C. Monma and S. Suri. Transitions in geometric spanning trees. 7th ACM Symp. Computational Geometry (1991) 239--249.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....distinct minimal 1 Steiner trees, that is, trees formed by fixing the position of the extra vertex and considering the minimum spanning tree of the resulting set of points. They give an O(n 2 ) bound on the number of such trees and construct a matching lower bound. Independently, Monma and Suri [3] proved the same bounds in the plane and gave bounds of O(n 2d ) and #(n d ) in any dimension d. We significantly improve Monma and Suri s upper bounds. We show that the maximum possible number of combinatorially distinct minimal 1 Steiner trees on n points in d space is O(n d log 2d 2 d ....

....an arrangement in which, for each cell, the # d possible edges specified by Yao s lemma are all determined. In the plane, this construction can be used to prove that O(n 2 ) topologies are possible. In higher dimensions, each cone Voronoi diagram has O(n 2 ) facets, giving an O(n 2d ) bound [3]. We cannot improve this directly because the cells in each diagram are not convex. Instead, we construct a more complicated structure to which Corollary 1 can be applied directly. Fix a single simplex of frame F d . A cone corresponding to the simplex is defined by d hyperplanes. We pass d ....

C. Monma and S. Suri. Transitions in geometric spanning trees. 7th ACM Symp. Computational Geometry (1991) 239--249.

....reduced to five. 4. NNG(V ) when considered as an undirected graph with the biroot treated as a single edge, is a subgraph of DT(V ) the Delaunay triangulation of V ) and of MST(V ) the minimum spanning tree of V ) The degree bound in (3) also holds for minimum spanning trees. Monma and Suri [13] showed that, conversely, any tree with vertex degree at most five is the minimum spanning tree of some point set; thus minimum spanning tree topologies of general position point sets are exactly characterized by their degrees. See [10] for complications arising from special position. We show ....

C. Monma and S. Suri. Transitions in geometric spanning trees. Proc. 7th ACM Symp. Computational Geometry (1991) 239--249.

....each point and its neighbor. This is a directed graph with outdegree one; thus it is a pseudo forest. Each component of the pseudo forest is a tree, with a length two directed cycle at the root. As with minimum spanning trees, the maximum degree in a nearest neighbor graph is five. Monma and Suri [1] showed that, conversely, any tree with vertex degree at most five is the minimum spanning tree of some point set; thus minimum spanning tree topologies are exactly characterized by their degrees. Paterson and Yao [2] considered the corresponding question for nearest neighbor graphs. They showed ....

C. Monma and S. Suri. Transitions in geometric spanning trees. 7th ACM Symp. Computational Geometry (1991) 239--249.

....how well can it be approximated in polynomial time 2. 3 Low degree spanning trees For points in the plane (with the Euclidean metric) any minimum spanning tree has degree at most six, and a perturbation argument shows that there always exists a minimum spanning tree with degree at most five [91]. In general the degree of a minimum spanning tree in any dimension is bounded by the kissing number (maximum number of disjoint unit spheres that can be simultanously tangent to a given unit sphere) 98] However it is interesting to consider the construction of trees with even smaller degree ....

C. Monma and S. Suri. Transitions in geometric spanning trees. Proc. 7th ACM Symp. Comp. Geom., 1991, pp. 239--249.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC