31,019 research outputs found
Covering Metric Spaces by Few Trees
A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995].
In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers
Real valued functions and metric spaces quasi-isometric to trees
We prove that if X is a complete geodesic metric space with uniformly
generated first homology group and is metrically proper on the
connected components and bornologous, then X is quasi-isometric to a tree.
Using this and adapting the definition of hyperbolic approximation we obtain
an intrinsic sufficent condition for a metric space to be PQ-symmetric to an
ultrametric space.Comment: 12 page
Bounded distortion homeomorphisms on ultrametric spaces
It is well-known that quasi-isometries between R-trees induce power
quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper
investigates power quasi-symmetric homeomorphisms between bounded, complete,
uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising
up to similarity as the end spaces of bushy trees). A bounded distortion
property is found that characterizes power quasi-symmetric homeomorphisms
between such ultrametric spaces that are also pseudo-doubling. Moreover,
examples are given showing the extent to which the power quasi-symmetry of
homeomorphisms is not captured by the quasiconformal and bi-H\"older conditions
for this class of ultrametric spaces.Comment: 20 pages, 1 figure. To appear in Ann. Acad. Sci. Fenn. Mat
Hausdorff dimension in graph matchbox manifolds
In this paper, we study the Hausdorff and the box dimensions of closed
invariant subsets of the space of pointed trees, equipped with a pseudogroup
action. This pseudogroup dynamical system can be regarded as a generalization
of a shift space. We show that the Hausdorff dimension of the space of pointed
trees is infinite, and the union of closed invariant subsets with dense orbit
and non-equal Hausdorff and box dimensions is dense in the space of pointed
trees.
We apply our results to the problem of embedding laminations into
differentiable foliations of smooth manifolds. To admit such an embedding, a
lamination must satisfy at least the following two conditions: first, it must
admit a metric and a foliated atlas, such that the generators of the holonomy
pseudogroup, associated to the atlas, are bi-Lipschitz maps relative to the
metric. Second, it must admit an embedding into a manifold, which is a
bi-Lipschitz map. A suspension of the pseudogroup action on the space of
pointed graphs gives an example of a lamination where the first condition is
satisfied, and the second one is not satisfied, with Hausdorff dimension of the
space of pointed trees being the obstruction to the existence of a bi-Lipschitz
embedding.Comment: Proof of Theorem 1.1 simplified as compared to the previous version;
Sections 5 and 6 contain new result
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