Rumor Spreading with No Dependence on Conductance

Abstract

In this paper, we study how a collection of interconnected nodes can efficiently perform a global computation in the $\mathcal{GOSSIP}$ 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 $\mathcal{LOCAL}$ 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 $\mathcal{LOCAL}$ 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 $\mathcal{GOSSIP}$ 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 $\phi$ and graph size $n$, culminating in a bound of $\Theta(\phi^{-1} \log n)$. 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 $O(D+\text{polylog}{(n)})$ 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 $\mathcal{LOCAL}$ model can be simulated in $O(T +\mathrm{polylog}(n))$ rounds in the $\mathcal{GOSSIP}$ model. We thus prove that these two models of distributed computation are equivalent up to an additive polylogarithmic term.