[Paper Review] Optimal Gossip with Direct Addressing

Bernhard Haeupler, Dahlia Malkhi|arXiv (Cornell University)|Feb 12, 2014

Cooperative Communication and Network Coding12 references4 citations

TL;DR

This paper presents a novel gossip algorithm in the direct addressing model that achieves optimal O(log log n) rounds for information dissemination, matching a proven lower bound. It simultaneously minimizes message complexity (O(1) per node) and bit complexity (O(log n) per message), outperforming prior work that required O(√log n) rounds or messages, while maintaining robustness against oblivious node failures.

ABSTRACT

Gossip algorithms spread information by having nodes repeatedly forward information to a few random contacts. By their very nature, gossip algorithms tend to be distributed and fault tolerant. If done right, they can also be fast and message-efficient. A common model for gossip communication is the random phone call model, in which in each synchronous round each node can PUSH or PULL information to or from a random other node. For example, Karp et al. [FOCS 2000] gave algorithms in this model that spread a message to all nodes in $Θ(\log n)$ rounds while sending only $O(\log \log n)$ messages per node on average. Recently, Avin and Elsässer [DISC 2013], studied the random phone call model with the natural and commonly used assumption of direct addressing. Direct addressing allows nodes to directly contact nodes whose ID (e.g., IP address) was learned before. They show that in this setting, one can "break the $\log n$ barrier" and achieve a gossip algorithm running in $O(\sqrt{\log n})$ rounds, albeit while using $O(\sqrt{\log n})$ messages per node. We study the same model and give a simple gossip algorithm which spreads a message in only $O(\log \log n)$ rounds. We also prove a matching $Ω(\log \log n)$ lower bound which shows that this running time is best possible. In particular we show that any gossip algorithm takes with high probability at least $0.99 \log \log n$ rounds to terminate. Lastly, our algorithm can be tweaked to send only $O(1)$ messages per node on average with only $O(\log n)$ bits per message. Our algorithm therefore simultaneously achieves the optimal round-, message-, and bit-complexity for this setting. As all prior gossip algorithms, our algorithm is also robust against failures. In particular, if in the beginning an oblivious adversary fails any $F$ nodes our algorithm still, with high probability, informs all but $o(F)$ surviving nodes.

Motivation & Objective

• To close the gap between the best-known upper and lower bounds for gossip in the direct addressing model.
• To design a gossip algorithm that simultaneously optimizes round, message, and bit complexity.
• To prove that O(log log n) rounds is information-theoretically optimal for this setting.
• To maintain fault tolerance under oblivious node failures while achieving optimal complexity.

Proposed method

• The algorithm uses a hierarchical clustering approach with three phases: Cluster1, Cluster2, and Cluster3, each progressively merging clusters to form larger, uniformly sized groups.
• Cluster1 and Cluster2 use randomized cluster formation and ClusterPUSH to grow clusters to size Θ(Δ/C''), where Δ is the maximum degree and C'' is a constant.
• Cluster3 employs a probabilistic activation of clusters followed by uniform random merging of inactive clusters into active ones, ensuring even load distribution.
• A BoundedClusterPush phase recruits unclustered nodes into clusters, with dynamic resizing to maintain cluster size within a constant factor.
• The UnclusteredNodesPull phase allows unclustered nodes to join clusters via random PULLs, ensuring all but o(F) nodes are informed after O(log log n) rounds.
• The algorithm maintains O(n) total message complexity by ensuring each node sends and receives only a constant number of messages on average.

Experimental results

Research questions

• RQ1 Can gossip in the direct addressing model achieve O(log log n) round complexity, and is this optimal?
• RQ2 Is it possible to simultaneously achieve optimal round, message, and bit complexity in this model?
• RQ3 How does the algorithm maintain fault tolerance under oblivious node failures while minimizing communication costs?
• RQ4 What are the tight bounds on the number of rounds required for information dissemination in this setting?

Key findings

• The proposed gossip algorithm completes in O(log log n) rounds with high probability, matching a proven Ω(log log n) lower bound, establishing optimality.
• The algorithm sends only O(1) messages per node on average, achieving optimal message complexity.
• The bit complexity is O(log n) per message, resulting in O(n log n) total bit complexity, which is optimal for the setting.
• The algorithm maintains fault tolerance: after F oblivious node failures, all but o(F) surviving nodes are informed with high probability.
• The algorithm’s O(n) message complexity is preserved even under failures, ensuring efficiency and scalability.
• The analysis confirms that no algorithm can achieve sub-log log n rounds with high probability, proving the round complexity is information-theoretically optimal.