Rumor Spreading and Vertex Expansion

Giakkoupis, George; Sauerwald, Thomas

doi:10.1137/1.9781611973099.129

Cited by 43 publications

(47 citation statements)

References 30 publications

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“…Similar results were obtained for the spreading time of uniform gossip on networks with vertex expansion α. In particular, [14,13] proved an O( log 2 n α ) bound and showed it to be tight as well. These results demonstrate very nicely that uniform gossip performs at least as good as the worst bottleneck that can be found in the network.…”

Section: Related Workmentioning

confidence: 93%

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Simple, Fast and Deterministic Gossip and Rumor Spreading

Haeupler

2015

J. ACM

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We study gossip algorithms for the rumor spreading problem, which asks each node to deliver a rumor to all nodes in an unknown network. Gossip algorithms allow nodes only to call one neighbor per round and have recently attracted attention as message efficient, simple, and robust solutions to the rumor spreading problem.A long series of papers analyzed the performance of uniform random gossip in which nodes repeatedly call a random neighbor to exchange all rumors with. A main result of this line of work was that uniform gossip completes in O( log n Φ ) rounds where Φ is the conductance of the network. More recently, non-uniform random gossip schemes were devised to allow efficient rumor spreading in networks with bottlenecks. In particular, STOC'12] gave an O(log 3 n) algorithm to solve the 1-local broadcast problem in which each node wants to exchange rumors locally with its 1-neighborhood. By repeatedly applying this protocol, one can solve the global rumor spreading quickly for all networks with small diameter, independently of the conductance.All these algorithms are inherently randomized in their design and analysis. A parallel research direction has been to reduce and determine the amount of randomness needed for efficient rumor spreading. This has been done via lower bounds for restricted models and by designing gossip algorithms with a reduced need for randomness, e.g., by using pseudorandom generators with short random seeds. The general intuition and consensus of these results has been that randomization plays a important role in effectively spreading rumors and that at least a polylogarithmic number of random bit are crucially needed.In this paper we improves over this state of the art in several ways by presenting a deterministic gossip algorithm that solves the the k-local broadcast problem in 2(k + log n) log n rounds 1 . Besides being the first efficient deterministic solution to the rumor spreading problem this algorithm is interesting in many aspects: It is simpler, more natural, more robust, and faster than its randomized pendant and guarantees success with certainty instead of with high probability. Its analysis is furthermore simple, self-contained, and fundamentally different from prior works. * The conference version of this paper was published in the ACM-SIAM Symposium for Discrete Algorithms [18] where it won a best student paper award.1 Throughout this paper log x denotes log 2 x , that is, the rounded up binary logarithm.

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Section: Related Workmentioning

confidence: 93%

“…A series of results showed that this algorithm performs well on well-connected graphs with no bottleneck(s) [11,22,6,5,12,14,13]. More precisely, the main result is a tight bound of O( log n Φ ) rounds, where Φ is the conductance of the network.…”

Section: Introductionmentioning

confidence: 99%

Simple, Fast and Deterministic Gossip and Rumor Spreading

Haeupler

2015

J. ACM

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“…The relationship between the time required for randomized information spreading and the vertex expansion of the underlying graph was studied in [24,25,40]. Feige et al [17] showed in their important paper a bound of O(∆(D + log n)) rounds for UniformGossip to spread a rumor in a graph of diameter D and maximum degree ∆.…”

Section: Related Workmentioning

confidence: 99%

Rumor Spreading with No Dependence on Conductance

Censor-Hillel¹,

Haeupler²,

Kelner³

et al. 2017

SIAM J. Comput.

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Abstract. In this paper, we study how a collection of interconnected nodes can efficiently perform a global computation in the GOSSIP model of communication. In this model nodes do not know the global topology of the network and may only initiate contact with a single neighbor in each round. This contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of Θ(φ −1 log n). In this paper, we give the first protocol that works efficiently on any topology. In particular we give an algorithm that solves the information dissemination problem in at most O(D +polylog(n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are equivalent up to an additive polylogarithmic term.

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“…While usually good expansion of the underlying graph implies fast rumor spreading [3,5,6,19,21,34], it was far from clear if graph expansion is the only reason for fast rumor spreading and more general gossip processes. For instance, most of these gossip algorithms are inherently randomized, and all of the analysis of these algorithms crucially rely on choosing neighbors independently and uniformly at random in each round.…”

Section: Introductionmentioning

confidence: 99%

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

Guo¹,

Sun²

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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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, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem:First, we establish a new connection between the rumor spreading process in an arbitrary graph and certain Markov chains. While most previous work analyzed the rumor spreading time in general graphs by studying the rate of the number of (un-)informed nodes after every round, we show that the mixing time of a certain Markov chain suffices to bound the rumor spreading time in an arbitrary graph.Second, we construct a reduction from rumor spreading processes to branching programs. This reduction gives us a general framework to derandomize the rumor spreading and other gossip processes. In particular, we show that, for any n-vertex expander graph, there is a protocol which informs every node in O(log n) rounds with high probability, and uses O(log n · log log n) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O(log n · log log n) random bits almost matches the lower bound of Ω(log n) random bits. We further show that, for many graph families (defined with respect to the expansion and the degree), O(poly log n) random bits in total suffice for fast rumor spreading. These results give us an almost complete understanding of the role of randomness in the rumor spreading process, which was extensively studied over the past years.

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Rumor Spreading and Vertex Expansion

Cited by 43 publications

References 30 publications

Simple, Fast and Deterministic Gossip and Rumor Spreading

Simple, Fast and Deterministic Gossip and Rumor Spreading

Rumor Spreading with No Dependence on Conductance

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

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