., Sysikaski, M.: Dynamic one-sided boundary labeling. In: Proceedings of the 18th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS’10), pp. 310–319 (2010)

  21. Raghavan, R., Cohoon, J., Sahni, S.: Single bend wiring. J. Algorithms 7(2), 232–257 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  22. van Kreveld, M., Strijk, T., Wolff, A.: Point labeling with sliding labels. Comput. Geom. Theory Appl. 13, 21–47 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  23. Zoraster, S.: Practical results using simulated annealing for point feature label placement. Cartogr. GIS 24(4), 228–238 (1997)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Benjamin Niedermann.

Additional information

A preliminary version of this paper has appeared in Proc. 13th Int. Algorithms Data Struct. Symp. (WADS’13), volume 8037 of Lect. Notes Comput. Sci., pages 463–474, Springer-Verlag. This research was initiated during the GraDr Midterm meeting at the TU Berlin in October 2012. The meeting was supported by the ESF EuroGIGA networking grant. Ph. Kindermann acknowledges support by the ESF EuroGIGA project GraDR (DFG Grant Wo 758/5-1).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kindermann, P., Niedermann, B., Rutter, I. et al. Multi-sided Boundary Labeling. Algorithmica 76, 225–258 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Computational geometry
  • Boundary labeling
  • Dynamic program