On the Cost of Waking Up

Abstract

Often, in a distributed system, a task must be performed in which all entities must be involved; however only some of them are active, while the others are inactive, unaware of the new computation that has to take place. In these situations, all entities must become active, a task known as Wake-Up. It is not difficult to see that Broadcast is just the special case of the Wake-Up problem, when there is only one initially active entity. Both problems can be solved with the same trivial but expensive solution: Flooding. More efficient broadcast protocols exist for some classes of dense interconnection networks. The research question we examine is whether also wake-up can be performed significantly better in three classes of regular interconnection networks: hypercubes, complete networks, and regular complete bipartite graphs.

In a d-dimensional hypercubenetwork of n nodes, the cost of broadcasting is Theta(n) even if the edge labeling is arbitrary and the network is asynchronous. We show that, instead, wake-up requires Omega(nlog n) message transmissions in the worst case, even if the network is synchronous and has sense of direction. Similarly, in a regular complete bipartitenetwork K_p,p of n=2panonymous entities the cost of broadcasting is Theta(n) even if the edge labeling is arbitrary and the network is asynchronous; instead, we show that wake-up requires Theta(n²) message transmissions in the worst case, even if the network is synchronous and has sense of direction.

In a completenetwork K_nof n entities, the cost of broadcasting is minimal: n-1 message transmissions suffice even if the entities are anonymous. In this paper we prove that the cost of wake-up is order of magnitude higher. In the case of anonymous entities, Omega(n²) message transmissions are needed in the worst case, even if the network is fully synchronous and has sense of direction. In the case of entities with distinct ids, Omega(nlog n) transmissions need to be performed and the bound is tight. This shows that, when the entities have Ids, Wake-Up is computationally as costly as the apparently more complex Election problem.