Tight bounds for algebraic gossip on graphs

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(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n²) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n²). 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.