 
Summary: EXPLORING UNKNOWN ENVIRONMENTS
SUSANNE ALBERSy AND MONIKA R. HENZINGERz
Abstract. We consider exploration problems where a robot has to construct a complete map
of an unknown environment. We assume that the environment is modeled by a directed, strongly
connected graph. The robot's task is to visit all nodes and edges of the graph using the minimum
number R of edge traversals. Koutsoupias 16] gave a lower bound for R of (d2m), and Deng and
Papadimitriou 12] showed an upper bound of dO(d)m, where m is the number of edges in the graph
and d is the minimum number of edges that have to be added to make the graph Eulerian. We give
the rst subexponential algorithm for this exploration problem, which achieves an upper bound of
dO(logd)m. We also show a matching lower bound of d (log d)m for our algorithm. Additionally, we
give lower bounds of 2 (d)m, resp. d (log d)m for various other natural exploration algorithms.
Key words. directed graph, exploration algorithm
AMS subject classi cations. 05C20, 68Q20, 68Q25, 68R10
1. Introduction. Suppose that a robot has to construct a complete map of an
unknown environment using a path that is as short as possible. In many situations
it is convenient to model the environment in which the robot operates by a graph.
This allows to neglect geometric features of the environment and to concentrate on
combinatorial aspects of the exploration problem. Deng and Papadimitriou 12] for
mulated thus the following exploration problem. A robot has to explore all nodes and
edges of an unknown, strongly connected directed graph. The robot visits an edge
