How to Realize a Graph on Random Points
Saad Quader, Alexander Russell

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Abstract
We are given an integer , a graph , and a uniformly random embedding of the vertices. We are interested in the probability that can be "realized" by a scaled Euclidean norm on , in the sense that there exists a non-negative scaling and a real threshold so that \[ (u,v) \in E \qquad \text{if and only if} \qquad \Vert f(u) - f(v) \Vert_w^2 < \theta\,, \] where . These constraints are similar to those found in the Euclidean minimum spanning tree (EMST) realization problem. A crucial difference is that the realization map is (partially) determined by the random variable . In this paper, we consider embeddings for arbitrary . We prove that arbitrary trees can be realized with high probability when $d =…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Stochastic processes and statistical mechanics · Graph Theory and Algorithms
