A \emph{$ν$-reliable spanner} of a metric space $(X,d)$, is a (dominating) graph $H$, such that for any possible failure set $B\subseteq X$, there is a set $B^+$ just slightly larger $|B^+|\le(1+ν)\cdot|B|$, and all distances between pairs in $X\setminus B^+$ are (approximately) preserved in $H\setminus B$. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of \emph{light} reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of $X$. We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an \emph{oblivious} reliable spanner is a distribution over spanners, and the bound on $|B^+|$ holds in expectation. We devise an oblivious $ν$-reliable $(2+\frac{2}{k-1})$-spanner for any $k$-HST, whose lightness is $\approx ν^{-2}$. We demonstrate a matching $Ω(ν^{-2})$ lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness. For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct {\em light} tree covers, in which every tree is a $k$-HST of low weight. Combining these covers with our results for $k$-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious $ν$-reliable $(1+\varepsilon)$-spanner with lightness $\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(ν^{-2}\cdot\log n)$, which is best possible (up to lower order terms).

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $$O(n^{3/2})$$ (resp., $$O(n^{7/5})$$ ) to the weighted setting, where the additive error is $$+2W$$ (resp., $$+4W$$ ). This means that for every pair u, v, the additive stretch is at most $$+2W_{u,v}$$ , where $$W_{u,v}$$ is the maximal edge weight on the shortest $$u-v$$ path (weights are normalized so that the minimum edge weight is 1). In addition, as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). showed a randomized algorithm yielding a $$+8W_{max}$$ spanner of size $$O(n^{4/3})$$ , here $$W_{max}$$ is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a $$+(6+\varepsilon )W$$ spanner for weighted graphs with size $$O(n^{4/3})$$ (for any constant $$\varepsilon >0$$ ), thus nearly matching the classical +6 spanner of size $$O(n^{4/3})$$ for unweighted graphs. Furthermore, we show a $$+(2+\varepsilon )W$$ subsetwise spanner of size $$O(n\cdot \sqrt{\vert S\vert })$$ , improving the $$+4W_{max}$$ result of as reported by Ahmed et al. (in: Adler I, Müller H (eds) Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK). (that had the same size). We also show a simple randomized algorithm for a $$+4W$$ emulator of size $${\tilde{O}}(n^{4/3})$$ . In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It was proved by Abboud A, Bodwin G (J ACM 64(4):28–12820 2017) that such spanners must suffer polynomially large stretches. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size $$+{\tilde{O}}(\sqrt{n}\cdot W)$$ spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of Dor D et al. (SIAM J Comput 29:1740–1759, 2000) for unweighted graphs, we devise an efficient randomized algorithm producing a $$+2W$$ spanner for weighted graphs of size $${\tilde{O}}(n^{3/2})$$ in $${\tilde{O}}(n^2)$$ time.

Abstract. This note points out an error in the proof of Theorem 4 in the article “Metric Embedding via Shortest Path Decompositions,” SIAM J. Comput., 51 (2022), pp. 290–314, by the authors, and withdraws the associated claim of Theorem 4.

Consider an undirected weighted graph G = (V,E,w). We study the problem of computing (1+ε)-approximate shortest paths for S × V, for a subset S ⊆ V of |S| = n^r sources, for some 0 < r ≤ 1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time Õ(|E| ⋅ n^{o(1)} + n^{ω(r)}), where n^{ω(r)} is the time required to multiply an n^r × n matrix by an n × n one. Our PRAM algorithm has polylogarithmic time (log n)^{O(1/ρ)}, and its work complexity is Õ(|E| ⋅ n^ρ + n^{ω(r)}), for any arbitrarily small constant ρ > 0. In particular, for r ≤ 0.313…, our centralized algorithm computes S × V (1+ε)-approximate shortest paths in n^{2 + o(1)} time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E| ⋅ n^ρ + n^{2+o(1)}), for any arbitrarily small constant ρ > 0. Previously existing solutions either require centralized time/parallel work of O(|E| ⋅ |S|) or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S| = n^r sources, for r ≤ 0.655, while previous state-of-the-art algorithms did so only for r ≤ 1/2. Moreover, it improves previous bounds for all r > 1/2. For unweighted graphs, the running time is improved further to poly(log log n) for r ≤ 0.655. Previously this running time was known for r ≤ 1/2.

