A Combinatorial Bound for Beacon-based Routing in Orthogonal Polygons

Beacon attraction is a movement system whereby a robot (modeled as a point in 2D) moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (also a point). This results in the robot moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle. When a robot can reach the activated beacon by this method, we say that the beacon attracts the robot. A beacon routing from p to q is a sequence b1, b2, . . . , bk of beacons such that activating the beacons in order will attract a robot from p to b1 to b2 . . . to bk, and where a beacon placed at q will attract bk. A routing set of beacons is a set B of beacons such that any two points p, q in the free space have a beacon routing with the intermediate beacons b1, b2, . . . bk all chosen from B. Here we address the question of “how large must such a B be?” in orthogonal polygons, and show that the answer is “sometimes as large as ⌊ n−4 3 ⌋ , but never larger.”