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# Locality-sensitive orderings and applications to reliable spanners

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... Filtser and Le [FL22] generalized Chan et al. [CHJ20] result to doubling spaces, 2 showing that every metric space with doubling dimension d admits a −O(d) , -LSO. Furthermore, they generalized the concept of LSO to other metric spaces, defining the two related notions of triangle-LSO (which turn to be useful for general metric spaces), and left-sided LSO (which turn to be useful for topologically restricted graphs). ...
... Furthermore, they generalized the concept of LSO to other metric spaces, defining the two related notions of triangle-LSO (which turn to be useful for general metric spaces), and left-sided LSO (which turn to be useful for topologically restricted graphs). Here, instead of presenting the left-sided LSO's of [FL22], we introduce the closely related notion of rooted-LSO, which has some additional structure. All the results and constructions for left-sided LSO in [FL22] hold for rooted LSO as well. ...
... Here, instead of presenting the left-sided LSO's of [FL22], we introduce the closely related notion of rooted-LSO, which has some additional structure. All the results and constructions for left-sided LSO in [FL22] hold for rooted LSO as well. We refer to [FL22] for a comparison between the different notions, and to Figure 1 for an illustration. ...
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Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A $(\tau,\rho)$-LSO is a collection $\Sigma$ of orderings such that for every $x,y\in\mathbb{R}^d$ there is an ordering $\sigma\in\Sigma$, where all the points between $x$ and $y$ w.r.t. $\sigma$ are in the $\rho$-neighborhood of either $x$ or $y$. In essence, LSO allow one to reduce problems to the $1$-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, $\ell_p$, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model.
... Another related problem is h-hop t-spanners. Here we are given a metric space (X, d), and the goal is to construct a graph G over X such that ∀x, y ∈ X, d X (x, y) ≤ d Sol13,HIS13] for Euclidean metrics, [FN18] for different metric spaces, [FL21b] for reliable 2-hop spanners, and [ASZ20] for low-hop emulators (where d (O(log log n)) G respects the triangle inequality). The idea of one-to-many embedding of graphs was originated by Bartal and Mendel [BM04], who for k ≥ 1 constructed embedding into ultrametric with O(n 1+ 1 k ) nodes and path distortion O(k⋅log n⋅ log log n) (see Definition 8, and ignore all hop constrains). ...
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In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,\beta,h)$ (here $t,\beta,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $\beta$ is called the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops. Haeupler {\em et al.} constructed embedding where $M$ contains $1-\epsilon$ fraction of the vertices and $\beta=t=O(\frac{\log^2 n}{\epsilon})$. They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems. In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.
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A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation. In other words, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.
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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.
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Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minor-free graphs of diameter $D$ into graphs of treewidth $O_{\epsilon}(\log n)$ with expected additive distortion $+\epsilon D$. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to $O_{\epsilon}(\log\log n)^2$. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from $n^{O_{\epsilon}(\log(n))}$ to $n^{O_{\epsilon}(\log\log(n))^3}$. Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter $D$ into graphs of treewidth $O((\log\log n)^2)/\epsilon)$ and additive distortion $+\epsilon D$ that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time. A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth $O(\log\log n)$ and hop-diameter $O(\log \log n)$. This result may be of independent interest.
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This survey provides a guiding reference to researchers seeking an overview of the large body of literature about graph spanners. It surveys the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.
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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 [29] 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 article, 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 that 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 n 1−1/ k that embeds into an ultrametric with distortion 8 k . 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 n 1−1/ 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 .
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We present a streaming algorithm for constructing sparse spanners and show that our algorithm significantly outperforms the state-of-the-art algorithm for this task (due to Feigenbaum et al.). Specifically, the processing time per edge of our algorithm is O (1), and it is drastically smaller than that of the algorithm of Feigenbaum et al., and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters of the state-of-the-art algorithm. We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has incremental update time of O (1), and a nontrivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides nontrivial bounds on both incremental and decremental update time for a wide range of stretch parameter t .
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In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour and M. Yannakakis [SIAM J. Comput. 23, 864–894 (1994; Zbl 0809.68075)] and is closely related to the metric labeling problem introduced by J. Kleinberg and E. Tardos [Proc. 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 14–23 (1999)]. We present approximation algorithms for 0-Extension. In arbitrary graphs, we present a O(logk)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by A. V. Karzanov [Eur. J. Comb. 19, 71–101 (1998; Zbl 0894.90147)]. It is similar in flavor to the linear programming relaxation of N. Garg, V. V. Vazirani, and M. Yannakakis [SIAM J. Comput. 25, 235–251 (1996; Zbl 0844.68061)] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least clgk for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations.
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It is shown that anyn point metric space is up to logn lipeomorphic to a subset of Hilbert space. We also exhibit an example of ann point metric space which cannot be embedded in Hilbert space with distortion less than (logn)/(log logn), showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance, to construct our example, we make use of random graphs.
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We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn).
Article
The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, d Y) embeds (1 +ε)-isomorphically into the Hilbert spacel 2.
Article
The main result is a metrical characterization of superreflexivity in Banach spaces. A Banach spaceX is not superreflexive if and only ifX contains hyperbolic trees as a metric space. The notion of non-linear cotype in discussed.
Article
In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is . This improves upon the result of Bartal who gave a bound of . Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion -distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
Conference Paper
We consider the problem of computing bounded-degree lightweight plane spanners of unit disk graphs in the local distributed model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for efficient unicasting and multicasting in wireless distributed systems. We present the first local distributed algorithm that computes a bounded-degree plane lightweight spanner of a given unit disk graph. The upper bounds on the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply a local distributed algorithm that computes a plane spanner of a given unit disk graph U, whose degree is at most 14, stretch factor at most 8.81, and weight at most 8.81 times the weight of a Euclidean Minimum Spanning Tree of V(U). We show a wider application of our techniques by giving an O(nlogn) time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight.
Book
Aimed at an audience of researchers and graduate students in computational geometry and algorithm design, this book uses the Geometric Spanner Network Problem to showcase a number of useful algorithmic techniques, data structure strategies, and geometric analysis techniques with many applications, practical and theoretical. The authors present rigorous descriptions of the main algorithms and their analyses for different variations of the Geometric Spanner Network Problem. Though the basic ideas behind most of these algorithms are intuitive, very few are easy to describe and analyze. For most of the algorithms, nontrivial data structures need to be designed, and nontrivial techniques need to be developed in order for analysis to take place. Still, there are several basic principles and results that are used throughout the book. One of the most important is the powerful well-separated pair decomposition. This decomposition is used as a starting point for several of the spanner constructions. © Giri Narasimhan, Michiel Smid 2007 and Cambridge University Press, 2009. All rights reserved.
Article
The problem of computing spanners of unweighted graphs in streaming model is presented. The streaming model has two characteristics, firstly the input data can be accessed only sequentially in the form of a stream, and secondly, the working memory is smaller than the size of the entire input stream. An algorithm in this model is allowed to make one or more passes over the input stream to solve a given computational problem. Single pass and linear time streaming algorithm for computing a spanner of size O(min(m, kn1+1/k)) for any unweighted graph. Ausiello and colleagues designed such an algorithm for spanners of stretch. Another interesting open problem is to design streaming algorithm for spanners of weighted graphs. The real challenge is to design a single pass streaming algorithm for weighted graphs without affecting the optimal bound on the spanner size and constant processing time per edge.