The (maximum receiver-centric) interference of a geometric graph (von (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^ 1 3 ); (2) no connected graph with vertex set V has interference o((log n)^ 1 4 ); and (3) the minimum spanning tree of @math has interference Theta(( n)^ 1 2 ).
In higher dimensions, Khabbazian, Durocher, and Haghnegahdar @cite_0 use their version of log to show that minimum spanning trees of random point sets have at most logarithmic interference. Part 1 of main improves the upper bound in durocher to @math and Part 1 of lower-bound gives a matching lower bound.
@cite_0
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Using the theory of negative association for measures and the notion of unimodularity for random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree-like graphs. Specifically, the normalized logarithm of the associated partition function (free energy) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton–Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide a new asymptotic formula for the maximum size of a b-matching in the Erdős–Renyi random graph with fixed average degree and diverging size, for any . To the best of our knowledge, this is the first time that correlation inequalities and unimodularity are combined together to yield a general proof of uniqueness of Gibbs measures on infinite trees. We believe that a similar argument is applicable to other Gibbs measures than those over spanning subgraphs considered here. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)1+e for some fixed e > 0). Both our bounds improve previously known bounds for the problem. This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as ( ( [3]n B ) ) and ( ( [4]n B ) ), respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of ( ( [2]n B) ) for graphs of diameter Ω(log n). We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in 0,1 d under the Hamming distance. Recall that H is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y ∈ 0,1 d with dist(x, y) ≤ r have probability at least p of collision under a randomly chosen h ∈ H, whereas all pairs x, y ∈ 0, 1 d with dist(x, y) ≥ cr have probability at most q of collision. Typically, one considers d → ∞, with c > 1 fixed and q bounded away from 0. For its applications to approximate nearest-neighbor search in high dimensions, the quality of an LSH family H is governed by how small its ρ parameter ρ = ln(1 p) ln(1 q) is as a function of the parameter c. The seminal paper of Indyk and Motwani [1998] showed that for each c ≥ 1, the extremely simple family H = x ↦ xi : i ∈ [d] achieves ρ ≤ 1 c. The only known lower bound, due to [2007], is that ρ must be at least ( e1 c - 1) (e1 c + 1) ≥ .46 c (minus od(1)). The contribution of this article is twofold. (1) We show the “optimal” lower bound for ρ: it must be at least 1 c (minus od(1)). Our proof is very simple, following almost immediately from the observation that the noise stability of a boolean function at time t is a log-convex function of t. (2) We raise and discuss the following issue: neither the application of LSH to nearest-neighbor search nor the known LSH lower bounds hold as stated if the q parameter is tiny. Here, “tiny” means q = 2-Θ(d), a parameter range we believe is natural. We prove the first logarithmic lower bounds for the approximability of the Minimum Dominating Set problem for the case of connected ( α , β ) -power law graphs for α being a size parameter and β the power law exponent. We give also a best up to now upper approximation bound for this problem in the case of the parameters β 2 . We develop also a new functional method for proving lower approximation bounds and display a sharp approximation phase transition area between approximability and inapproximability of the underlying problems. Our results depend on a method which could be also of independent interest. We present the first poly-logarithmic competitive online algorithm for minimum metric bipartite matching. Via induction and a careful use of potential functions, we show that a simple randomized greedy algorithm is competitive on a hierarchically separated tree. Application of recent results on randomized embedding of metrics into trees yield the poly-logarithmic result for general metrics. Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞. Let (X ) be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ( |X| 2 ) points. We show the existence of a non-crossing properly colored geometric perfect matching on (X ) (if (|X| ) is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most (3 ) on (X ). Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points. We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the trade-off between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is an O(log n)1+o(1)-time algorithm for finding a (2O(log* n)log n)-spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs a O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α,β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n)φ) where φ = 1+☂5 2 is the golden ratio. As the distance increases the Fibonacci spanner's multiplicative distortion passes through four discrete stages, moving from logarithmic to doubly logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion. The interference at a wireless node s can be modelled by the number of wireless nodes whose transmission ranges cover s. Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (equivalently, a power level) to each node such that the resulting communication graph is connected, while minimizing the maximum interference. We consider the model introduced by von (2005), in which each transmission range is represented by a ball and edges in the communication graph are symmetric. The problem is NP-complete in two dimensions (Buchin 2008) and no polynomial-time approximation algorithm is known. Furthermore, even in one dimension (the highway model), the problem's complexity is unknown and the maximum interference of a set of n wireless nodes can be as high as Theta(sqrt(n)) (von 2005). In this paper we show how to solve the problem efficiently in settings typical for wireless ad hoc networks. In particular, we show that if node positions are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, for any fixed d, then the topology given by the closure of the Euclidean minimum spanning tree of P has maximum interference O(log n) with high probability. We extend this bound to a general class of communication graphs over a broad set of probability distributions. Next we present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on the expected maximum interference. Finally, we discuss an empirical evaluation of our algorithm with a suite of simulation results. 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 We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). We show that the largest density of factor of i.i.d. independent sets in the dd-regular tree is asymptotically at most (logd) d(log⁡d) d as d→∞d→∞. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random dd-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically 2(logd) d2(log⁡d) d. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Renyi graphs. This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of O( m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O ( p log 3 n ) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. The worst-case time for one query is O (log n log log n ). For the k -edge witness problem (“Does the removal of k given edges disconnect the graph?”) the expected time for p updates is O ( p log 3 n ) and the expected time for q queries is O ( qk log 3 n ). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O ( pk log 3 n ). This implies an algorithm to maintain a 1 + e-approximation of the minimum spanning tree in expected time O (( p log 3 n log U ) e) for p updates, where the weights of the edges are between 1 and U . Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal's algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution spl mu on them such that the expected stretch of any edge of G in a tree chosen according to spl mu is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. We consider the maximum independent set problem on sparse graphs with maximum degree d. The best known result for the problem is an SDP based O(d log log d log d) approximation due to Halperin. It is also known that no o(d log2 d) approximation exists assuming the Unique Games Conjecture. We show the following two results: (i) The natural LP formulation for the problem strengthened by O(log4(d)) levels of the mixed-hierarchy has an integrality gap of O(d log2 d), where O(·) ignores some log log d factors. However, our proof is non-constructive, in particular it uses an entropy based approach due to Shearer, and does not give a O(d log2 d) approximation algorithm with sub-exponential running time. (ii) We give an O(d log d) approximation based on polylog(d)-levels of the mixed hierarchy that runs in nO(1) exp(logO(1) d) time, improving upon Halperin's bound by a modest log log d factor. Our algorithm is based on combining Halperin's approach together with an idea used by Ajtai, Erdos, Komlos and Szemeredi to show that Kr-free, degree-d graphs have independent sets of size Ωr(n log log d d). Given @math wireless transceivers located in a plane, a fundamental problem in wireless communications is to construct a strongly connected digraph on them such that the constituent links can be scheduled in fewest possible time slots, assuming the SINR model of interference. In this paper, we provide an algorithm that connects an arbitrary point set in @math slots, improving on the previous best bound of @math due to Moscibroda. This is complemented with a super-constant lower bound on our approach to connectivity. An important feature is that the algorithms allow for bi-directional (half-duplex) communication. One implication of this result is an improved bound of @math on the worst-case capacity of wireless networks, matching the best bound known for the extensively studied average-case. We explore the utility of oblivious power assignments, and show that essentially all such assignments result in a worst case bound of @math slots for connectivity. This rules out a recent claim of a @math bound using oblivious power. On the other hand, using our result we show that @math slots suffice, where @math is the ratio between the largest and the smallest links in a minimum spanning tree of the points. Our results extend to the related problem of minimum latency aggregation scheduling, where we show that aggregation scheduling with @math latency is possible, improving upon the previous best known latency of @math . We also initiate the study of network design problems in the SINR model beyond strong connectivity, obtaining similar bounds for biconnected and @math -edge connected structures. We consider distance labeling schemes for trees: given a tree with n nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by [, J. Alg., 2004] and an upper bound by Peleg [Peleg, J. Graph Theory, 2000] establish that labels must use Theta(log^2(n)) bits. [, ESA, 2001] show that for very small approximate stretch, labels use Theta(log(n) log(log(n))) bits. Several other papers investigate various variants such as, for example, small distances in trees [, SODA, 2003]. We improve the known upper and lower bounds of exact distance labeling by showing that 1 4*log^2(n) bits are needed and that 1 2*log^2(n) bits are sufficient. We also give (1 + epsilon)-stretch labeling schemes using Theta(log(n)) bits for constant epsilon > 0. (1 + epsilon)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar [Talwar, STOC, 2004]. In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size 2*log(n) - Theta(log(log(n))). For simple paths with k nodes and edge weights in [1,n], we show that labels must have size (k - 1) k*log(n) + Theta(log(k)). This paper introduces the concept of low-congestion shortcuts for (near-)planar networks, and demonstrates their power by using them to obtain near-optimal distributed algorithms for problems such as Minimum Spanning Tree (MST) or Minimum Cut, in planar networks. Consider a graph G = (V, E) and a partitioning of V into subsets of nodes S1, . . ., SN, each inducing a connected subgraph G[Si]. We define an α-congestion shortcut with dilation β to be a set of subgraphs H1, . . ., HN ⊆ G, one for each subset Si, such that 1. For each i ∈ [1, N], the diameter of the subgraph G[Si] + Hi is at most β. 2. For each edge e ∈ E, the number of subgraphs G[Si] + Hi containing e is at most α. We prove that any partition of a D-diameter planar graph into individually-connected parts admits an O(D log D)-congestion shortcut with dilation O(D log D), and we also present a distributed construction of it in O(D) rounds. We moreover prove these parameters to be near-optimal; i.e., there are instances in which, unavoidably, max α, β = Ω(D[EQUATION]). Finally, we use low-congestion shortcuts, and their efficient distributed construction, to derive O(D)-round distributed algorithms for MST and Min-Cut, in planar networks. This complexity nearly matches the trivial lower bound of Ω(D). We remark that this is the first result bypassing the well-known Ω(D + [EQUATION]) existential lower bound of general graphs (see Peleg and Rubinovich [FOCS'99]; Elkin [STOC'04]; and Das [STOC'11]) in a family of graphs of interest. The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and @math . By employing a subroutine that solves the post office problem, we show that, for fixed k @math 3, such a minimum spanning tree can be found in time O( @math ), where a(k) = @math . The bound can be improved to O( @math ) for points in the 3-dimensional Euclidean space. We also obtain o( @math ) algorithms for finding a farthest pair in a set of n points and for other related problems.