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har–Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with $$\tilde{O}(n^{1+1/t^2})$$ edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of $$\ell _p$$ with $$1<p\le 2$$ . Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of $$\ell _p$$ for $$1

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y∈X has a low distortion path in one of the trees. If it has the stronger property that every point x∈X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of shortest path decomposition (SPD) depth is inductively defined: A (weighed) path graph has SPD depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\nicefrac{1}{p},\nicefrac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\nicefrac{1}{p},\nicefrac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor.

Given metric spaces $(X,d)$ and $(Y,\rho)$ and an ordering $x_1,x_2,\ldots,x_n$ of $(X,d)$, an embedding $f: X \rightarrow Y$ is said to have a prioritized distortion $\alpha(\cdot)$, for a function $\alpha(\cdot)$, if for any pair $x_j,x'$ of distinct points in $X$, the distortion provided by $f$ for this pair is at most $\alpha(j)$. If $Y$ is a normed space, the embedding is said to have prioritized dimension $\beta(\cdot)$ if $f(x_j)$ may have at most $\beta(j)$ nonzero coordinates. The notion of prioritized embedding was introduced by Filtser and the current authors in [M. Elkin, A. Filtser, and O. Neiman, SIAM J. Comput., 47 (2018), pp. 829--858], where a rather general methodology for constructing such embeddings was developed. Though this methodology enabled [M. Elkin, A. Filtser, and O. Neiman, SIAM J. Comput., 47 (2018), pp. 829--858] to come up with many prioritized embeddings, it typically incurs some loss in the distortion. In other words, in the worst case, prioritized embeddings obtained via this methodology incur distortion which is at least a constant factor off compared to the distortion of the classical counterparts of these embeddings. This constant loss is problematic for isometric embeddings. It is also troublesome for Matoušek's embedding of general metrics into $\ell_\infty$, which, for a parameter $k = 1,2,\ldots$, provides distortion $2k-1$ and dimension $O(k \log n \cdot n^{1/k})$. All logarithms in this paper are base 2. In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into $\ell_\infty$ with dimension $O(\log j)$, matching the worst-case guarantee of $O(\log n)$ of the classical embedding of [N. Linial, E. London, and Y. Rabinovich, Combinatorica, 15 (1995), pp. 215--245]. The second one is a prioritized Matoušek embedding of general metrics into $\ell_\infty$, which, for a parameter $k=1,2,\ldots$, provides prioritized distortion $2 \lceil k {{\log j} \over {\log n}} \rceil - 1$ and dimension $O(k \log n \cdot n^{1/k})$, again matching the worst-case guarantee $2k-1$ in the distortion of the classical Matoušek embedding. We also provide a dimension-prioritized variant of Matoušek's embedding. Finally, we devise prioritized embeddings of general metrics into a single ultrametric and of general graphs into a single spanning tree, with asymptotically optimal distortion.

Given a metric space (X, d), a set of terminals $$K\subseteq X$$ , and a parameter $$0<\epsilon <1$$ , we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in $$K\times X$$ up to a factor of $$1+\epsilon$$ , 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 in several settings, no terminal spanner or embedding with distortion close to 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion $$1+\epsilon$$ and space s(|X|) has its terminal counterpart, with distortion $$1+O(\epsilon )$$ and space $$s(|K|)+n$$ . In particular, for any doubling metric on n points, a set of k terminals, and constant $$0<\epsilon <1$$ , there exists Moreover, surprisingly, the last two results apply if only the metric space on K is doubling, while the metric on X can be arbitrary.

A t-spanner H of a weighted graph G = (V, E, w) is a subgraph that approximates all pairwise distances up to a factor of t. The lightness of H is defined as the ratio between the weight of H to that of the minimum spanning tree. An (α, β)-Shallow Light Tree (SLT) is a tree of lightness β, that approximates all distances from a designated root vertex up to a factor of α. A long line of works resulted in efficient algorithms that produce (nearly) optimal light spanners and SLTs.

