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Greedy Spanners in Euclidean Spaces Admit Sublinear Separators
The greedy spanner in low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness
Self-avoiding walk is ballistic on graphs with more than one end
We prove that on any transitive graph with infinitely many ends, a self-avoiding walk of length is ballistic with extremely high probability, in the sense that there exist constants c,t>0 such that for every . Furthermore, we show that the number of self-avoiding walks of length grows asymptotically like , in the sense that there exists C>0 such that for every . Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of which does not fix an end of
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
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