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13
Quasirandom Rumor Spreading
 In Proc. of SODA’08
, 2008
"... We propose and analyse a quasirandom analogue to the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round each informed node chooses a neighbor at random and informs it. Results of Frieze and Grimmett (Discrete Appl. Mat ..."
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Cited by 24 (10 self)
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We propose and analyse a quasirandom analogue to the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round each informed node chooses a neighbor at random and informs it. Results of Frieze and Grimmett (Discrete Appl. Math. 1985) show that this simple protocol succeeds in spreading a rumor from one node of a complete graph to all others within O(log n) rounds. For the network being a hypercube or a random graph G(n, p) with p ≥ (1+ε)(log n)/n, also O(log n) rounds suffice (Feige, Peleg, Raghavan, and Upfal, Random Struct. Algorithms 1990). In the quasirandom model, we assume that 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. Surprisingly, irrespective of the orders of the lists, the above mentioned bounds still hold. In addition, we also show a O(log n) bound for sparsely connected random graphs G(n, p) with p = (log n+f(n))/n, where f(n) → ∞ and f(n) = O(log log n). Here, the classical model needs Θ(log 2 (n)) rounds. Hence the quasirandom model achieves similar or better broadcasting times with a greatly reduced use of random bits.
Quasirandom Rumor Spreading: Expanders, Push vs. Pull, and Robustness
"... Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading ..."
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Cited by 18 (6 self)
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Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom rumor spreading inform all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and ErdősRényi random graphs. For all network topologies, we show that if one of the push or pull model works well, so does the other. We also show that quasirandom rumor spreading is robust against transmission failures. If each message sent out gets lost with probability f, then the runtime increases only by a factor of O(1/(1 − f)).
Broadcasting vs. mixing and information dissemination on Cayley graphs
 In 24th Int. Symp. on Theor. Aspects of Computer Science (STACS
, 2007
"... Abstract. One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succe ..."
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Cited by 13 (6 self)
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Abstract. One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. First, we consider the relationship between randomized broadcasting and random walks on graphs. In particular, we prove that the runtime of the algorithm described above is upper bounded by the corresponding mixing time, up to a logarithmic factor. One key ingredient of our proofs is the analysis of a continuoustype version of the afore mentioned algorithm, which might be of independent interest. Then, we introduce a general class of Cayley graphs, including (among others) Star graphs, Transposition graphs, and Pancake graphs. We show that randomized broadcasting has optimal runtime on all graphs belonging to this class. Finally, we develop a new proof technique by combining martingale tail estimates with combinatorial methods. Using this approach, we show the optimality of our algorithm on another Cayley graph and obtain new knowledge about the runtime distribution on several Cayley graphs. 1
On the Randomness Requirements of Rumor Spreading
, 2011
"... We investigate the randomness requirements of the classical rumor spreading problem on fully connected graphs with n vertices. In the standard random protocol, where each node that knows the rumor sends it to a randomly chosen neighbor in every round, each node needs O((log n) 2) random bits in orde ..."
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Cited by 7 (1 self)
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We investigate the randomness requirements of the classical rumor spreading problem on fully connected graphs with n vertices. In the standard random protocol, where each node that knows the rumor sends it to a randomly chosen neighbor in every round, each node needs O((log n) 2) random bits in order to spread the rumor in O(log n) rounds with high probability (w.h.p.). For the simple quasirandom rumor spreading protocol proposed by Doerr, Friedrich, and Sauerwald (2008), ⌈log n ⌉ random bits per node are sufficient. A lower bound by Doerr and Fouz (2009) shows that this is asymptotically tight for a slightly more general class of protocols, the socalled gatemodel. In this paper, we consider general rumor spreading protocols. We provide a simple pushprotocol that requires only a total of O(n log log n) random bits (i.e., on average O(log log n) bits per node) in order to spread the rumor in O(log n) rounds w.h.p. We also investigate the theoretical minimal randomness requirements of efficient rumor spreading. We prove the existence of a (nonuniform) pushprotocol for which a total of 2 log n + log log n + o(log log n) random bits suffice to spread the rumor in log n + ln n + O(1) rounds with probability 1−o(1). This is contrasted by a simple timerandomness tradeoff for the class of all rumor spreading protocols, according to which any protocol that uses log n − log log n − ω(1) random bits requires ω(log n) rounds to spread the rumor.
Quasirandom rumor spreading: An experimental analysis
 In Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX
, 2009
"... We empirically analyze two versions of the wellknown “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. At SODA 2008, three of the authors proposed a quasirandom variant. Here, ..."
