We consider the problem of a robot which has to find a target in an unknown simple polygon, based only on what it has seen so far. A street is a polygon for which the two boundary chains from start to target are mutually weakly visible. A target inside a street can be found by walking a path that is at most a constant times longer than the shortest path in the street from start to target. We define a strictly larger class of polygons, called generalized streets or G-streets, which are characterized by the property that every point on the boundary of a G-street is visible from a point on a horizontal line segment connecting the two boundary chains. We present an on-line strategy for a robot to find the target in an unknown rectilinear G-street; the length of its path is at most 9 times the length of the shortest path in the L 1 metric, and 9.06 times the length of the L 2 -shortest path. These bounds are optimal. Key words: Simple polygon, street, searching, doubling, competitive...

Abstract. We consider all planar oriented curves that have the following property depending on a fixed angle ϕ. ForeachpointB on the curve, the rest of the curve lies inside a wedge of angle ϕ with apex in B. This property restrains the curve’s meandering, and for ϕ ≤ π 2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ϕ<π, we provide an upper bound c(ϕ) for the length of such a curve, divided by the distance between its endpoints, and prove this bound to be tight. A main step is in proving that the curve’s length cannot exceed the perimeter of its convex hull, divided by 1 + cos ϕ. Keywords: Self-approaching curves, convex hull, detour, arc length. 1

We consider the problem of on-line searching for the kernel of a unknown star-shaped polygon. In this motion planning problem, the robot starts from a point s inside a simple star-shaped polygon P , and aims to reach the kernel of P . The robot has no knowledge of P (apart from the fact that is a star-shaped polygon) but it is equipped with an on-board vision system that allows it to see its surrounding space. We prove that any strategy for this purpose in the worst case must traverse at least 1:515 times the shortest distance from s to the kernel of P . The lower bound is obtained by numerically computing a lower bound and then bounding the error of the computed solution algebraically. This improves over the best previously known lower bound of 1:44 by Lopez-Ortiz and Schuierer. 1 Introduction In recent years on-line searching has been an active area of research in Computer Science (e.g. [1, 2, 3]). In its full generality, an on-line search problem consists of an agent sear...