Given metric spaces (X, d) and (Y, ρ) and an ordering x1,x2,…,xn of (X, d), an embedding f: X → Y is said to have a prioritized distortion α(·), for a function α(·), if for any pair xj,x′ of distinct points in X, the distortion provided by f for this pair is at most a(j). If Y is a normed space, the embedding is said to have prioritized dimension β(·), if f(xj) may have at most β(j) nonzero coordinates. The notion of prioritized embedding was introduced by Filtser and the current authors in [EFN18], where a rather general methodology for constructing such embeddings was developed. Though this methodology enabled [EFN18] to come up with many prioritized embeddings, it typically incurs some loss in the distortion. In other words, in the worst-case, prioritized embeddings obtained via this methodology incur distortion which is at least a constant factor off, compared to the distortion of the classical counterparts of these embeddings. This constant loss is problematic for isometric embeddings. It is also troublesome for Matousek's embedding of general metrics into ℓ∞, which for a parameter k = 1, 2, …, provides distortion 2k–1 and dimension O(k log n·n1/k). In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into with dimension O(log j), matching the worst-case guarantee of O(log n) of the classical embedding of Linial et al. [LLR95]. The second one is a prioritized Matousek's embedding of general metrics into ℓ∞, which for a parameter k = 1,2, …, provides prioritized distortion and dimension O(k log n · n1/k), again matching the worst-case guarantee 2k – 1 in the distortion of the classical Matousek's embedding. We also provide a dimension-prioritized variant of Matousek's embedding. Finally, we devise prioritized embeddings of general metrics into (single) ultra-metric and of general graphs into (single) spanning tree with asymptotically optimal distortion.

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995]. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Dimensionality reduction plays a central role in real-world applications for Machine Learning, among many fields. In particular, metric dimensionality reduction where data from a general metric is mapped into low dimensional space, is often used as a first step before applying machine learning algorithms. In almost all these applications the quality of the embedding is measured by various average case criteria. Metric dimensionality reduction has also been studied in Math and TCS, within the extremely fruitful and influential field of metric embedding. Yet, the vast majority of theoretical research has been devoted to analyzing the worst case behavior of embeddings and therefore has little relevance to practical settings. The goal of this paper is to bridge the gap between theory and practice view-points of metric dimensionality reduction, laying the foundation for a theoretical study of more practically oriented analysis. This paper can be viewed as providing a comprehensive theoretical framework addressing a line of research initiated by VL [NeuroIPS' 18] who have set the goal of analyzing different distortion measurement criteria, with the lens of Machine Learning applicability, from both theoretical and practical perspectives. We complement their work by considering some important and vastly used average case criteria, some of which originated within the well-known Multi-Dimensional Scaling framework. While often studied in practice, no theoretical studies have thus far attempted at providing rigorous analysis of these criteria. In this paper we provide the first analysis of these, as well as the new distortion measure developed by [VL18] designed to possess Machine Learning desired properties. Moreover, we show that all measures considered can be adapted to possess similar qualities. The main consequences of our work are nearly tight bounds on the absolute values of all distortion criteria, as well as first approximation algorithms with provable guarantees.

For a positive parameter β, the β-bounded distance between a pair of vertices u,v in a weighted undirected graph G = (V,E,ømega) is the length of the shortest u-v path in G with at most β edges, aka hops. For β as above and ε > 0, a (β,ε)-hopset of G = (V,E,ømega) is a graph GH =(V,H,ømegaH) on the same vertex set, such that all distances in G are (1+ε)-approximated by β-bounded distances in G ∪ GH. Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Ømega(n łog n) edges, or with a hopbound nØmega(1). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ε) (łog łog n)łog łog n + O(1). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [11] for computing hopsets with a constant (i.e., independent of n) hopbound requires nØmega(1) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [11]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

We prove that any graph excluding $K_r$ as a minor can be partitioned into clusters of diameter at most $\Delta$ while removing at most $O(r/\Delta)$ fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who, building on the work of Klein, Plotkin, and Rao, gave a partitioning that required removing $O(r^2/\Delta)$ fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter $O(r)$ and strong-diameter partitions with padding parameter $O(r^2)$ for $K_r$-minor-free graphs, $O(k)$ for treewidth-$k$ graphs, and $O(\log g)$ for graphs with (Euler) genus $g$.

We prove that any weighted graph $G=(V,E,w)$ with $n$ points and $m$ edges has a spanning tree $T$ such that $\sum_{\{u,v\}\in E}\frac{d_T(u,v)}{w(u,v)}=O(m\log n\log\log n)$. Moreover, such a tree can be found in time $O(m\log n\log\log n)$. Our result is obtained using our new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most four times the radius of the induced subgraph of the cluster in the original graph.

