On the termination of flooding
2020-02-12T14:19:33Z (GMT) by
Flooding is among the simplest and most fundamental of all graph/network algorithms. Consider a (distributed network in the form of a) finite undirected graph G with a distinguished node v that begins flooding by sending copies of the same message to all its neighbours and the neighbours, in the next round, forward the message to all and only the neighbours they did not receive the message from in that round and so on. We assume that nodes do not keep a record of the flooding event, thus, raising the possibility that messages may circulate infinitely even on a finite graph. We call this history-less process amnesiac flooding (to distinguish from a classic distributed implementation of flooding that maintains a history of received messages to ensure a node never sends the same message again). Flooding will terminate when no node in G sends a message in a round, and, thus, subsequent rounds. As far as we know, the question of termination for amnesiac flooding has not been settled - rather, non-termination is implicitly assumed.
In this paper, we show that surprisingly synchronous amnesiac flooding always terminates on any arbitrary finite graph and derive exact termination times which differ sharply in bipartite and non-bipartite graphs. In particular, synchronous flooding terminates in e rounds, where e is the eccentricity of the source node, if and only if G is bipartite, and, otherwise, in j rounds where e < j ≤ e + d + 1 and d is the diameter of G. Since e is bounded above by d, this implies termination times of at most d and of at most 2d + 1 for bipartite and non-bipartite graphs respectively. This suggests that if communication/broadcast to all nodes is the motivation, the history-less amnesiac flooding is asymptotically time optimal and obviates the need for construction and maintenance of spanning structures like spanning trees. Moreover, the clear separation in the termination times of bipartite and non-bipartite graphs may suggest possible mechanisms for distributed discovery of the topology/distances in an arbitrary graph.
For comparison, we also show that, for asynchronous networks, however, an adaptive adversary can force the process to be non-terminating.