Termination of amnesiac flooding

Abstract

We consider a stateless ‘amnesiac’ variant of the stateful distributed network flooding algorithm, expanding on our conference papers [PODC’19, STACS’20]. Flooding begins with a set of source ‘initial’ nodes I seeking to broadcast a message M in rounds, in a network represented by an undirected graph (G, E) with set of nodes G and edges E. In every round, nodes with M send M to all neighbours from which they did not receive M in the previous round. Nodes do not remember earlier rounds, raising the possibility that M circulates indefinitely. Stateful flooding overcomes this by nodes recording every message circulated and ignoring M if received again. This overhead was assumed to be necessary. We show that almost optimal broadcast can still be achieved without this overhead. We prove that amnesiac flooding terminates on every finite graph and derive sharp bounds for termination times. Define (G, E) to be I-bipartite if the quotient graph, contracting all nodes in I to a single node, is bipartite. We prove that, if d is the diameter and e(I) the eccentricity of the set I, flooding terminates in e(I) rounds if (G, E) is I-bipartite and j rounds with e(I) < j ≤ e(I)+d+1≤2d+1 if (G, E) is non I-bipartite. The separation in the termination times can be used for distributed discovery of topologies/distances in graphs. Termination is guaranteed if edges are lost during flooding but not, in general, if there is a delay at an edge. However, the cases of single-edge fixed delays of duration τ rounds in single-source bipartite graphs terminate by round 2d+τ−1 , and all cases of multiple-edge fixed delays in multiple-source cycles terminate.