Local Computations on Closed Unlabelled Edges: The Election Problem and the Naming Problem

Abstract

The different local computations mechanisms are very useful for delimiting the borderline between positive and negative results in distributed computations. Indeed, they enable to study the importance of the synchronization level and to understand how important is the initial knowledge. A high level of synchronization involved in one atomic computation step makes a model powerful but reduces the degree of parallelism. Charron-Bost et al. [1] study the difference between synchronous and asynchronous message passing models. The model studied in this paper involves more synchronization than the message passing model: an elementary computation step modifies the states of two neighbours in the network, depending only on their current states. The information the processors initially have can be global information about the network, such as the size, the diameter or the topology of the network. The initial knowledge can also be local: each node can initially know its own degree for example. Another example of local knowledge is the existence of a port numbering: each processor locally gives numbers to its incident edges and in this way, it can consistently distinguish its neighbours. In Angluin’s model [2], it is assumed that a port numbering exists, whereas it is not the case in our model. In fact, we obtain a model with a strictly lower power of computation by relaxing the hypothesis on the existence of a port numbering.