Bounds for Algebraic Gossip on Graphs

TL;DR

This paper establishes tight bounds for the stopping times of algebraic gossip algorithms on graphs, showing they are proportional to the number of nodes and dependent on graph degree, using novel queuing theory methods.

Contribution

It provides the first tight bounds for algebraic gossip on general graphs, introducing a new queuing theory approach for analyzing network coding algorithms.

Findings

Stopping time is O(D*n) for any graph, where D is maximum degree.

Bounded degree graphs have a tight Theta(n) stopping time.

Constructed example shows an Omega(n^2) stopping time for some graphs.

Abstract

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(D*n) where D is the maximum degree of the graph. This leads to a tight bound of Theta(n) for bounded degree graphs and an upper bound of O(n^2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Omega(n^2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.