An Optimal Algorithm for the 1-Searchability of Polygonal Rooms
The 1-searcher is a mobile guard who can see only along a ray emanating from his position and can continuously change the direction of the ray with bounded speed. A polygonal region P with a specified point d on its boundary is called a room, and denoted by (P, d). The room (P, d) is said to be 1-searchable if the searcher, starting at the point d, can eventually see a mobile intruder who moves arbitrarily fast inside P, without allowing the intruder to touch d. We present an optimal O(n) time algorithm to determine whether there is a point x on the boundary of P such that the room (P, x) is 1-searchable. This improves upon the previous O(n log n) time bound, which was established for determining whether or not a room (P, d) is 1-searchable, where d is a given point on the boundary of P.
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