Groupings and Pairings in Anonymous Networks

We consider a network of processors in the absence of unique identities, and study the k-Grouping problem of partitioning the processors into groups of size k and assigning a distinct identity to each group. The case k=1 corresponds to the well known problems of leader election and enumeration for which the conditions for solvability are already known. The grouping problem for k≥2 requires to break the symmetry between the processors partially, as opposed to problems like leader election or enumeration where the symmetry must be broken completely (i.e. a node has to be distinguishable from all other nodes). We determine what properties are necessary for solving these problems, characterize the classes of networks where it is possible to solve these problems, and provide a solution protocol for solving them. For the case k=2 we also consider a stronger version of the problem, called Pairing where each processor must also determine which other processor is in its group. Our results show that the solvable class of networks in this case varies greatly, depending on the type of prior knowledge about the network that is available to the processors. In each case, we characterize the classes of networks where Pairing is solvable and determine the necessary and sufficient conditions for solving the problem.