Miller et al. [48] devised a distributed 1 algorithm in the CONGEST model that, given a parameter k = 1,2,…, constructs an O ( k )-spanner of an input unweighted n -vertex graph with O ( n 1+1/ k ) expected edges in O ( k ) rounds of communication. In this article, we improve the result of Reference [48] by showing a k -round distributed algorithm in the same model that constructs a (2 k −1)-spanner with O ( n 1+1/ k }/ϵ) edges, with probability 1−ϵ for any ϵ>0. Moreover, when k =ω(log n ), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n (1+ o (1)), with probability 1− o (1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that are sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k , it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for (1+ϵ, β)-spanners and emulators. In particular, we show that for any unweighted n -vertex graph and any ϵ > 0, there exists a (1+ ϵ, (log log n / ϵ) log log n )-emulator with O ( n ) edges. All previous constructions of (1+ϵ, β)-spanners and emulators employ a superlinear number of edges for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths’ computation in unweighted graphs.

Distributed \em routing is one of the most central and fundamental problems in the area of Distributed Graph Algorithms. It was extensively studied for almost thirty years. Nevertheless, the currently existing solutions for this problem require either prohibitively large construction (aka preprocessing) time, or prohibitively large memory usage either during the construction or during the routing phase, and suffer from suboptimal labels and tables' sizes. We devise a distributed routing scheme that enjoys the best of all worlds. Specifically, its construction time and memory requirements during the construction phase are near-optimal, and so is also the tradeoff between the sizes of routing tables and labels on the one hand, and the stretch on the other. On the way to this result, we also improve upon existing solutions for the distributed exact \em tree routing problem. Previous solutions require Ω(√ ) memory, and provide tables and labels of size O(log n) and O(log^2 n), respectively. Our solution, on the other hand, requires just O(log n) memory, and has tables of size O(1), and labels of size O(log n). These bounds match the bounds of the best-known centralized solution.

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.

We study the problem of embedding weighted graphs of pathwidth k into ℓp spaces. Our main result is an O(kmin{1p,12})-distortion embedding. For p=1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p >1.

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. Bansal et al., 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. Khan et al., 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 we alleviate this limitation by devising a suite of prioritized metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,\ldots,x_n)$ of the graph vertices (resp., metric points) one can devise a metric data structure (resp., embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex $x_j$ will depend on the rank $j$ of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on $j$. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) simultaneously. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than of their nonprioritized counterparts.

Miller et al. [43] devised a distributed1 algorithm in the CONGEST model, that given a parameter k = 1, 2,…, constructs an O(k)-spanner of an input unweighted n-vertex graph with O(n1+1/k) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k — 1)- spanner with O(n1+1/k/∊) edges, with probability 1 — ∊, for any ∊ > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 — o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for (1 + ∊, ß)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any ∊ > 0, there exists a with O(n) edges. All previous constructions of (1 + ∊, β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths’ computation in unweighted graphs.

In this paper we study terminal embeddings , in which one is given a finite metric ( X , d X ) (or a graph G = ( V , E ) ) and a subset K ⊆ X of its points are designated as terminals . The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve ≈ | K | ⋅ | X | pairs, the distortion depends only on | K | , rather than on | X | . We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X × X and with respect to K × X . Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [10] devised an O ˜ ( log ⁡ r ) -approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an O ˜ ( log ⁡ | K | ) - approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since | K | ≤ r , our bound generalizes that of [10] .

A $(\beta,\epsilon)$-hopset for a weighted undirected n-vertex graph $G=(V,E)$ is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most $\beta$ edges, whose length is within $1+\epsilon$ of the shortest path. In her seminal paper, Cohen [ J. ACM, 47 (2000), pp. 132--166] introduced the notion of hopsets in the context of parallel computation of approximate shortest paths, and since then it has found numerous applications in various settings, such as dynamic graph algorithms, distributed computing, and the streaming model. Cohen [ J. ACM, 47 (2000), pp. 132--166] devised efficient algorithms for constructing hopsets with polylogarithmic in n number of hops. Her constructions remain the state of the art since the publication of her paper in the proceedings of STOC'94, i.e., for more than two decades. In this paper we exhibit the first construction of sparse hopsets with a constant number of hops. We also find efficient algorithms for hopsets in various computational settings, improving the best-known constructions. Generally, our hopsets strictly outperform the hopsets of [ J. ACM, 47 (2000), pp. 132--166] in terms of both their parameters and the resources required to construct them. We demonstrate the applicability of our results for the fundamental problem of computing approximate shortest paths from $s$ sources. Our results improve the running time for this problem in the parallel, distributed, and streaming models for a vast range of s.

