Line-of-Sight Pursuit in Monotone and Scallop Polygons

Abstract

We study a turn-based game in a simply connected polygonal environment Q between a pursuer P and an adversarial evader E. Both players can move in a straight line to any point within unit distance during their turn. The pursuer P wins by capturing the evader, meaning that their distance satisfies d(P, E) ≤ 1, while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader E always knows the location of P. Meanwhile, P only has line-of-sight visibility: P observes the evader's position only when the line segment connecting them lies entirely within the polygon. Therefore P must search for E when the evader is hidden from view.We provide a winning strategy for P in the family of strictly sweepable polygons, meaning that a straight line L can be moved continuously over Q so that (1) L ∩ Q is always convex and (2) every point on the boundary ∂Q is swept exactly once. This strict sweeping requires that L moves along Q via a sequence of translations and rotations. We develop our main result by first considering pursuit in the subfamilies of monotone polygons (where L moves by translation) and scallop polygons (where L moves by a single rotation). Our algorithm uses rook strategy during its active pursuit phase, rather than the well-known lion strategy. The rook strategy is crucial for obtaining a capture time that is linear in the area of Q. For monotone and scallop polygons, our algorithm has a capture time of O(n(Q) + area(Q)), where n(Q) is the number of polygon vertices. The capture time bound for strictly sweepable polygons is O(n(Q) • area(Q)).