Distributed Spanner Construction in Doubling Metric Spaces
نویسندگان
چکیده
This paper presents a distributed algorithm that runs on an n-node unit ball graph (UBG) G residing in a metric space of constant doubling dimension, and constructs, for any ε > 0, a (1 + ε)-spanner H of G with maximum degree bounded above by a constant. In addition, we show that H is “lightweight”, in the following sense. Let ∆ denote the aspect ratio of G, that is, the ratio of the length of a longest edge in G to the length of a shortest edge in G. The total weight of H is bounded above by O(log ∆)·wt(MST ), where MST denotes a minimum spanning tree of the metric space. Finally, we show that H satisfies the so called leapfrog property, an immediate implication being that, for the special case of Euclidean metric spaces with fixed dimension, the weight of H is bounded above by O(wt(MST )). Thus, the current result subsumes the results of the authors in PODC 2006 that apply to Euclidean metric spaces, and extends these results to metric spaces with constant doubling
منابع مشابه
An Optimal Dynamic Spanner for Doubling Metric Spaces
For a set S of points in a metric space, a t-spanner is a graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the actual distance between the points. In this paper, we consider points residing in a metric space of doubling dimension λ, and show how to construct a dynamic (1+ ε)-spanner with constant degree and O(log ...
متن کاملNear Isometric Terminal Embeddings for Doubling Metrics
Given a metric space (X, d), a set of terminals K ⊆ X , and a parameter t ≥ 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs inK ×X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist...
متن کاملt-Spanners for metric space searching
The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G′(V,E) as the representation of the metric database. A t-spanner is a subgraph G′(V,E) of a graph G(V,A), such that E ⊆ A and G′ approximates the s...
متن کاملNote on Bounded Degree Spanners for Doubling Metrics
We focus on obtaining sparse representations of metrics: these are called spanners, and they have been studied extensively both for general and Euclidean metrics. Formally, a t-spanner for a metric M = (V, d) is an undirected graph G = (V,E) such that the distances according to dG (the shortest-path metric of G) are close to the distances in d: i.e., d(u, v) ≤ dG(u, v) ≤ t d(u, v). Clearly, one...
متن کاملMathematische Zeitschrift Modulus and the Poincaré inequality on metric measure spaces
The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging se...
متن کامل