Line-of-Sight Pursuit in Monotone and Scallop Polygons

We study a turn-based game in a simply connected polygonal environment [Formula: see text] between a pursuer [Formula: see text] and an adversarial evader [Formula: see text]. Both players can move in a straight line to any point within unit distance during their turn. The pursuer [Formula: see text] wins by capturing the evader, meaning that their distance satisfies [Formula: see text], while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader [Formula: see text] always knows the location of [Formula: see text]. Meanwhile, [Formula: see text] only has line-of-sight visibility: [Formula: see text] observes the evader’s position only when the line segment connecting them lies entirely within the polygon. Therefore [Formula: see text] must search for [Formula: see text] when the evader is hidden from view. We provide a winning strategy for [Formula: see text] in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line [Formula: see text] can be moved continuously over [Formula: see text] so that (1) [Formula: see text] is a line segment and (2) every point on the boundary [Formula: see text] 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 [Formula: see text]. For both monotone and scallop polygons, our algorithm has a capture time of [Formula: see text], where [Formula: see text] is the number of polygon vertices.

TRIANGULATING A REGION BETWEEN ARBITRARY POLYGONS

The paper presents an optimal algorithm for triangulating a region between arbitrary polygons on the plane with time complexity O(N log⁡N ). An efficient algorithm is received by reducing the problem to the triangulation of simple polygons with holes. A simple polygon with holes is triangulated using the method of monotone chains and keeping overall design of the algorithm simple. The problem is solved in two stages. In the first stage a convex hull for m polygons is constructed by Graham’s method. As a result, a simple polygon with holes is received. Thus, the problem of triangulating a region between arbitrary polygons is reduced to the triangulation of a simple polygon with holes. In the next stage the simple polygon with holes is triangulated using an approach based on procedure of splitting polygon onto monotone polygons using the method of chains [15]. An efficient triangulating algorithm is received. The proposed algorithm is characterized by a very simple implementation, and the elements (triangles) of the resulting triangulation can be presented in the form of simple and fast data structure: a tree of triangles [17].