A Combinatorial Framework for Map Labeling
The general map labeling problem consists in labeling a set of sites (points, lines, regions) given a set of candidates (rectangles, circles, ellipses, irregularly shaped labels) for each site. A map can be a classical cartographical map, a diagram, a graph or any other figure that needs to be labeled. A labeling is either a complete set of non-conflicting candidates, one per site, or a subset of maximum cardinality. Finding such a labeling is NP-hard.
We present a combinatorial framework to attack the problem in its full generality. The key idea is to separate the geometric from the combinatorial part of the problem. The latter is captured by the conflict graph of the candidates and by rules which successively simplify this graph towards a near-optimal solution.
We exemplify this framework at the problem of labeling point sets with axis-parallel rectangles as candidates, four per point. We do this such that it becomes clear how our concept can be applied to other cases. We study competing algorithms and do a thorough empirical comparison. The new algorithm we suggest is fast, simple and effective.
KeywordsSimulated Annealing Greedy Algorithm Constraint Satisfaction Problem Polynomial Time Approximation Scheme Local Consistency
- [AvKS97]Pankaj Agarwal, Marc van Kreveld, and Subhash Suri. Label placement by maximum independent set in rectangles. In Proceedings of the 9th Canadian Conference on Computational Geometry, pages 233–238, 1997.Google Scholar
- [CFMS97]Jon Christensen, Stacy Friedman, Joe Marks, and Stuart Shieber. Empirical testing of algorithms for variable-sized label placement. In Proceedings of the 13th Annual ACM Symposium on Computational Geometry, pages 415–417, 1997.Google Scholar
- [DMM+97]_Srinivas Doddi, Madhav V. Marathe, Andy Mirzaian, Bernard M.E. Moret, and Binhai Zhu. Map labeling and its generalizations. In Proceedings of the 8th ACM-SIAM Symposium on Discrete Algorithms, pages 148–157, 1997.Google Scholar
- [FW91]Michael Formann and Frank Wagner. A packing problem with applications to lettering of maps. In Proc. 7th Annu. ACM Sympos. Comput. Geom., pages 281–288, 1991.Google Scholar
- [Jam96]Michael B. Jampel. Over-Constrained Systems in CLP and CSP. PhD thesis, Dept. of Comp. Sci. City University, London, sept 1996.Google Scholar
- [JFM96]Michael Jampel, Eugene Freuder, and Michael Maher, editors. Over-Constrained Systems. Number 1106 in LNCS. Springer, August 1996.Google Scholar
- [KT98]Konstantinos G. Kakoulis and Ionnis G. Tollis. A unified approach to labeling graphical features. In Proc. 14th Annu. ACM Sympos. Comput. Geom., pages 347–356, June 1998.Google Scholar
- [SFV95]Thomas Schiex, Hélène Fargier, and Gérard Verfaillie. Valued constraint satisfaction problems: Hard and easy problems. In Proc. International Joint Conference on AI, aug 1995.Google Scholar
- [vKSW98]Marc van Kreveld, Tycho Strijk, and Alexander Wolff. Point set labeling with sliding labels. In Proc. 14th Annu. ACM Sympos. Comput. Geom., pages 337–346, June 1998.Google Scholar