On the Influence of Graph Density on Randomized Gossiping

Abstract

Information dissemination is a fundamental problem in parallel and distributed computing. In its simplest variant, known as the broadcasting problem, a single message has to be spread among all nodes of a graph. A prominent communication protocol for this problem is based on the so-called random phone call model (Karp et al., FOCS 2000). In each step, every node opens a communication channel to a randomly chosen neighbor, which can then be used for bi-directional communication. In recent years, several efficient algorithms have been developed to solve the broadcasting problem in this model.Motivated by replicated databases and peer-to-peer networks, Berenbrink et al., ICALP 2010, considered the so-called gossiping problem in the random phone call model. There, each node starts with its own message and all messages have to be disseminated to all nodes in the network. They showed that any O(log n)-time algorithm in complete graphs requires Ω(log n) message transmissions per node to complete gossiping, with high probability, while it is known that in the case of broadcasting the average number of message transmissions per node is O(log log n). Furthermore, they explored different possibilities on how to reduce the communication overhead of randomized gossiping in complete graphs.It is known that the O(n log log n) bound on the number of message transmissions produced by randomized broadcasting in complete graphs cannot be achieved in sparse graphs even if they have best expansion and connectivity properties. In this paper, we analyze whether a similar influence of the graph density also holds w.r.t. the performance of gossiping. We study analytically and empirically the communication overhead generated by gossiping algorithms w.r.t. the random phone call model in random graphs and also consider simple modifications of the random phone call model in these graphs. Our results indicate that, unlike in broadcasting, there seems to be no significant difference between the performance of randomized gossiping in complete graphs and sparse random graphs. Furthermore, our simulations illustrate that by tuning the parameters of our algorithms, we can significantly reduce the communication overhead compared to the traditional push-pull approach in the graphs we consider.

Self-stabilizing repeated balls-into-bins

“…The above result strongly improves over the best previous bounds [12][13][14] and it is almost tight, since the classical lower bound Ω(log n/ log log n) on the maximum load (see, e.g., [11]) clearly applies also in our repeated setting. Our result further implies that, under the FIFO queueing policy, any ball performs Ω(t/ log n) steps of its individual random walk over any sequence of t = poly(n) rounds w.h.p., so the parallel cover time is O n log 2 n w.h.p.…”

“…Their aim is to achieve a fast mixing time for every random walk in the case of good expander graphs. In particular, in [13], a logarithmic bound is shown for the complete graph when m = O(n/ log n) random walks are performed over a logarithmic time interval, while a similar bound is also given for some families of almost-regular random graphs in [14]. A new analysis is given in [12] for regular graphs and time intervals of arbitrary length, yielding the bound O √ t .…”

“…4), and [14] (see there Lemma 6 in Subsection 3.1), since it describes the process of performing parallel random walks in the (uniform) gossip model (also known as random phone-call model [15,16]) when every message can contain at most one token. Maximum load (i.e., node congestion), token delays, mixing and cover times are here the most crucial aspects.…”

“…This line of research is also motivated by several recent applications of parallel random walks in the (uniform) gossip model [8,13,14,40]. As mentioned in the introduction, previous analysis of this process provides a bound O √ t on the maximum load after t rounds on regular graphs [12].…”

Self-Stabilizing Repeated Balls-into-Bins

“…Previous results are thus not helpful to establish whether this process is stable (or, even more, self-stabilizing) or not. Moreover, the previous analyses of the maximum load in [7,9,20] are far from tight, since they rely on some rough approximations of the studied process via other, much simpler Markov chains: for instance, in [7], the authors consider the processwhich clearly dominates the original one -where, at every round, a new ball is inserted in every empty bin. That analysis thus does not exploit the global invariant (a fixed number n of balls) of the original process.…”