Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance

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 $O(D+\text{polylog}{(n)})$ rounds in a network of diameter $D$, withno dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of $D$, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires $T$ rounds in the LOCAL model can be simulated in $O(T +\mathrm{polylog}(n))$ rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent.

💡 Research Summary

The paper addresses the fundamental problem of information dissemination (also known as the rumor spreading problem) in the GOSSIP communication model, where each node may initiate contact with at most one neighbor per synchronous round and has no knowledge of the global network topology. Historically, the UniformGossip algorithm—where each node randomly selects a neighbor each round—has been the primary tool for this setting. Its convergence time, however, depends heavily on the graph’s conductance φ, yielding a bound of O(φ⁻¹·log n). In poorly connected graphs with small conductance, this leads to Θ(n) round complexities, far from optimal.

The authors break this conductance dependence by introducing a new subroutine called Superstep that solves the NeighborExchange problem (each node must deliver its current messages to all of its neighbors) in O(log³ n) rounds for any graph. Superstep repeatedly runs the UniformGossip protocol for log³ n rounds, then reverses the direction of the message exchanges to enforce symmetry. A key insight is that within any high‑conductance cluster, a single UniformGossip iteration suffices for every pair of vertices inside the cluster to exchange messages. The authors prove an existential decomposition: every graph can be partitioned into clusters of sufficiently high conductance such that a constant fraction of intra‑cluster edges belong to these clusters. Consequently, each Superstep iteration reduces the number of “unsatisfied” edges (edges whose endpoints have not yet exchanged messages) by a constant factor, leading to a logarithmic number of iterations overall.

Having a fast NeighborExchange primitive enables a simulation of any LOCAL‑model algorithm (where a node may broadcast to all neighbors each round) within the GOSSIP model. The simulation works by first using Superstep to emulate the LOCAL broadcast of a round, then performing the local computation of that round. This yields a (log³ n, 0)‑simulator: any T‑round LOCAL algorithm can be executed in O(T + log³ n) GOSSIP rounds. By leveraging known sparse‑graph spanner constructions, the multiplicative overhead can be reduced further for specific graph families. For graphs with bounded degree Δ, hereditary density δ, or other sparsity measures, the authors obtain simulators with overheads O(Δ), O(δ + δ·log n), etc.

Applying this simulation to the trivial LOCAL algorithm for rumor spreading—each node broadcasts its known messages to all neighbors each round—gives an O(D + polylog n) GOSSIP algorithm, where D is the network diameter. This matches the trivial lower bound of Ω(D) up to an additive polylogarithmic term and dramatically improves upon prior bounds that were Ω(n) for many low‑diameter graphs. The result shows that, despite the severe communication restriction of the GOSSIP model, it is essentially as powerful as the LOCAL model up to a small additive polylogarithmic penalty.

The paper also explores the relationship between NeighborExchange and graph spanners. Any fast NeighborExchange algorithm implicitly constructs a low‑stretch spanner, and conversely, any efficient spanner construction can be used to improve the simulator’s overhead. The Superstep algorithm itself can be viewed as a novel spanner construction that uses only O(n·log³ n) total messages, a first of its kind in the GOSSIP setting.

In summary, the contributions are:

1. A conductance‑independent O(D + polylog n) rumor‑spreading algorithm for the GOSSIP model.
2. The Superstep subroutine solving NeighborExchange in O(log³ n) rounds for arbitrary graphs.
3. A general simulation framework showing that any LOCAL algorithm can be run in the GOSSIP model with only an additive polylogarithmic overhead.
4. Connections to graph decomposition and spanner theory that yield refined simulators for sparse graph families.

These results close a long‑standing gap between the two most studied distributed communication models and provide a robust theoretical foundation for designing fast global algorithms in networks where nodes can only engage in limited, pairwise communications.