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.
Exact distance labelings were studied in the precession of bits (i.e. not asymptotically) see e.g. @cite_31 . Another type of labeling studied is adjacency labeling @cite_20 , where given two labels one can compute whether the vertices are adjacent. Efficiency of the labeling algorithms has also been studied @cite_5 .
@cite_5 @cite_31 @cite_20
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We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3) 2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n log n) decoding time, and (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n 2. We also give results for bipartite graphs and for exact and 1-additive distance oracles. A distance labeling scheme labels the n nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A D-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least D from each other. In this paper we consider distance labeling schemes for the classical case of unweighted and undirected graphs. We present the first distance labeling scheme of size opnq for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from general graphs which require pnq size Moon [Proc. of Glasgow Math. Association 1965]. As an intermediate result we give a Op n D log A distance labeling scheme is a distributed data-structure designed to answer queries about distance between any two vertices of a graph G. The data-structure consists in a label L(x,G) assigned to each vertex x of G such that the distance d"G(x,y) between any two vertices x and y can be estimated as a function f(L(x,G),L(y,G)). In this paper, by the use of split decomposition of graphs, we combine several types of distance labeling schemes. This yields to optimal label length schemes for the family of distance-hereditary graphs and for other families of graphs, allowing distance estimation in constant time once the labels have been constructed. A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node u stores its distance to all nodes from an appropriately chosen set of hubs (S(u) V ). For a queried pair of nodes (u, v), the length of a shortest (u !- !v )-path passing through a hub node from (S(u) S(v) ) is then used as an upper bound on the distance between u and v.