35 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

    Strong Robustness of Randomized Rumor Spreading Protocols

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    Randomized rumor spreading is a classical protocol to disseminate information across a network. At SODA 2008, a quasirandom version of this protocol was proposed and competitive bounds for its run-time were proven. This prompts the question: to what extent does the quasirandom protocol inherit the second principal advantage of randomized rumor spreading, namely robustness against transmission failures? In this paper, we present a result precise up to (1±o(1))(1 \pm o(1)) factors. We limit ourselves to the network in which every two vertices are connected by a direct link. Run-times accurate to their leading constants are unknown for all other non-trivial networks. We show that if each transmission reaches its destination with a probability of p∈(0,1]p \in (0,1], after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n) rounds the quasirandom protocol has informed all nn nodes in the network with probability at least 1-n^{-p\e/40}. Note that this is faster than the intuitively natural 1/p1/p factor increase over the run-time of approximately log⁥2n+ln⁥n\log_2 n + \ln n for the non-corrupted case. We also provide a corresponding lower bound for the classical model. This demonstrates that the quasirandom model is at least as robust as the fully random model despite the greatly reduced degree of independent randomness.Comment: Accepted for publication in "Discrete Applied Mathematics". A short version appeared in the proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009). Minor typos fixed in the second version. Proofs of Lemma 11 and Theorem 12 fixed in the third version. Proof of Lemma 8 fixed in the fourth versio

    Quasirandom Rumor Spreading

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    We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within O (log n ) rounds on complete graphs, hypercubes, random regular graphs, ErdƑs-RĂ©nyi random graphs, and Ramanujan graphs with probability 1 − o (1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown. </jats:p

    Randomized rounding and rumor spreading with stochastic dependencies

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    Randomness is an important ingredient of modern computer science. The present thesis is concerned with two uses of randomness, viz. randomized roundings and randomized rumor spreading algorithms. The theorem of Beck and Fiala (1981) asserts that for every hypergraph and every set of vertex weights there is a rounding of the vertex weights such that the additive rounding error for all hyperedges is bounded by the maximum degree. In Chapter 2 this theorem will be extended to randomized roundings, that is, to roundings that are efficiently generated at random in such a way that each value is rounded up with probability equal to its fractional part. The larger part of this thesis deals with randomized rumor spreading algorithms. These are protocols for disseminating information on graphs. The classical randomized rumor spreading was introduced and first investigated by Frieze and Grimmett on the complete graph (1985). In Chapter 3 a generalization of their results both in terms of the model used and in terms of the underlying graph will be shown. In Chapter 4 a quasirandom rumor spreading protocol introduced by Doerr, Friedrich, and Sauerwald (2008) will be considered. We present a detailed analysis of its evolution and show that its performance and robustness match performance and robustness of the randomized rumor spreading protocol. The unifying idea is to use dependencies so as to obtain results that are superior or equal to those obtained via independent randomness.Die Verwendung von Zufallselementen ist ein wichtiger Bestandteil der modernen Informatik. Die vorliegende Arbeit untersucht zwei Bereiche, in denen randomisierte Methoden Verwendung finden, nĂ€mlich randomisierte Rundungen und randomisierte Algorithmen zur GerĂŒchteverbreitung. Der Satz von Beck und Fiala (1981) sagt aus, dass es fĂŒr jeden Hypergraphen und fĂŒr jeden Satz von Knotengewichten eine Rundung gibt derart, dass der Rundungsfehler pro Kante vom Maximalgrad beschrĂ€nkt wird. Im ersten Teil der Arbeit wird dieser Satz auf den Fall randomisierter Rundungen verallgemeinert, das heißt auf zufĂ€llige Rundungen, bei denen jede Zahl mit der Wahrscheinlichkeit entsprechend ihren Nachkommastellen aufgerundet wird. Der zweite, grĂ¶ĂŸere Teil der Arbeit handelt von randomisierten Algorithmen zur GerĂŒchteverbreitung. Das klassische "Randomized Rumor Spreading" wurde von Frieze und Grimmett (1985) eingefĂŒhrt. Ihre Ergebnisse werden in Kapitel 3 sowohl hinsichtlich des Modells als auch hinsichtlich des zugrundegelegten Graphen verallgemeinert. In Kapitel 4 wird ein quasizufĂ€lliges Modell zur GerĂŒchteverbreitung betrachtet und gezeigt, dass es bezĂŒglich Laufzeit und Robustheit dem klassischen Modell gleichwertig ist. Gemeinsam liegt beiden Teilen der Arbeit die Idee zugrunde, stochastische AbhĂ€ngigkeiten zu nutzen um Ergebnisse zu erzielen, die den unter Verwendung stochastischer UnabhĂ€ngigkeit erzielten gleichwertig oder ĂŒberlegen sind

    Tight bounds for rumor spreading in graphs of a given conductance

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    We study the connection between the rate at which a rumor spreads throughout a graph and the conductance of the graph -- a standard measure of a graph\u27s expansion properties. We show that for any n-node graph with conductance phi, the classical PUSH-PULL algorithm distributes a rumor to all nodes of the graph in O(phi^(-1) log(n)) rounds with high probability (w.h.p.). This bound improves a recent result of Chierichetti, Lattanzi, and Panconesi [STOC 2010], and it is tight in the sense that there exist graphs where Omega(phi^(-1)log(n)) rounds of the PUSH-PULL algorithm are required to distribute a rumor w.h.p. We also explore the PUSH and the PULL algorithms, and derive conditions that are both necessary and sufficient for the above upper bound to hold for those algorithms as well. An interesting finding is that every graph contains a node such that the PULL algorithm takes O(phi^(-1) log(n)) rounds w.h.p. to distribute a rumor started at that node. In contrast, there are graphs where the PUSH algorithm requires significantly more rounds for any start node

    Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading

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    We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network. We present the first protocol for any expander graph GG with nn nodes such that, the protocol informs every node in O(log⁥n)O(\log n) rounds with high probability, and uses O~(log⁥n)\tilde{O}(\log n) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O~(log⁥n)\tilde{O}(\log n) random bits almost matches the lower bound of Ω(log⁥n)\Omega(\log n) random bits for dense graphs. We further show that, for many graph families, polylogarithmic number of random bits in total suffice to spread the rumor in O(polylog⁥n)O(\mathrm{poly}\log n) rounds. These results together give us an almost complete understanding of the randomness requirement of this fundamental gossip process. Our analysis relies on unexpectedly tight connections among gossip processes, Markov chains, and branching programs. First, we establish a connection between rumor spreading processes and Markov chains, which is used to approximate the rumor spreading time by the mixing time of Markov chains. Second, we show a reduction from rumor spreading processes to branching programs, and this reduction provides a general framework to derandomize gossip processes. In addition to designing rumor spreading protocols, these novel techniques may have applications in studying parallel and multiple random walks, and randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1304.135

    Low Randomness Rumor Spreading via Hashing

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    International audienceWe consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple push-based protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudo-random generator, respectively. For several well-studied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability. Previous explicit algorithms require Omega(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashing-based algorithm is within an O(log n)-factor of the theoretical minimum determined by [Giakkoupis and Woelfel, 2011]
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