SCG '86 Proceedings of the second annual symposium on Computational geometry
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Information Processing Letters
How long can a euclidean traveling salesman tour be?
SIAM Journal on Discrete Mathematics
Shortest watchman routes in simple polygons
Discrete & Computational Geometry
Ray shooting and lines in space
Handbook of discrete and computational geometry
Approximation algorithms for lawn mowing and milling
Computational Geometry: Theory and Applications
Fast computation of shortest watchman routes in simple polygons
Information Processing Letters
Ray Shooting, Depth Orders and Hidden Surface Removal
Ray Shooting, Depth Orders and Hidden Surface Removal
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Touring a sequence of polygons
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Information Processing Letters
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Note: Approximation algorithms for art gallery problems in polygons
Discrete Applied Mathematics
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Optimal exploration of terrains with obstacles
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Watchman routes for lines and segments
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Optimal patrolling of fragmented boundaries
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
Watchman routes for lines and line segments
Computational Geometry: Theory and Applications
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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^2^/^3@?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.