Time and space optimality of rotor-router graph exploration

We consider the problem of exploration of an anonymous, port-labelled, undirected graph with n nodes, m edges, by a single mobile agent. Initially the agent does not know the topology of the graph nor any of the global parameters. Moreover, the agent does not know the incoming port when entering to a vertex thus it cannot backtrack its moves. We study a (Ma,Mw)-agent model with two types of memory: Ma bits of internal memory at the agent and Mw bits of local memory at each node that is modifiable by the agent upon its visit to that node. In such a model, condition Ma +Mw ≥ log2 d has to be satisfied at each node of degree d for the agent to be able to traverse each edge of the graph. As our main result we show an algorithm for (1,O(log d))-agent exploring any graph with return in optimal time O(m). We also show that exploration using (0,∞)-agent sometimes requires Ω(n) steps. On the other hand, any algorithm for (∞, 0)-agent is a subject to known lower bounds for Universal Traversal Sequence thus requires Ω(n) steps in some graphs. We also observe that neither (0,∞)-agent nor (∞, 0)agent can stop after completing the task without the knowledge of some global parameter of the graph. This shows separation between the model with two types of memory and the models with only one in terms of exploration time and the stop property.