Label-Guided Graph Exploration by a Finite Automaton
Résumé
A finite automaton, simply referred to as a {\em robot}, has to explore a graph, i.e., visit all the nodes of the graph. The robot has no a priori knowledge of the topology of the graph or of its size. It is known that, for any $k$-state robot, there exists a $(k+1)$-node graph of maximum degree~3 that the robot cannot explore. This paper considers the effects of allowing the system designer to add short labels to the graph nodes in a preprocessing stage, and using these labels to guide the exploration by the robot. We describe an exploration algorithm that given appropriate 2-bit labels (in fact, only 3-valued labels) allows a robot to explore all graphs. Furthermore, we describe a suitable labeling algorithm for generating the required labels, in linear time. We also show how to modify our labeling scheme so that a robot can explore all graphs of bounded degree, given appropriate 1-bit labels. In other words, although there is no robot able to explore all graphs of maximum degree~3, there is a robot $\cR$, and a way to color in black or white the nodes of any bounded-degree graph $G$, so that $\cR$ can explore the colored graph $G$. Finally, we give impossibility results regarding graph exploration by a robot with no internal memory ({\it i.e.}, a single state automaton).
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