A major open problem in the field of metric embedding is the existence of dimension reduction for @math -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 @math spaces with @math . In particular, we introduce an @math -point subset of @math with doubling constant O(1), and demonstrate that any embedding of the set into @math with distortion @math must have @math .
The instances used by the papers mentioned above are based on recursive graph constructions. The papers of @cite_48 @cite_44 used the diamond graph, which has high doubling constant, but @cite_38 showed that their proof can be extended to the Laakso graph, yielding essentially the same result but for a subset of low doubling dimension. For the @math space, there are also strong lower bounds, For instance, the metric induced by an @math -point expander graph (which is in @math as any other finite metric), requires dimension at least @math in any @math distortion embedding. which are based on large girth graphs @cite_3 measure concentration @cite_24 and geometric arguments @cite_38 .
@cite_38 @cite_48 @cite_3 @cite_44 @cite_24
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This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber-Galil expander. Our results give an expansion factor of @math for subgraphs of these three-regular graphs with @math inputs for @math prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is bounded by @math , where a is the load factor. This assumes a double hashing scheme in which two hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman’s result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber-Galil graph expander of focus. This is related to any key distribution having expected @math probes for unsuccessful search for double hashing given the initial random, independent, and uniform choice of two universal hash functions. We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the l1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in l1, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is acomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space. We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. 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. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more). We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses the adjacency matrices of expander graphs, which often leads to recovery algorithms much more efficient than @math minimization. However, to date, constructions based on expanders have required very high expansion coefficients which can potentially make the construction of such graphs difficult and the size of the recoverable sets small. In this paper, we construct sparse measurement matrices for the recovery of non-negative vectors, using perturbations of the adjacency matrix of an expander graph with much smaller expansion coefficient. We present a necessary and sufficient condition for @math optimization to successfully recover the unknown vector and obtain expressions for the recovery threshold. For certain classes of measurement matrices, this necessary and sufficient condition is further equivalent to the existence of a "unique" vector in the constraint set, which opens the door to alternative algorithms to @math minimization. We further show that the minimal expansion we use is necessary for any graph for which sparse recovery is possible and that therefore our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much faster than @math optimization. Finally, we demonstrate through theoretical bounds, as well as simulation, that our method is robust to noise and approximate sparsity. We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l2n, while (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3).It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space--that is defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n (λ2+1)), where λ2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0<δ≤½ is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quais-random graphs of logarithmic sphericity (Lemma 7).For the above bound to be linear, λ2 must be constant. We show that if the second eigenvalue of an n 2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2) entries (Theorem 5). Furthermore, for any 0<δ<½, and λ2, there are only finitely many δn-regular graphs with second eigenvalue at most λ2 (Corollary 4). 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 devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion (O ( _ X n ) , ) where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an (O ( ( _ X ) n ) ) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved. We give a O(slog n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow. We look at a model of random graphs suggested by Gilbert: given an integer @math and @math , scatter @math vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than @math apart.We consider the asymptotics when the metric space is the interval [0, 1], and @math is a function of @math , for @math . We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.) A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494…)n-many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximately @math n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Preliminary version of this paper appeared in Proceedings of the Seventeenth ACM-SIAM Symposium on Discrete Algorithms 2006. ArXive version. Given a metric M = (V, d), a graph G = (V, E) is a t-spanner for M if every pair of nodes in V has a "short" path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every such short path also uses at most D edges. We consider the problem of constructing sparse (1 + e)-spanners with small hop diameter for metrics of low doubling dimension.In this paper, we show that given any metric with constant doubling dimension k, and any 0 < e < 1, one can find a (1 + e)-spanner for the metric with nearly linear number of edges (i.e., only O(n log* n + ne-O(k)) edges) and a constant hop diameter, and also a (1 + e)-spanner with linear number of edges (i.e., only ne-O(k) edges) which achieves a hop diameter that grows like the functional inverse of the Ackermann's function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114.