Abstract
We study graph augmentation under the dilation criterion. In our case, we consider a plane geometric graph \(G = (V,E)\) and a set C of edges. We aim to add to G a minimal number of nonintersecting edges from C to bound the ratio between the graph-based distance and the Euclidean distance for all pairs of vertices described by C. Motivated by the problem of decomposing a polygon into natural subregions, we present an optimal linear-time algorithm for the case that P is a simple polygon and C models an internal triangulation of P. The algorithm admits some straightforward extensions. Most importantly, in pseudopolynomial time, it can approximate a solution of minimum total length or, if C is weighted, compute a solution of minimum total weight. We show that minimizing the total length or the total weight is weakly NP-hard.
Finally, we show how our algorithm can be used for two well-known problems in GIS: generating variable-scale maps and area aggregation.
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In this paper “edge” always indicates an element of \(P \cup C\)—a node in \(\mathcal {T}\)—and never an edge between nodes (parent-child relation) in \(\mathcal {T}\).
References
Aronov, B., Buchin, K., Buchin, M., Jansen, B., de Jong, T., van Kreveld, M., Löffler, M., Luo, J., Silveira, R.I., Speckmann, B.: Connect the dot: computing feed-links for network extension. J. Spat. Inf. Sci. 3, 3–31 (2011)
Aronov, B., de Berg, M., Cheong, O., Gudmundsson, J., Haverkort, H., Smid, M., Vigneron, A.: Sparse geometric graphs with small dilation. Comput. Geom. 40, 207–219 (2008)
Bose, P., Keil, J.: On the stretch factor of the constrained Delaunay triangulation. In: Proceedings 3rd International Symposium on Voronoi Diagrams in Science and Engineering, pp. 25–31 (2006)
Bose, P., Smid, M.: On plane geometric spanners: a survey and open problems. Comput. Geom. 47(7), 818–830 (2013)
Chazelle, B., Dobkin, D.: Decomposing a polygon into its convex parts. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing, pp. 38–48 (1979)
Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4(1–4), 97–108 (1989)
Farshi, M., Giannopoulos, P., Gudmundsson, J.: Improving the stretch factor of a geometric network by edge augmentation. SIAM J. Comput. 38(1), 226–240 (2008)
Feng, H.Y.F., Pavlidis, T.: Decomposition of polygons into simpler components: feature generation for syntactic pattern recognition. IEEE Trans. Comput. 24(6), 636–650 (1975)
Giannopoulos, P., Klein, R., Knauer, C., Kutz, M., Marx, D.: Computing geometric minimum-dilation graphs is NP-hard. Int. J. Comput. Geom. Appl. 20(2), 147–173 (2010)
Harrie, L., Sarjakoski, T.: Simultaneous graphic generalization of vector data sets. GeoInformatica 6(3), 233–261 (2002)
Haunert, J.-H., Sering, L.: Drawing road networks with focus regions. IEEE Trans. Vis. Comput. Graph. 17(12), 2555–2562 (2011)
Haunert, J.-H., Wolff, A.: Area aggregation in map generalisation by mixed-integer programming. Int. J. Geogr. Inf. Sci. 24(12), 1871–1897 (2010)
Keil, J.M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Int. J. Comput. Geom. Appl. 12(3), 181–192 (2002)
Klein, R., Levcopoulos, C., Lingas, A.: A PTAS for minimum vertex dilation triangulation of a simple polygon with a constant number of sources of dilation. Comput. Geom. 34, 28–34 (2006)
Lien, J.-M., Amato, N.M.: Approximate convex decomposition of polygons. Comput. Geom. Theory Appl. 35(1), 100–123 (2006)
Lingas, A.: The power of non-rectilinear holes. In: Nielsen, M., Schmidt, E.M. (eds.) Automata, Languages and Programming. LNCS, vol. 140, pp. 369–383. Springer, Heidelberg (1982)
Siddiqi, K., Kimia, B.B.: Parts of visual form: computational aspects. IEEE Trans. Pattern Anal. Mach. Intell. 17(3), 239–251 (1995)
van Dijk, T.C., van Goethem, A., Haunert, J.-H., Meulemans, W., Speckmann, B.: Accentuating focus maps via partial schematization. In: Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 418–421 (2013)
Voisard, A., Scholl, M.O., Rigaux, P., Databases, S.: With Application to GIS. Morgan Kaufmann, Burlington (2002)
Wulff-Nilsen, C.: Computing the dilation of edge-augmented graphs in metric spaces. Comput. Geom. 43(2), 68–72 (2010)
Acknowledgments
The authors would like to thank Johannes Oehrlein for helpful discussions on the topic of this paper. W. Meulemans is supported by Marie Skłodowska-Curie Action MSCA-H2020-IF-2014 656741.
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Haunert, JH., Meulemans, W. (2016). Partitioning Polygons via Graph Augmentation. In: Miller, J., O'Sullivan, D., Wiegand, N. (eds) Geographic Information Science. GIScience 2016. Lecture Notes in Computer Science(), vol 9927. Springer, Cham. https://doi.org/10.1007/978-3-319-45738-3_2
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