  A mobile robot represented by a point moving in the plane has to explore an
unknown terrain with obstacles. Both the terrain and the obstacles are modeled
as arbitrary polygons. We consider two scenarios: the unlimited vision, when
the robot situated at a point p of the terrain explores (sees) all points q of
the terrain for which the segment pq belongs to the terrain, and the limited
vision, when we require additionally that the distance between p and q be at
most 1. All points of the terrain (except obstacles) have to be explored and
the performance of an exploration algorithm is measured by the length of the
trajectory of the robot. For unlimited vision we show an exploration algorithm
with complexity O(P + D?k), where P is the total perimeter of the terrain
(including perimeters of obstacles), D is the diameter of the convex hull of
the terrain, and k is the number of obstacles. We do not assume knowledge of
these parameters. We also prove a matching lower bound showing that the above
complexity is optimal, even if the terrain is known to the robot. For limited
vision we show exploration algorithms with complexity O(P + A + ?Ak), where A
is the area of the terrain (excluding obstacles). Our algorithms work either
for arbitrary terrains, if one of the parameters A or k is known, or for c-fat
terrains, where c is any constant (unknown to the robot) and no additional
knowledge is assumed. (A terrain T with obstacles is c-fat if R/r ? c, where R
is the radius of the smallest disc containing T and r is the radius of the
largest disc contained in T .) We also prove a matching lower bound ?(P + A +
?Ak) on the complexity of exploration for limited vision, even if the terrain
is known to the robot.
