65,923 research outputs found

    Quasirandom Rumor Spreading: An Experimental Analysis

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    We empirically analyze two versions of the well-known "randomized rumor spreading" protocol to disseminate a piece of information in networks. In the classical model, in each round each informed node informs a random neighbor. In the recently proposed quasirandom variant, each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list. While for sparse random graphs a better performance of the quasirandom model could be proven, all other results show that, independent of the structure of the lists, the same asymptotic performance guarantees hold as for the classical model. In this work, we compare the two models experimentally. This not only shows that the quasirandom model generally is faster, but also that the runtime is more concentrated around the mean. This is surprising given that much fewer random bits are used in the quasirandom process. These advantages are also observed in a lossy communication model, where each transmission does not reach its target with a certain probability, and in an asynchronous model, where nodes send at random times drawn from an exponential distribution. We also show that typically the particular structure of the lists has little influence on the efficiency.Comment: 14 pages, appeared in ALENEX'0

    Perfect Matchings as IID Factors on Non-Amenable Groups

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    We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.Comment: 16 pages; corrected missing reference in v

    Geographic Gossip: Efficient Averaging for Sensor Networks

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    Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste in energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of nn and n\sqrt{n} respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy ϵ\epsilon using O(n1.5lognlogϵ1)O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1}) radio transmissions, which yields a nlogn\sqrt{\frac{n}{\log n}} factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields.Comment: To appear, IEEE Transactions on Signal Processin

    Hyperbolic intersection graphs and (quasi)-polynomial time

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    We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in dd-dimensional hyperbolic space, which we denote by Hd\mathbb{H}^d. Using a new separator theorem, we show that unit ball graphs in Hd\mathbb{H}^d enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in 2O(n11/(d1))2^{O(n^{1-1/(d-1)})} time for any fixed d3d\geq 3, while the same problems need 2O(n11/d)2^{O(n^{1-1/d})} time in Rd\mathbb{R}^d. We also show that these algorithms in Hd\mathbb{H}^d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H2\mathbb{H}^2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial (nO(logn)n^{O(\log n)}) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and 33-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in H2\mathbb{H}^2 have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2Ω(n)2^{\Omega(\sqrt{n})} time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching nΩ(logn)n^{\Omega(\log n)} lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

    Developments on Spectral Characterizations of Graphs

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    In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs
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