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.
For distortion @math in planar graphs, Thorup @cite_3 and Klein @cite_17 constructed distance labels of size @math . Abraham and Gavoille @cite_25 generalized this result to @math -minor-free graphs, achieving label size @math . @math is a function that depends on @math only, taken from the structure theorem of Robertson and Seymour @cite_16 . No low-dimension embedding into @math with distortion @math is known for planar graphs or even treewidth- @math graphs. If one allows larger distortion, Krauthgamer al @cite_2 proved that planar graphs embed with distortion @math into @math , or more generally that @math -minor-free graphs embed with distortion @math into @math . Abraham al @cite_30 showed that @math -minor-free graphs embed with distortion @math into @math . Turning to priorities, Elkin al @cite_29 constructed prioritized versions of distance labeling with distortion @math . Specifically, for planar and @math -minor-free graphs they achieve label sizes of @math and @math , respectively. No prioritized embeddings are known, nor lower bounds thereof.
@cite_30 @cite_29 @cite_3 @cite_2 @cite_16 @cite_25 @cite_17
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Indyk and Sidiropoulos (2007) proved that any orientable graph of genus @math can be probabilistically embedded into a graph of genus @math with constant distortion. Viewing a graph of genus @math as embedded on the surface of a sphere with @math handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion @math , by iteratively removing the handles. By removing all @math handles at once, we present a probabilistic embedding with distortion @math for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-cut graph of Erickson and Har Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009). This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of @math , with the guarantee that for each @math simultaneously, the distortion of a fraction @math of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion @math . For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion @math . These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the @math distortion is @math . For probabilistic embedding into spanning trees we prove a scaling distortion of @math , which implies constant @math -distortion for every fixed @math . It has been shown by Indyk and Sidiropoulos [IS07] that any graph of genus g>0 can be stochastically embedded into a distribution over planar graphs with distortion 2^O(g). This bound was later improved to O(g^2) by Borradaile, Lee and Sidiropoulos [BLS09]. We give an embedding with distortion O(log g), which is asymptotically optimal. Apart from the improved distortion, another advantage of our embedding is that it can be computed in polynomial time. In contrast, the algorithm of [BLS09] requires solving an NP-hard problem. Our result implies in particular a reduction for a large class of geometric optimization problems from instances on genus-g graphs, to corresponding ones on planar graphs, with a O(log g) loss factor in the approximation guarantee. We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1 12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1 (22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c d , for a constant c > 0. The exponent c d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2 d log 3 2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically.