Essential Constraints of Edge-Constrained Proximity Graphs
Given a plane forest \(F = (V, E)\) of \(|V| = n\) points, we find the minimum set \(S \subseteq E\) of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F. We present an \(O(n \log n)\)-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, \(\beta \)-skeleton and Delaunay triangulation.
We present an algorithm that identifies the minimum set \(S\subseteq E\) of edges of a given plane graph \(I=(V,E)\) such that \(I \subseteq CG_\beta (V, S)\) for \(1 \le \beta \le 2\), where \(CG_\beta (V, S)\) is the constraint \(\beta \)-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.
KeywordsProximity graphs Constraints Visibility MST Delaunay \(\beta \)-skeletons
- 5.Jaromczyk, J.W., Kowaluk, M.: A note on relative neighborhood graphs. In: SoCG, pp. 233–241 (1987)Google Scholar
- 6.Jaromczyk, J.W., Kowaluk, M., Yao, F.: An optimal algorithm for constructing \(\beta \)-skeletons in \(l_p\) metric. Manuscript (1989)Google Scholar
- 7.Kirkpatrick, D.G., Radke, J.D.: A framework for computational morphology. In: Computational Geometry, vol. 2, pp. 217–248. Machine Intelligence and Pattern Recognition, North-Holland (1985)Google Scholar
- 9.Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. In: STOC, pp. 114–122. ACM (1981)Google Scholar