Mobile vs. point guards

We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.Comment: This version covers a previously missing case in both Phase 2 &

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

On rr-Guarding Thin Orthogonal Polygons

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.Comment: 18 page