Wang tilings and distributed orientation on anonymous torus networks

Abstract

This paper investigates the orientation problem on anonymous n×n torus networks. In the orientation problem, the goal is to reach an agreement, among all processors, on a consistent channel labelling. This paper shows that, if processors can label their channels arbitrarily, then there is no distributed orientation algorithm. Therefore the orientation problem is studied in various orientation models M that restrict local channel labellings. For many M, there are network labellings that, to the processors, are indistinguishable from orientations. These pseudo-orientations give rise to two problems related to orientation. In the verification problem, processors in an n×n torus distributively verify whether a network labelling in M is an orientation. The verification problem is shown to be equivalent to a problem of tiling the torus with Wang tiles. In the pseudo-orientation problem, the goal is to distributively find a pseudo-orientation. The pseudo-orientation problem is shown to be equivalent to a problem in graph theory.