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Cited by 6 (4 self)
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We empirically analyze two versions of the wellknown “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. At SODA 2008, three of the authors proposed a quasirandom variant. Here, 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 (which was expected, though maybe not to this extent), but also that the runtime is more concentrated around the mean value (which 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 the particular structure of the lists has little influence on the efficiency. In particular, there is no problem if all nodes use an identical order to inform their neighbors. 1
How Efficient Can Gossip Be? (On the Cost of Resilient Information Exchange)
"... Gossip, also known as epidemic dissemination, is becoming an increasingly popular technique in distributed systems. Yet, it has remained a partially open question: how robust are such protocols? We consider a natural extension of the random phonecall model (introduced by Karp et al. [1]), and we an ..."
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Cited by 2 (0 self)
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Gossip, also known as epidemic dissemination, is becoming an increasingly popular technique in distributed systems. Yet, it has remained a partially open question: how robust are such protocols? We consider a natural extension of the random phonecall model (introduced by Karp et al. [1]), and we analyze two different notions of robustness: the ability to tolerate adaptive failures, and the ability to tolerate oblivious failures. For adaptive failures, we present a new gossip protocol, TrickleGossip, which achieves nearoptimal O(n log 3 n) message complexity. To the best of our knowledge, this is the first epidemicstyle protocol that can tolerate adaptive failures. We also show a direct relation between resilience and message complexity, demonstrating that gossip protocols which tolerate a large number of adaptive failures need to use a superlinear number of messages with high probability. For oblivious failures, we present a new gossip protocol, CoordinatedGossip, that achieves optimal O(n) message complexity. This protocol makes novel use of the universe reduction technique to limit the message complexity.
Algorithmica DOI 10.1007/s004530129710y Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions
, 2012
"... Abstract A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n1/d] d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω( ..."
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Abstract A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n1/d] d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean log n distance of ω( rd−1), their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n1/d /r) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n1/d /r + log n) rounds.
Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions
"... Abstract A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n 1/d] d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω ..."
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Abstract A random geometric graph (RGG) is defined by placing n points uniformly at random in [0,n 1/d] d, and joining two points by an edge whenever their Euclidean distance is at most some fixed r. We assume that r is larger than the critical value for the emergence of a connected component with Ω(n) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a minimum Euclidean distance of ω(logn), their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is Θ(n 1/d /r) w.h.p. We also analyze the following randomized broadcast algorithm on RGGs. Atthe beginning, onlyone node from the largest connectedcomponent of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within Θ(n 1/d /r +logn) rounds. 1
Fast Fault Tolerant Rumor Spreading with Minimum Message Complexity
, 2012
"... We study fault tolerant rumor spreading algorithms in the complete graph topology. Our focus is on algorithms that use minimum communication both in a global and local sense: they establish the minimum possible number of interprocessor connections in total, and in each round each processor is invol ..."
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We study fault tolerant rumor spreading algorithms in the complete graph topology. Our focus is on algorithms that use minimum communication both in a global and local sense: they establish the minimum possible number of interprocessor connections in total, and in each round each processor is involved in at most one connection. The challenge is designing such algorithms that have an asymptotically optimal, that is, logarithmic, time complexity even in the presence of failed nodes. We first show that if nodes are crashed not adversarially, but independently at random with constant probability less than one, then already the basic GP algorithm of Gasieniec and Pelc (Parallel Computing 22:903–912, 1996) with high probability has an asymptotically optimal O(log n) time complexity. This improves significantly over the worstcase guarantee of f + O(log n) given there for f crashed nodes. We then show that by adding randomization to the algorithm, these time and communication complexities can be maintained also against adversarial failures. This is easily achieved by running the GPalgorithm with randomly permuted node labels, at the price, however, that this permutation (or at least significant parts of it) also have to be disseminated. To overcome this, we show that the random permutation can be chosen from a set of only ω(n / log n) permutations. Consequently, the permutation can be communicated by adding Θ(log n) bits to each message, which is an overhead produced by many communication protocols including the GP algorithm. Naturally, this requires all processors to know this set of permutations, which needs ω(n 2) space at each processor and some preliminary communication to set up the system.
Fast Generation and Mixing of Random Graphs in PeertoPeer Networks
, 2011
"... Abstract—In this work we show how to generate and rapidly mix uniform random graphs in a model where incoming and outgoing degrees of nodes are defined in advance. We show how to use a previous result on Dating Service working on top of any Distributed Hash Table so that a random graph is generated ..."
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Abstract—In this work we show how to generate and rapidly mix uniform random graphs in a model where incoming and outgoing degrees of nodes are defined in advance. We show how to use a previous result on Dating Service working on top of any Distributed Hash Table so that a random graph is generated in logarithmic number of rounds and mixed so that two snapshots of the graph taken in logarithmic time distance are independent with high probability. We consider two models of graphs: directed graphs and undirected graphs where some nodes are behind firewalls. We consider a synchronous model of computation but show how to adapt it to a highly dynamic and asynchronous environment such as peertopeer networks. Keywordspeertopeer, distributed hash tables, heterogeneous P2P, random graph generation and mixing I.