Neighbourhood Graphs and Locally Minimal Triangulations
Neighbourhood (or proximity) graphs, such as nearest neighbour graph, closest pairs, relative neighbourhood graph and k-nearest neighbour graph are useful tools in many tasks inspecting mutual relations, similarity and closeness of objects. Some of neighbourhood graphs are subsets of Delaunay triangulation (DT) and this relation can be used for efficient computation of these graphs. This paper concentrates on relation of neighbourhood graphs to the locally minimal triangulation (LMT) and shows that, although generally these graphs are not LMT subgraphs, in most cases LMT contains all or many edges of these graphs. This fact can also be used for the neighbourhood graphs computation, namely in kinetic problems, because LMT computation is easier.
KeywordsNearest neighbour graph K-nearest neighbour graph Locally minimal triangulation Delaunay triangulation Kinetic problem
This work was supported by the Czech Science Foundation, the project number 17-07690S, and by the Ministry of Education, Youth and Sports of the Czech Republic, project number LO1506 (PUNTIS). 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.
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