881 research outputs found
A simple approach for lower-bounding the distortion in any Hyperbolic embedding
International audienceWe answer open questions of [Verbeek and Suri, SOCG'14] on the relationships between Gromov hyperbolicity and the optimal stretch of graph embeddings in Hyperbolic space. Then, based on the relationships between hyperbolicity and Cops and Robber games, we turn necessary conditions for a graph to be Cop-win into sufficient conditions for a graph to have a large hyperbolicity (and so, no low-stretch embedding in Hyperbolic space). In doing so we derive lower-bounds on the hyperbolicity in various graph classes – such as Cayley graphs, distance-regular graphs and generalized polygons, to name a few. It partly fills in a gap in the literature on Gromov hyperbolicity, for which few lower-bound techniques are known
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Volume distortion in groups
Given a space in , a cycle in may be filled with a chain in two
ways: either by restricting the chain to or by allowing it to be anywhere
in . When the pair acts on , we define the -volume
distortion function of in to measure the large-scale difference between
the volumes of such fillings. We show that these functions are quasi-isometry
invariants, and thus independent of the choice of spaces, and provide several
bounds in terms of other group properties, such as Dehn functions. We also
compute the volume distortion in a number of examples, including characterizing
the -volume distortion of in , where is a
diagonalizable matrix. We use this to prove a conjecture of Gersten.Comment: 27 pages, 10 figure
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