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 @math rounds in a network of diameter @math , withno dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of @math , which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires @math rounds in the LOCAL model can be simulated in @math rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent.
One fundamental property of the distributed system that affects the number of rounds required for information spreading is the communication model. The model was introduced by @cite_1 , allowing every node to contact one other node in each round. In our setting, this corresponds to the complete graph. This model alone received much attention, such as in bounding the number of calls @cite_7 , bounding the number of random bits used @cite_8 , bounding the number of bits @cite_11 , and more.
@cite_11 @cite_1 @cite_7 @cite_8
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We study the communication complexity of rumor spreading in the random phone-call model. Suppose nplayers communicate in parallel rounds, where in each round every player calls a randomly selected communication partner. A player u is allowed to exchange messages during a round only with the player that u called, and with all the players that @math received calls from, in that round. In every round, a (possibly empty) set of rumors to be distributed among all players is generated, and each of the rumors is initially placed in a subset of the players. Karp et. al Karp2000 showed that no rumor-spreading algorithm that spreads a rumor to all players with constant probability can be both time-optimal, taking O(lg n) rounds, and message-optimal, using O(n) messages per rumor. For address-oblivious algorithms, in particular, they showed that Ω(n lg lg n) messages per rumor are required, and they described an algorithm that matches this bound and takes O(lg n) rounds. We investigate the number of communication bits required for rumor spreading. On the lower-bound side, we establish that any address-oblivious algorithm taking O(lg n) rounds requires Ω(n (b+ lg lg n)) communication bits to distribute a rumor of size b bits. On the upper-bound side, we propose an address-oblivious algorithm that takes O(lg n) rounds and uses O(n(b+ lg lg n lg b)) bits. These results show that, unlike the case for the message complexity, optimality in terms of both the running time and the bit communication complexity is attainable, except for very small rumor sizes b Investigates the class of epidemic algorithms that are commonly used for the lazy transmission of updates to distributed copies of a database. These algorithms use a simple randomized communication mechanism to ensure robustness. Suppose n players communicate in parallel rounds in each of which every player calls a randomly selected communication partner. In every round, players can generate rumors (updates) that are to be distributed among all players. Whenever communication is established between two players, each one must decide which of the rumors to transmit. The major problem is that players might not know which rumors their partners have already received. For example, a standard algorithm forwarding each rumor form the calling to the called players for spl Theta (ln n) rounds needs to transmit the rumor spl Theta (n ln n) times in order to ensure that every player finally receives the rumor with high probability. We investigate whether such a large communication overhead is inherent to epidemic algorithms. On the positive side, we show that the communication overhead can be reduced significantly. We give an algorithm using only O(n ln ln n) transmissions and O(ln n) rounds. In addition, we prove the robustness of this algorithm. On the negative side, we show that any address-oblivious algorithm needs to send spl Omega (n ln ln n) messages for each rumor, regardless of the number of rounds. Furthermore, we give a general lower bound showing that time and communication optimality cannot be achieved simultaneously using random phone calls, i.e. every algorithm that distributes a rumor in O(ln n) rounds needs spl omega (n) transmissions. We propose a new protocol for the fundamental problem of disseminating a piece of information to all members of a group of n players. It builds upon the classical randomized rumor spreading protocol and several extensions. The main achievements are the following: Our protocol spreads a rumor from one node to all other nodes in the asymptotically optimal time of (1 + o(1)) log2 n. The whole process can be implemented in a way such that only O(nf(n)) calls are made, where f(n) = ω(1) can be arbitrary. In spite of these quantities being close to the theoretical optima, the protocol remains relatively robust against failures; for random node failures, our algorithm again comes arbitrarily close to the theoretical optima. The protocol can be extended to also deal with adversarial node failures. The price for that is only a constant factor increase in the run-time, where the constant factor depends on the fraction of failing nodes the protocol is supposed to cope with. It can easily be implemented such that only O(n) calls to properly working nodes are made. In contrast to the push-pull protocol by [FOCS 2000], our algorithm only uses push operations, i.e., only informed nodes take active actions in the network. On the other hand, we discard address-obliviousness. To the best of our knowledge, this is the first randomized push algorithm that achieves an asymptotically optimal running time. 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 so-called gate-model. In this paper, we consider general rumor spreading protocols. We provide a simple push-protocol 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 (non-uniform) push-protocol 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 time-randomness 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.