Transitions in geometric minimum spanning trees (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved embeddings of graph metrics into random trees
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Succinct Greedy Graph Drawing in the Hyperbolic Plane
Graph Drawing
Resilient routing for sensor networks using hyperbolic embedding of universal covering space
INFOCOM'10 Proceedings of the 29th conference on Information communications
Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces
INFOCOM'10 Proceedings of the 29th conference on Information communications
Revisiting Hyperbolic Voronoi Diagrams from Theoretical, Applied and Generalized Viewpoints
ISVD '10 Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and Engineering
Low distortion delaunay embedding of trees in hyperbolic plane
GD'11 Proceedings of the 19th international conference on Graph Drawing
Low distortion delaunay embedding of trees in hyperbolic plane
GD'11 Proceedings of the 19th international conference on Graph Drawing
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This paper considers the problem of embedding trees into the hyperbolic plane. We show that any tree can be realized as the Delaunay graph of its embedded vertices. Particularly, a weighted tree can be embedded such that the weight on each edge is realized as the hyperbolic distance between its embedded vertices. Thus the embedding preserves the metric information of the tree along with its topology. The distance distortion between non adjacent vertices can be made arbitrarily small --- less than a (1+ε) factor for any given ε. Existing results on low distortion of embedding discrete metrics into trees carry over to hyperbolic metric through this result. The Delaunay character implies useful properties such as guaranteed greedy routing and realization as minimum spanning trees.