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On the Complexity of Distributed Wake-Up

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. Another typical occurrence of this problem, known also as Reset, is when some entities, upon detecting the occurrence of an anomalous condition, decide that the ongoing computation must be restarted. 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. In this paper we study the message complexity of performing a wake-up in hypercubes and in complete networks. In a d-dimensional hypercube network H_d of n=2d anonymous entities, the cost of broadcasting is Θ(n) even if the edge labeling is arbitrary. We show that, instead, wake-up requires Ω(n log n) message transmissions in the worst case, even if the network is fully synchronous and has sense of direction. In a complete network Kn of 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, Ω(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, we prove that Ω(n log 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.