Graph Exploration by a Finite Automaton
Résumé
A finite automaton, simply referred to as a {\em robot}, has to explore a graph whose nodes are unlabeled and whose edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the graph or of its size. Its task is to traverse all the edges of the graph. We first show that, for any $K$-state robot and any $d\geq 3$, there exists a planar graph of maximum degree $d$ with at most $K+1$ nodes that the robot cannot explore. This bound improves all previous bounds in the literature. More interestingly, we show that, in order to explore all graphs of diameter $D$ and maximum degree $d$, a robot needs $\Omega(D\log{d})$ memory bits, even if we restrict the exploration to planar graphs. This latter bound is tight. Indeed, a simple DFS at depth $D+1$ enables a robot to explore any graph of diameter $D$ and maximum degree $d$ using a memory of size $O(D\log{d})$ bits. We thus prove that the worst case space complexity of graph exploration is $\Theta(D\log{d})$ bits.
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