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.
@cite_18: This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in BFM86 ). We then proceed to construct optimal Ramsey partitions, and use them to show that for every @math , every @math -point metric space has a subset of size @math which embeds into Hilbert space with distortion @math . This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor BLMN05 , in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in TZ05 . Namely, we show that for every @math point metric space @math , and @math , there exists an @math -approximate distance oracle whose storage requirement is @math , and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.
@cite_26: An approximate distance oracle is a succinct data structure that provides fast answers to distance queries between any two nodes of a given graph. In this paper we consider approximate distance oracles for general undirected graphs with non-negative edge weights with constant query time. We present a distance oracle of size O(kn1+1 k), with 2k --- 1 stretch and O(1) query time. This improves the O(log k) query time of Wulff-Nilsen's distance oracle [SODA '13], which in turn improved the O(k) query time of Thorup and Zwick's distance oracle [J. ACM '05].
@cite_8: 
@cite_28: The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [MN07] devised the so called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset S ⊆ V of a given graph G = (V, E), such that there exists a spanning tree of G that has small stretch for S. Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these trees serves as a special type of spanner, a tree-padding spanner. We use this spanner to devise the first compact stateless routing scheme with O(1) routing decision time, and labels which are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem, and provide a new deterministic construction. We prove that for every k, any n-point metric space has a subset S of size at least n1--1 k which embeds into an ultrametric with distortion 8k. We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k, any n-vertex graph G = (V, E) has a subset S of size at least n1--1 k, and a spanning tree of G, that has stretch O(k log log n) between any point in S and any point in V.
@cite_29: Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [N. Linial, E. London, and Y. Rabinovich, Combinatorica, 15 (1995), pp. 215--245], online algorithms [N. , Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, IEEE Computer Society, Washington, DC, 2011, pp. 267--276], distributed algorithms [M. , Distrib. Comput., 25 (2012), pp. 189--205], and for computing sparsifiers [Y. Shavitt and T. Tankel, IEEE ACM Trans. Netw., 12 (2004), pp. 993--1006]. However, this methodology appears to have a limitation: the worst-case performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices points should enjoy a better service (i.e., stretch distortion, label size dimension) than that given by the worst-case guarantee. In this paper...
@cite_1: 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.
@cite_6: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1 k) expected time, constructing a data structure of size O(kn1p1 k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1 k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1 k) space had a query time of Ω(n1 k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
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 .