Competitive searching in polygons—Beyond generalised streets

Abstract

We consider a robot inside an unknown polygon P which has to find a path from a starting point s to a target point t. It is equipped with on-board cameras through which it can get the visibility map of its immediate surroundings. We define two new classes of polygons and provide strategies for searching these classes of polygons.

The first class of polygons is called horizontal-vertical streets or HV-streets and for a polygon in this class, every point on the boundary is visible from either a vertical or a horizontal line segment connecting the two polygonal chains from s to t. We provide a strategy under which the robot walks at most 14.5 times the distance of the shortest path from s to t to reach the point t. We also prove that this is an optimal strategy for searching such polygons. The second class of polygons is called θ-generalized-streets or θ-G-streets and every point on such a polygon is visible from at least one line segment at an angle θ connecting the two polygonal chains between s and t. Here, θ is an arbitrary but fixed angle with respect to the x axis. We provide a strategy for searching an isothetic polygon of this class and the robot travels at most 19.97 times the shortest distance between s and t under our strategy.

This research is supported by the DFG-Project ”Diskrete Probleme”, No. Ot 64/8-1.