Label-Guided Graph Exploration by a Finite Automaton

Abstract

A finite automaton, simply referred to as a 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 ${\mathcal R}$, and a way to color in black or white the nodes of any bounded-degree graph G, so that ${\mathcal R}$ can explore the colored graph G. Finally, we give impossibility results regarding graph exploration by a robot with no internal memory (i.e., a single state automaton).