Bounds for algebraic gossip on graphs

We study the stopping times of gossip algorithms fornetwork coding. We analyze algebraic gossip (i.e., random linearcoding) and consider three gossip algorithms for informationspreading: Pull, Push, and Exchange. The stopping time ofalgebraic gossip is known to be linear for the complete graph, butthe question of determining a tight upper bound or lower boundsfor general graphs is still open. We take a major step in solvingthis question, and prove that algebraic gossip on any graph ofsize n is O(∆n) where ∆ is the maximum degree of the graph.This leads to a tight bound of Θ(n) for bounded degree graphsand an upper bound of O(n) for general graphs. We show thatthe latter bound is tight by providing an example of a graphwith a stopping time of Ω(n). Our proofs use a novel methodthat relies on Jackson’s queuing theorem to analyze the stoppingtime of network coding; this technique is likely to become usefulfor future research.