Going home through an unknown street

Abstract

We consider the problem of a robot traversing an unknown polygon with the aid of standard visibility. The robot has to find a path from a starting point s to a target point t. We provide upper and lower bounds on the ratio of the distance traveled by the robot in comparison to the length of a shortest path.

We consider two problems in this context. First we assume that the location of the target t is known to the robot. We prove a lower bound of √2 on the competitive ratio of any deterministic algorithm that solves this problem. This bound matches the competitive ratio for searches in a rectilinear streets with an unknown target point which implies that, for rectilinear streets, such knowledge provides no advantage for the robot. In addition, we obtain a lower bound of 9 for the competitive ratio of searching in generalized streets with known target which closely matches the upper bound for an unknown target. Secondly, we consider a new strategy for searching in an arbitrarily oriented street where the location of t is unknown. We show that our strategy achieves a competitive ratio of √1+(1+π/4)²(∼ 2.05) which significantly improves the best previously known ratio of 2√1+1/√2(∼ 2.61).

This research is partially supported by the DPG-Ptoject ”Diskrete Probleme”, No. Ot 64/8-1.