Searching a Simple Polygon by a k-Searcher

The polygon search problem is the problem of searching for a mobile intruder in a simple polygon by the mobile searcher who holds flashlights and whose visibility is limited to the rays emanating from his flashlights. The goal is to decide whether there exists a search schedule for the searcher to detect the intruder, no matter how fast he moves, and if so, generate such a schedule. A searcher is called the k-searcher if he can see along k rays emanating from his position, and the ∞-searcher if he has a 360° field of vision. We present necessary and sufficient conditions for a polygon to be searchable by a k-searcher (for k = 1 or 2), and give O(n2) time algorithms for testing the k-searchability of simple polygons and generating a search schedule if it exists. We also show that any polygon that is searchable by an ∞-searcher is searchable by a 2-searcher. Our results solve a long-standing open problem in computational geometry and robotics, and confirm a conjecture due to Suzuki and Yamashita.

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