Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents

We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n-vertex graph using O(log log n) pebbles, even when n is not known a priori. The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O(log log n) agents with constant memory can explore any n-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n) when we allow each agent to have at most O((log n)1-ε) bits of memory for any ε > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n) pebbles to explore any n-vertex graph.