Optimal mobile agent algorithms for finding size and orientation of an unknown torus

In this paper we consider an agent initially located in an arbitrary node of an anonymous, unoriented torus of unknown size. Each node contains a small whiteboard that the agent can mark. The first goal of the agent is to efficiently find the size of the torus, i.e., to discover the torus dimensions without having to traverse the entire topology. The second goal of the agent is to give a compass orientation to the torus. Efficiency is measured in terms of number of moves performed by the agent. We propose an algorithm for finding the size of the torus in Theta(max{m, n}) moves (where n and m are the dimensions of the torus), and an algorithm for orienting the torus in Theta(m x n) moves.