61 research outputs found
Quasirandom Rumor Spreading
We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (ârandomized rumor spreadingâ). 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, ErdĆs-RĂ©nyi random graphs, and Ramanujan graphs with probability 1 â
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.
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Quasirandom Rumor Spreading: An Experimental Analysis
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
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 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 , after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n)
rounds the quasirandom protocol has informed all nodes in the network with
probability at least 1-n^{-p\e/40}. Note that this is faster than the
intuitively natural factor increase over the run-time of approximately
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
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Communication Complexity of Quasirandom Rumor Spreading
We consider rumor spreading on random graphs and hypercubes in the quasirandom phone call model. In this model, every node has a list of neighbors whose order is specified by an adversary. In step i every node opens a channel to its ith neighbor (modulo degree) on that list, beginning from a randomly chosen starting position. Then, the channels can be used for bi-directional communication in that step. The goal is to spread a message efficiently to all nodes of the graph.For random graphs (with sufficiently many edges) we present an address-oblivious algorithm with runtime O(logn) that uses at most O(nloglogn) message transmissions. For hypercubes of dimension logn we present an address-oblivious algorithm with runtime O(logn) that uses at most O(n(loglogn)2) message transmissions.Together with a result of ElsĂ€sser (Proc. of SPAAâ06, pp. 148â157, 2006), our results imply that for random graphs the communication complexity of the quasirandom phone call model is significantly smaller than that of the standard phone call model
Quasi-Random Rumor Spreading: Reducing Randomness Can Be Costly
We give a time-randomness tradeoff for the quasi-random rumor spreading
protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete
graphs. In this protocol, the goal is to spread a piece of information
originating from one vertex throughout the network. Each vertex is assumed to
have a (cyclic) list of its neighbors. Once a vertex is informed by one of its
neighbors, it chooses a position in its list uniformly at random and then
informs its neighbors starting from that position and proceeding in order of
the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron.~J.~Combin.~2009]
showed that after rounds, the rumor will have been
broadcasted to all nodes with probability .
We study the broadcast time when the amount of randomness available at each
node is reduced in natural way. In particular, we prove that if each node can
only make its initial random selection from every -th node on its list,
then there exists lists such that steps are needed to inform every vertex with
probability at least . This shows that a further reduction of the amount of
randomness used in a simple quasi-random protocol comes at a loss of
efficiency
Randomized rounding and rumor spreading with stochastic dependencies
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 heiĂt auf zufĂ€llige Rundungen, bei denen jede Zahl mit der Wahrscheinlichkeit entsprechend ihren Nachkommastellen aufgerundet wird. Der zweite, gröĂere 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
Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance
In this paper, we study the question of how efficiently a collection of
interconnected nodes can perform a global computation in the widely studied
GOSSIP model of communication. In this model, nodes do not know the global
topology of the network, and they may only initiate contact with a single
neighbor in each round. This model 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 this paper, we give an algorithm that solves the information
dissemination problem in at most rounds in a network
of diameter , withno dependence on the conductance. This is at most an
additive polylogarithmic factor from the trivial lower bound of , which
applies even in the LOCAL model. In fact, we prove that something stronger is
true: any algorithm that requires rounds in the LOCAL model can be
simulated in rounds in the GOSSIP model. We thus
prove that these two models of distributed computation are essentially
equivalent
Bit-complexité des protocoles de gossip
National audienceNous Ă©tudions le problĂšme du \emph{gossip} (i.e., diffusion de rumeurs) dans le modĂšle des appels alĂ©atoires. ConsidĂ©rons noeuds communiquant en parallĂšle par Ă©tape. A chaque Ă©tape, un ensemble (potentiellement vide) de \emph{rumeurs} est gĂ©nĂ©rĂ© Ă chaque noeud, la mĂȘme rumeur pouvant ĂȘtre gĂ©nĂ©rĂ©e simultanĂ©ment par plusieurs noeuds. L'objectif est de diffuser ces rumeurs Ă tous les noeuds. Pour ce faire, Ă chaque Ă©tape, chaque noeud appelle un autre noeud choisi uniformĂ©ment alĂ©atoirement parmi l'ensemble de tous les noeuds, et un noeud ne peut alors communiquer qu'avec le noeud qu'il a appelĂ©, et les noeuds qui l'ont potentiellement appelĂ©. Dans ce modĂšle, Karp et ses co-auteurs~\cite{Karp2000} ont montrĂ© qu'aucun algorithme de gossip ne peut ĂȘtre Ă la fois optimal en temps (i.e., s'exĂ©cuter en Ă©tapes) et en volume de communication (i.e., s'exĂ©cuter en transmettant au plus messages). En particulier, ils ont montrĂ© que tout algorithme de gossip n'utilisant pas les IDs des noeuds et diffusant toute rumeur en Ă©tapes doit Ă©changer messages par rumeur. Karp et ses co-auteurs ont Ă©galement montrĂ© que ce compromis peut ĂȘtre atteint. Dans cet article, nous Ă©tudions le volume de communication estimĂ© en nombre de bits Ă©changĂ©s plutĂŽt qu'en nombre de messages. Nous montrons tout d'abord que tout algorithme de gossip n'utilisant pas les IDs des noeuds et diffusant toute rumeur en Ă©tapes doit Ă©changer bits pour diffuser une rumeur de bits. Nous proposons alors un algorithme de gossip n'utilisant pas les IDs des noeuds qui diffuse toute rumeur en Ă©tapes, en Ă©changeant bits pour une rumeur de bits. Ces rĂ©sultats dĂ©montrent que contrairement Ă ce qu'il peut sembler lorsque l'on mesure le volume de communication en nombre de messages, il est possible d'ĂȘtre Ă la fois optimal en temps (i.e., s'exĂ©cuter en Ă©tapes) et en volume de communication (i.e., s'exĂ©cuter en transmettant au plus bits), sauf pour des rumeurs extrĂȘmement petites, de taille bits
Low Randomness Rumor Spreading via Hashing
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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