Rumor Spreading with No Dependence on Conductance

Abstract

Abstract. In this paper, we study how a collection of interconnected nodes can efficiently perform a global computation in the GOSSIP model of communication. In this model nodes do not know the global topology of the network and may only initiate contact with a single neighbor in each round. This 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 the LOCAL model this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of Θ(φ −1 log n). In this paper, we give the first protocol that works efficiently on any topology. In particular we give an algorithm that solves the information dissemination problem in at most O(D +polylog(n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are equivalent up to an additive polylogarithmic term.

Random Gossip Processes in Smartphone Peer-to-Peer Networks

“…Notice, because our algorithm only transfers a constant number of tokens in each call to Communicate, each such call requires at most O(b max /R b ) time. 4 The δ update and δ conn parameters upper bound the time required to get through the update and blockedForConn methods, respectively. It follows that each iteration of our gossip algorithm's main aMTM loop requires at most O(δ max ) time, making δ max a useful aggregate parameter for bounding asynchronous time complexity.…”

“…Similarly, we can adapt our notions of the productive subgraph G(t), and minimum productive subgraph G min (t), for a time t using the values of s u (t) and N u (t). That being said, 4 We omit for now the time required for two connected nodes to determine which token to transfer. Our algorithm simply specifies that they transfer some token in the set difference of their token sets.…”

“…By comparison, the best-known gossip solution in the classical telephone model requires O(D + polylog(n)) rounds [4]. This result was considered a breakthrough as it removed the dependence on graph properties such as expansion or conductance.…”

“…This result was considered a breakthrough as it removed the dependence on graph properties such as expansion or conductance. The solution in [4], however, requires unbounded concurrent connections and unbounded message size (allowing all rumors in the set difference between two nodes to be delivered during a given one-round connection 2 ).…”

How to Spread a Rumor

“…The above two protocols have been studied extensively over the past 15 years, and have also found several applications, including data aggregation [7,28,31], resource discovery [24], failure detection [35], and even efficient simulation of arbitrary distributed computations [9].…”