Wang tilings and distributed verification on anonymous torus networks

Consider n² processors arranged in an n x n torus network in which each processor is connected by direct communication channels with its four neighbours. This paper studies the following verification problem on anonymous n x n torus networks: verify whether the network is oriented; that is, verify whether there is an agreement, among all processors, on a consistent channel labelling. The problem is to be solved by a distributed algorithm executed by the processors themselves. If processors can label their channels arbitrarily, then there are network labellings that are not oriented but, to the processors, are indistinguishable from ones that are oriented. Hence there is no deterministic distributed verification algorithm. However, a verification algorithm does exist if the initial labellings are suitably restricted. We describe the restrictions placed on the initial labellings by subsets of the permutation group S₄. We show that the existence of an algorithm for verification is equivalent to the existence of certain tilings of the torus with Wang tiles. Using this equivalence, we have determined the existence of a distributed algorithm for the verification problem for all n x n torus networks for an important class of restrictions, the subgroups of S₄.