Guarding curvilinear art galleries with vertex or point guards

Abstract

One of the earliest and most well known problems in computational geometry is the so-called 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. In this paper 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, $\lfloor\frac{4n}{7}\rfloor-1$ guards are sometimes necessary, and $\lfloor\frac{2n}{3}\rfloor$ guards are always sufficient. Moreover, an $O(n\log{}n)$ time and O(n) space algorithm is described that produces a vertex guarding set of size at most $\lfloor\frac{2n}{3}\rfloor$. When we allow point guards the afore-mentioned lower bound drops down to $\lfloor\frac{n}{2}\rfloor$. In the special case of monotone piecewise-convex polygons we can show that $\lfloor\frac{n}{2}\rfloor$ vertex guards are always sufficient and sometimes necessary; these bounds remain valid even if we allow point guards. In the case of piecewise-concave polygons, we show that $2n-4$ 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.