Space lower bounds for graph exploration via reduced automata

We consider the task of exploring graphs with anonymous nodes by a team of non-cooperative robots modeled as finite automata. These robots have no a priori knowledge of the topology of the graph, or of its size. Each edge has to be traversed by at least one robot. We first show that, for any set of q non-cooperative K-state robots, there exists a graph of size O(qK) that no robot of this set can explore. This improves the O(K-O(q)) bound by Rollik (1980). Our main result is an application of this improvement. It concerns exploration with stop, in which one robot has to explore and stop after completing exploration. For this task, the robot is provided with a pebble, that it can use to mark nodes. We prove that exploration with stop requires Omega(log n) bits for the family of graphs with at most n nodes. On the other hand, we prove that there exists an exploration with stop algorithm using a robot with O(D log Delta) bits of memory to explore all graphs of diameter at most D and degree at most Delta.