OPTIMAL CONSTRUCTION OF SENSE OF DIRECTION IN A TORUS BY A MOBILE AGENT

Abstract

Sense of direction is a property of the edge-labeling of a network whose availability facilitates computations and often decreases their complexity. In this paper we consider the problem of providing a sense of direction to an anonymous, unoriented torus. The edges of the torus are initially labeled arbitrarily, and they have to be relabeled with a classical compass sense of direction. We first describe an algorithm where the relabeling of the edges is performed by a mobile agent that carries a token. The algorithm is optimal both in terms of number of tokens (with no tokens the task cannot be performed), and in terms of number of moves. We then show that the same technique can be used to orient the torus in the classical message-passing environment, thus obtaining an orientation algorithm that improves all the existing ones in this setting.