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.

For a pair of positive parameters D,Χ, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,Χ) network decomposition}, if the supergraph G(P), obtained by contracting each of the clusters of P, can be properly Χ-colored. The decomposition P is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most D, i.e., if for every cluster C ∈ P and every two vertices u,v ∈ C, the distance between them in the induced graph G(C) of C (resp., in G) is at most D.

Given a distributed network represented by a weighted undirected graph G=(V,E) on n vertices, and a parameter k, we devise a distributed algorithm that computes a routing scheme in O(n1/2+1/k+D)⋅ no(1) rounds, where D is the hop-diameter of the network. The running time nearly matches the lower bound of Ω(n1/2+D) rounds (which holds for any scheme with polynomial stretch). The routing tables are of size Õ(n1/k), the labels are of size O(k log2n), and every packet is routed on a path suffering stretch at most 4k-5+o(1). Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by [LP13a, STOC 2013] and [LP15, PODC 2015]. The former has similar properties but suffers from substantially larger routing tables of size O(n1/2+1/k), while the latter has sub-optimal running time of Õ(min{(nD)1/2 ⋅ n1/k,n2/3+2/(3k)+D}).

Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion O(1/ρ2).1 The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n -vertex graphs using a trivial deterministic algorithm with a worst-case update time of O ( n ). No deterministic algorithm that outperforms the naïve O ( n ) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd, who in 1993 devised a deterministic algorithm with an amortized update time of O (( n + m ) √2/2 ), where m is the number of edges. In this article, we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O (√ m ). Moreover, our algorithm maintains a matching, which in fact is a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 − ϵ)-approximate MCM improving upon the naïve O ( n ) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sublogarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O (log n /log log n ). This result addresses an open question of Onak and Rubinfeld [2010]. We also show a deterministic algorithm with optimal space usage, which for arbitrary graphs maintains a maximal matching in amortized O (√ m ) time and uses only O ( n + m ) space.

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 [21], online algorithms [7], distributed algorithms [19] and for computing sparsifiers [28]. 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.

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 $\epsilon$, with the guarantee that for each $\epsilon$ simultaneously, the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. 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 $\ell_2$ distortion is $O(\sqrt{\log n})$. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies constant $\ell_q$-distortion for every fixed $q<\infty$.

A major open problem in the field of metric embedding is the existence of dimension reduction for $n$-point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the point set, and not on its cardinality. In this paper, we negate this possibility for $\ell_p$ spaces with $p>2$. In particular, we introduce an $n$-point subset of $\ell_p$ with doubling constant $O(1)$, and demonstrate that any embedding of the set into $\ell_p^d$ with distortion $D$ must have $D\ge\Omega((\frac{\log n}{d})^{\frac{1}{2}-\frac{1}{p}})$.

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality; in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\varepsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\varepsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\varepsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm and improves previous work by a factor of $O(\log k)$.

A major open problem in the field of metric embedding is the existence of dimension reduction for n-point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for ℓp spaces with p > 2. In particular, we introduce an n-point subset of ℓp with doubling constant O(1), and demonstrate that any embedding of the set into ℓdp with distortion D must have D ≥ Ω ((c log n/d)1/2−1/p).

We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r/Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2/Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n-vertex graphs using a trivial deterministic algorithm with a worst-case update time of O(n). No deterministic algorithm that outperforms the naive O(n) one was reported up to this date. The only progress in this direction is due to Ivkovic and Lloyd [14], who in 1993 devised a deterministic algorithm with an amortized update time of O((n+m)√2/2), where m is the number of edges.

