3,266 research outputs found
Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs
A (1 + eps)-approximate distance oracle for a graph is a data structure that
supports approximate point-to-point shortest-path-distance queries. The most
relevant measures for a distance-oracle construction are: space, query time,
and preprocessing time. There are strong distance-oracle constructions known
for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs
(Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n)
space for n-node graphs. We argue that a very low space requirement is
essential. Since modern computer architectures involve hierarchical memory
(caches, primary memory, secondary memory), a high memory requirement in effect
may greatly increase the actual running time. Moreover, we would like data
structures that can be deployed on small mobile devices, such as handhelds,
which have relatively small primary memory. In this paper, for planar graphs,
bounded-genus graphs, and minor-excluded graphs we give distance-oracle
constructions that require only O(n) space. The big O hides only a fixed
constant, independent of \epsilon and independent of genus or size of an
excluded minor. The preprocessing times for our distance oracle are also faster
than those for the previously known constructions. For planar graphs, the
preprocessing time is O(n lg^2 n). However, our constructions have slower query
times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our
linear-space results, we can in fact ensure, for any delta > 0, that the space
required is only 1 + delta times the space required just to represent the graph
itself
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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