Rapport (Rapport De Recherche) Année : 2007

Guarding curvilinear art galleries with vertex or point guards

Résumé

One of the earliest and most well known problems in computational geometry is the so-called \emph{art gallery problem}. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. We consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our main focus is on polygons the edges of which are convex arcs pointing towards the exterior or interior of the polygon (but not both), named piecewise convex and piecewise concave polygons. We prove that, in the case of piecewise convex polygons, if we only allow vertex guards, 4n71 guards are sometimes necessary, and 2n3 guards are always sufficient. Moreover, an O(nlogn) time and O(n) space algorithm is described that produces a vertex guarding set of size at most 2n3. When we allow point guards the afore-mentioned lower bound drops down to n2. In the special case of monotone piecewise convex polygons we can show that n2+1 vertex or n2 point guards are always sufficient and sometimes necessary. In the case of piecewise concave polygons, we show that 2n4 point guards are always sufficient and sometimes necessary, whereas it might not be possible to guard such polygons by vertex guards. We conclude with bounds for other types of curvilinear polygons and future work.
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Dates et versions

inria-00132988 , version 1 (23-02-2007)
inria-00132988 , version 2 (28-02-2007)
inria-00132988 , version 3 (20-04-2007)
inria-00132988 , version 4 (24-04-2007)
inria-00132988 , version 5 (22-06-2007)
inria-00132988 , version 6 (26-06-2007)
inria-00132988 , version 7 (16-02-2008)
inria-00132988 , version 8 (18-02-2008)

Identifiants

  • HAL Id : inria-00132988 , version 8

Citer

Menelaos Karavelas, Elias Tsigaridas. Guarding curvilinear art galleries with vertex or point guards. [Research Report] RR-6132, INRIA. 2007. ⟨inria-00132988v8⟩
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