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. In 1982, Beck conjectured that the discrepancy of this set system is O(1). In other words, the conjecture says that each integer from 1 through n can be colored either red or blue so that the number of red and blue integers in each interval of each permutations differs only by a constant. (The discrepancy of a set system based on two permutations is at most two.) Our main result is a counterexample to this conjecture: for any positive integer n = 3k, we construct three permutations whose corresponding set system has discrepancy Ω(log n). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This construction also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set √ system corresponding to £ permutations has discrepancy O(√ℓ). Our work was inspired by an intriguing paper from SODA 2011 by Eisenbrand, Palvolgyi and Rothvoß, who show a surprising connection between the discrepancy of three permutations and the bin packing problem: They show that Beck's conjecture implies a constant worst-case bound on the additive integrality gap for the Gilmore-Gomory LP relaxation for bin packing in the special case when all items have sizes strictly between 1/4 and 1/2, also known as the three partition problem. Our counterexample shows that this approach to bounding the additive integrality gap for bin packing will not work. We can, however, prove an interesting implication of our construction in the reverse direction: there are instances of bin packing and corresponding optimal basic feasible solutions for the Gilmore-Gomory LP relaxation such that any packing that contains only patterns from the support of these solutions requires at least opt + Ω(log m) bins, where m is the number of items. Finally, we discuss some implications that our construction has for other areas of discrepancy theory.

We prove that any graph G=(V,E) with n points and m edges has a spanning tree T such that ∑(u,v)∈ E(G)dT(u,v) = O(m log n log log n). Moreover such a tree can be found in time O(m log n log log n). Our result is obtained using a new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most 4 times the radius of the induced subgraph of the cluster in the original graph.

Given a set of n points in ℓ1, how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the ℓ2norm, where O((log n)/ϵ2) dimensions suffice to achieve 1+ϵ distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in ℓ1. A recent result shows that distortion 1+ϵ can be achieved with n/ϵ2dimensions. On the other hand, the only lower bounds known are that distortion δ requires nΩ(1/δ2)dimensions and that distortion 1+ϵ requires n1/2-O(ϵ log(1/ϵ))dimensions. In this work, we show the first near linear lower bounds for dimension reduction in ℓ1. In particular, we show that 1+ϵ distortion requires at least n1-O(1/log(1/ϵ))dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in ℓ1.

We study the problem of embedding metric spaces into low dimensional ℓp spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into with k-local distortion of O((logk)/p). We also show that any ultrametric can be embedded into with k-local distortion 1 + ∊. Our embedding results have immediate applications to local Distance Oracles. We show how to preprocess a graph in polynomial time to obtain a data structure of O(nk1/t log2 k) bits, such that distance queries from any node to its k nearest neighbors can be answered with stretch O(t).

We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET~T[dT(u, nu)/dG(u, nu)] is bounded by Otilde(log n). Our result is obtained via a new approach of building "highways" between portals and a new strong diameter probabilistic decomposition theorem.

In many application areas, complex data sets are often representedby some metric space and metric embedding is used to provide a more structured representation of the data. In many of these applications much greater emphasis is put on the preserving the local structure of the original space than on maintaining its complete structure. This is also the case in some networking applications where "small world" phenomena in communication patterns has been observed. Practical study of embedding has indeed involved with finding embeddings with this property. In this paper we initiate thestudy of local embeddings of metric spaces and provide embeddings with distortion depending solely on the local structureof the space.

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few.The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications.A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(log n) distortion.Bourgain's result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion.Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain's theorem.In fact, our embedding possesses a much stronger property. We define the lq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible lq-distortion for all 1 ≤ q ≤ ∞ simultaneously.These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension.The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain's embedding.We prove that any metric space on n points embeds into Lp with distortion O(log n) in dimension O(log n). This provides an optimal bound on the dimension of the embedding.Somewhat surprisingly, we show that a further small improvement is possible at a small price in the distortion, obtaining an embedding with distortion O(log1+θ n) in optimal dimension O(θ-1 log n/log log n), for any θ > 0. It is worth noting that with the small loss in the distortion this improves upon the best known embedding of arbitrary spaces into Euclidean space, where dimension reduction is used.Our techniques also allow to obtain the optimal distortion for embedding into Lp with nearly tight dimension. For any 1 ≤ p ≤ ⊂ and any 1 ≤ k ≤ p, we give an embedding into Lp with distortion O(⌈ log n/k ⌉) in dimension 2O(k)log n.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results. These metric decompositions seem of independent interest.

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler (2004), we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into /spl lscr//sub 1/ which exhibit gracefully degrading distortion: these is a single embedding into /spl lscr//sub 1/ that achieves distortion at most O(log 1//spl epsi/) on all but at most an /spl epsi/ fraction of distances, simultaneously for all /spl epsi/ > 0. We extend this with distortion O(log 1//spl epsi/)/sup 1/p/ to maps into general /spl lscr//sub p/, p /spl ges/ 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for /spl epsi/-slack embeddings from lower bounds for low-distortion embeddings.