Watchman tours for polygons with holes
نویسندگان
چکیده
A watchman tour in a polygonal domain (for short, polygon) is a closed curve in the polygon such that every point in the polygon is visible from at least one point of the tour. We show that the length of a minimum watchman tour in a polygon P with k holes is O(per(P )+ √ k·diam(P )), where per(P ) and diam(P ) denote the perimeter and the diameter of P , respectively. Apart from the multiplicative constant, this bound is tight in the worst case. We then generalize our result to watchman tours in polyhedra with holes in 3-space. We obtain an upper bound of O(per(P )+ √ k · per(P ) · diam(P )+k ·diam(P )), which is again tight in the worst case. Our methods are constructive and lead to efficient algorithms for computing such tours. We also revisit the NP-hardness proof of the Watchman Tour Problem for polygons with holes.
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