Altitude terrain guarding and guarding uni-monotone polygons

Altitude Terrain Guarding and Guarding Uni-Monotone Polygons

We show that the problem of guarding an x-monotone terrain from an altitude line and the problem of guarding a uni-monotone polygon are equivalent. We present a polynomial time algorithm for both problems, and show that the cardinality of a minimum guard set and the cardinality of a maximum witness set coincide. Thus, uni-monotone polygons are perfect; this result also extends to monotone mount...

Optimal guarding of polygons and monotone chains

In this paper we study several problems concerning the guarding of a polygon or a x-monotone polygonal chain P with n vertices from a set of points lying on it. Our results are: (1) An O(n logn) time sequential algorithm for computing the shortest guarding boundary chain of a polygon P. (2) An O(n logn) time sequential algorithm for computing the smallest set of consecutive edges guarding a pol...

The Complexity of Guarding Monotone Polygons

A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction fromMonotone 3SAT, we prove that interior...

Approximate Guarding of Monotone and Re tilinear Polygons

On k-Guarding Polygons

We describe a polynomial time O(k log log OPTk(P ))approximation algorithm for the k-guarding problem of finding a minimum number, OPTk(P ), of vertex guards of an n-vertex simple polygon P so that for every point p ∈ P , the number of guards that see p is at least the minimum of k and the number of vertices that see p. Our approach finds O ( k ε log log 1 ε ) size (k, ε)-nets for instances of ...