On Gallery Watchmen in grids

Abstract

In the Gallery Watchman Problem (posed by Klee) we want to determine the minimum number of watchmen needed so that every point in the gallery is seen by at least one watchman at any time. The watchmen are not allowed to move but it is assumed that they can see in all directions. The gallery is modeled as a polygon which may be multi-connected, i.e., it may contain polygonal holes. Holes may represent obstacles, interior structures, etc. Chvatal [3] showed that if the polygon is simple (i.e., it contains no holes) and has n sides, then [1⁄2n] watchmen are sufficient. This result can be obtained by three-coloring the vertices of a triangulation of the polygon [4]. If w e further restrict the polygon to be rectilinear (i.e., all edges are horizontal or vertical), then [ l n l watchmen are sufficient [8]. Both of these results are bounds on the number of watchmen needed and do not give optimum solutions in general. In [1], a fast algorithm that obtains solutions of size no more than l nl for simple polygons is given. Again, the algorithm does not always produce optimum solutions. The problem of minimizing the number of watchmen is related to a polygon decomposition problem. A polygon is star-shaped if there is a point in its interior from which all points are visible. A point y is visible from point x if the open line segment connecting x and y is totally in the interior of the polygon. The set of points from which all points in a star-shaped polygon are visible is the kernel of the star-shaped polygon. The gallery watchman problem is equivalent to that of covering a polygon with star-shaped polygons. This follows from the fact that the area seen by each watchman must be a star-shaped polygon and, given a star-shaped polygon, a watchman anywhere in its kernel can see all points in it. If the polygon contains holes, the problem was shown to be intractable in [11]. Recently, it was shown that the problem remains intractable even for simple polygons [10]. In this paper we consider the gallery watchman problem in grids. In this case, the space that we want to supervise is a grid-like network of corridors. Both movement and visibility are along the corridors. The problem becomes interesting when we allow obstacles in the grid. Obstacles are modeled as missing segments of the full grid. In the next section we show that the gallery watchman problem remains intractable for three-dimensional grids and we give an efficient (polynomial time) solution for two-dimensional grids.