Berry, Lindsay, Beveridge, Andrew, Butterfield, Jane, Isler, Volkan, Keller, Zachary, Shine, Alana, and Wang, Junyi
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 two families of polygons: monotone polygons and scallop polygons. In both families, a straight line L can be moved continuously over Q so that (1) L ∩ Q is a line segment and (2) every point on the boundary ∂ Q is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook's strategy during its pursuit phase, rather than the well-known lion's strategy. The rook's strategy is crucial for obtaining a capture time that is linear in the area of Q. For both 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. [ABSTRACT FROM AUTHOR]