@math rounds has been a well known upper bound for rumor spreading using push&pull in the random phone call model (i.e., uniform gossip in the complete graph). A matching lower bound of @math is also known for this special case. Under the assumption of this model and with a natural addition that nodes can call a partner once they learn its address (e.g., its IP address) we present a new distributed, address-oblivious and robust algorithm that uses push&pull with pointer jumping to spread a rumor to all nodes in only @math rounds, w.h.p. This algorithm can also cope with @math node failures, in which case all but @math nodes become informed within @math rounds, w.h.p.
Beside the basic random phone call model, gossip algorithms and rumor spreading were generalized in several different ways. The basic extension was to study (i.e., the called partner is selected uniformly at random from the neighbors lists) on graphs other than the clique. Feige et. al. @cite_25 studied randomized broadcast in networks and extended the result of @math rounds for to different types of graphs like hypercubes and random graphs models. Following the work of @cite_31 , and in particular in recent years the protocol was studied intensively, both to give tight bounds for general graphs and to understand its performance advantages on specific families of graphs. A lower bound of @math for uniform gossip on the clique can be conclude from @cite_24 that studies the sequential case. We are not aware of a lower bound for general, address-oblivious .
@cite_24 @cite_31 @cite_25
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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. We show three results. For random graphs (with sufficiently many edges) we present an address-oblivious algorithm with runtime O(log n) that uses at most O(n log log n) message transmissions. For hypercubes of dimension log n we present an address-oblivious algorithm with runtime O(log n) that uses at most O(n(log log n)2) message transmissions. For hypercubes we also show a lower bound of Ω(n log n log log n) on the total number of message transmissions required by any O(log n) time address-oblivious algorithm in the standard random phone call model. Together with a result of [8], our results imply that for random graphs and hypercubes the communication complexity of the quasirandom phone call model is significantly smaller than that of the standard phone call model. This seems to be surprising given the small amount of randomness used in our model. 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 +e)(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(log2(n)) rounds. Hence the quasirandom model achieves similar or better broadcasting times with a greatly reduced use of random bits. In this paper, we provide a detailed comparison between a fully randomized protocol for rumor spreading on a complete graph and a quasirandom protocol introduced by Doerr, Friedrich, and Sauerwald [Quasirandom rumor spreading, in Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 773-781]. In the former, initially there is one vertex which holds a piece of information, and during each round every one of the informed vertices chooses uniformly at random and independently one of its neighbors and informs it. In the quasirandom version of this method (cf. Doerr, Friedrich, and Sauerwald) each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor, and at each subsequent round it picks the next neighbor from its list and informs it. We give a precise analysis of the evolution of the quasirandom protocol on the complete graph with @math vertices and show that it evolves essentially in the same way as the randomized protocol. In particular, if @math denotes the number of rounds that are needed until all vertices are informed, we show that for any slowly growing function @math , we have @math , with probability @math . 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. Recently, non-uniform random gossip schemes were devised to allow efficient rumor spreading in networks with bottlenecks. In particular, [Censor-, 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. This and all prior gossip algorithms for the rumor spreading problem have been inherently randomized in their design and analysis. This resulted in a parallel research direction trying 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. The general intuition and consensus of these results has been that randomization plays a important role in effectively spreading rumors. 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. 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.