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

Keren Censor-Hillel, Bernhard Haeupler+2 more|arXiv (Cornell University)|Apr 14, 2011

Complexity and Algorithms in Graphs28 references4 citations

TL;DR

This paper presents a novel gossip-based algorithm for global computation in distributed networks that achieves fast rumor spreading in O(D + polylog n) rounds, independent of graph conductance. By recursively decomposing graphs into high-conductance clusters and simulating LOCAL model algorithms with only polylogarithmic overhead, the method enables efficient global information dissemination even in poorly connected networks.

ABSTRACT

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 $O(D+ ext{polylog}{(n)})$ 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.

Motivation & Objective

• Address the inefficiency of prior gossip algorithms that depend on graph conductance, especially in poorly connected or bottleneck-ridden networks.
• Overcome the limitation of uniform gossip (random neighbor selection) which can stall at bottlenecks due to repeated communication within well-connected regions.
• Design a protocol that ensures fast global information dissemination regardless of network connectivity, achieving near-optimal performance.
• Demonstrate that any algorithm running in T rounds in the LOCAL model can be simulated in O(T + polylog n) rounds in the GOSSIP model, establishing near-equivalence between the two models.

Proposed method

• Introduce a recursive graph decomposition into clusters with sufficiently high conductance to enable fast internal information exchange.
• Use a modified gossip protocol—Superstep—where nodes exchange information with neighbors in a structured, multi-phase manner to reduce redundant communication.
• Implement a weight-decay mechanism on edges to discourage repeated communication between the same nodes, promoting exploration of under-connected paths.
• Construct sparse spanners with low hereditary density to simulate LOCAL model algorithms efficiently in the GOSSIP model.
• Leverage the existence of (2^log*n log n, log^4 n)- and (O(1), polylog n)-simulators to enable efficient simulation of LOCAL algorithms with bounded overhead.
• Apply a nearly-additive spanner construction to further reduce simulation cost, achieving a (O(1), polylog n)-simulator for general graphs.

Experimental results

Research questions

• RQ1 Can global computation be performed efficiently in the GOSSIP model without relying on graph conductance?
• RQ2 Is it possible to simulate any T-round LOCAL model algorithm in the GOSSIP model with only polylogarithmic overhead?
• RQ3 How can a gossip protocol be designed to avoid bottlenecks caused by repeated communication between well-connected nodes?
• RQ4 What structural properties of a graph enable fast information dissemination even when conductance is low?
• RQ5 Can asymmetric edge weights and dynamic neighbor selection improve rumor spreading performance in the GOSSIP model?

Key findings

• The proposed Superstep algorithm ensures that all nodes learn all information in O(D + polylog n) rounds, where D is the network diameter, with no dependence on conductance.
• The algorithm achieves a simulation overhead of O(polylog n) when emulating any T-round LOCAL model algorithm, making the two models essentially equivalent in terms of computational power.
• The method constructs a (O(1), polylog n)-simulator for the GOSSIP model, enabling efficient simulation of a broad class of LOCAL algorithms.
• The recursive cluster decomposition ensures that the number of node pairs that have not exchanged information decreases by a constant factor in each phase.
• The use of spanners with constant hereditary density allows for efficient simulation of LOCAL algorithms with bounded multiplicative and additive overhead.
• The results show that even in graphs with polylogarithmic diameter, prior conductance-dependent algorithms could require Ω(n) rounds, whereas the new algorithm remains efficient.