Line-of-Sight Pursuit in Monotone and Scallop Polygons

TL;DR

This paper presents pursuit strategies for a pursuer with line-of-sight visibility to capture an evader in certain polygonal environments, achieving linear capture time in monotone and scallop polygons.

Contribution

It introduces a pursuit algorithm using rook's strategy in monotone and scallop polygons, with linear capture time, expanding pursuit-evasion theory in these environments.

Findings

Pursuer can guarantee capture in monotone polygons.

Pursuer can guarantee capture in scallop polygons.

Capture time is linear in polygon area and number of vertices.

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 $d(P, E) \leq 1$, 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) $L \cap Q$ is a line segment and (2) every point on the boundary $\partial Q$ 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 $O(n(Q) + \mbox{area}(Q))$, where $n(Q)$ is the number of polygon vertices.