Nearest Neighbour Graph and Locally Minimal Triangulation
Nearest neighbour graph (NNG) is a useful tool namely for collision detection tests. It is well known that NNG, when considered as an undirected graph, is a subgraph of Delaunay triangulation (DT) and this relation can be used for efficient NNG computation. This paper concentrates on relation of NNG to the locally minimal triangulation (LMT) and shows that, although NNG can be proved not to be a LMT subgraph, in most cases LMT contains all or nearly all NNG edges. This fact can also be used for NNG computation, namely in kinetic problems, because LMT computation is easier.
KeywordsNearest Neighbour Graph Locally Minimal Triangulation Delaunay triangulation Kinetic problem
This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic, the project SGS-2016-013 Advanced Graphical and Computing Systems and the project PUNTIS (LO1506) under the program NPU I. We would like to thank to T. Bayer from the Charles University in Prague, Czech Republic for supplying us the real terrain data for the experiments.
- 3.Bayer, T.: Department of Applied Geoinformatics and Cartography, Faculty of Science, Charles University, Prague, Czech Republic. https://web.natur.cuni.cz/~bayertom. Accessed 15 May 2017
- 4.Beirouti, R., Snoeyink, J.: Implementations of the LMT heuristic for minimum weight triangulation. In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG 1998, pp. 96–105. ACM, New York (1998)Google Scholar
- 6.Cho, H.G.: On the expected number of common edges in Delaunay and greedy triangulation. J. WSCG 5(1–3), 50–59 (1997)Google Scholar
- 9.Dickerson, M.T., Montague, M.H.: A (usually?) connected subgraph of the minimum weight triangulation. In: Proceedings of the Twelfth Annual Symposium on Computational Geometry, SCG 1996, pp. 204–213. ACM, New York (1996)Google Scholar
- 12.Guibas, L., Russel, D.: An empirical comparison of techniques for updating Delaunay triangulations. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry, SCG 2004, pp. 170–179. ACM, New York (2004)Google Scholar
- 13.Kim, Y.S., Park, D.G., Jung, H.Y., Cho, H.G., Dong, J.J., Ku, K.J.: An improved TIN compression using Delaunay triangulation. In: Proceedings of Seventh Pacific Conference on Computer Graphics and Applications (Cat. No. PR00293), pp. 118–125 (1999)Google Scholar
- 17.Špelič, D., Novak, F., Žalik, B.: Delaunay triangulation benchmarks. J. Electr. Eng. 59(1), 49–52 (2008)Google Scholar