We investigate for which metric spaces the performance of distance labeling and of @math -embeddings differ, and how significant can this difference be. Recall that a distance labeling is a distributed representation of distances in a metric space @math , where each point @math is assigned a succinct label, such that the distance between any two points @math can be approximated given only their labels. A highly structured special case is an embedding into @math , where each point @math is assigned a vector @math such that @math is approximately @math . The performance of a distance labeling or an @math -embedding is measured via its distortion and its label-size dimension. We also study the analogous question for the prioritized versions of these two measures. Here, a priority order @math of the point set @math is given, and higher-priority points should have shorter labels. Formally, a distance labeling has prioritized label-size @math if every @math has label size at most @math . Similarly, an embedding @math has prioritized dimension @math if @math is non-zero only in the first @math coordinates. In addition, we compare these their prioritized measures to their classical (worst-case) versions. We answer these questions in several scenarios, uncovering a surprisingly diverse range of behaviors. First, in some cases labelings and embeddings have very similar worst-case performance, but in other cases there is a huge disparity. However in the prioritized setting, we most often find a strict separation between the performance of labelings and embeddings. And finally, when comparing the classical and prioritized settings, we find that the worst-case bound for label size often translates'' to a prioritized one, but also a surprising exception to this rule.
Another way to represent distances is a distance oracle @cite_6 . This is a data structure that, given a pair of points, returns an estimate of their pairwise distance. The properties of interest are distortion, space and query time. A distance labeling can be viewed as a distributed version of a distance oracles, see also @cite_18 @cite_26 @cite_1 @cite_29 @cite_28 @cite_8 .
@cite_18 @cite_26 @cite_8 @cite_28 @cite_29 @cite_1 @cite_6
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Distributed distance oracles consist of a labeling scheme which assigns a label to each node and a local data structure deployed to each node. When a node v wants to know the distance to a node u, it queries its local data structure with the label of u. The data structure returns an estimated distance to u, which must be larger than the actual distance but can be overestimated. The accuracy of the distance oracle is measured by stretch, which is defined as the maximum ratio between actual distances and estimated distances over all pairs (u, v). For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n 5 3)-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space O(n 2- ) for some constant 0) either required query time polynomial in n or could only answer approximate distance queries.Furthermore, we show how to trade-off time and space: for any S &#x2265; n 3 2, we show how to obtain an S-space distance 5 2 oracle that answers queries in time O(S n 3 2 log n). This is a polynomial improvement over the previous planar distance oracles with o(n 1 4) query time. A (1+e)-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 Ω(e-1n lg n) space for n-node graphs. 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 e 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(nlg2 n). However, our constructions have slower query times. For planar graphs, the query time is O(e-2 lg2 n). For all our linear-space results, we can in fact ensure, for any δ > 0, that the space required is only 1 + δ times the space required just to represent the graph itself. A distance oracle is a compact data structure capable of quickly estimating distances in a given graph. In this paper we provide a new construction for distance oracles in general undirected weighted graphs. Our data structure, for any integer k, requires O( n1+1 k) space, guarantees a stretch of 2k-1, and answers any query in only O(1) time.