Affiliation:
1. Boston University, Boston, MA
2. CNRS and Université Paris Diderot, Paris, France
3. CNRS and Université Bordeaux I, France
4. The Weizmann Institute, Rehovot, Israel
Abstract
A finite automaton, simply referred to as a
robot
, has to explore a graph, that is, visit all the nodes of the graph. The robot has no a priori knowledge of the topology of the graph, nor of its size. It is known that for any
k
-state robot, there exists a graph of maximum degree 3 that the robot cannot explore. This article considers the effects of allowing the system designer to add short labels to the graph nodes in a preprocessing stage, for helping 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 R, and a way to color in black or white the nodes of any bounded-degree graph
G
, so that 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).
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
65 articles.
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