XX Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI 2007) (2007)
Belo Horizonte, Minas Gerais, Brazil
Oct. 7, 2007 to Oct. 10, 2007
Marcelo C. Couto , State University of Campinas, Brazil
Cid C. de Souza , State University of Campinas, Brazil
Pedro J. de Rezende , State University of Campinas, Brazil
In this paper, we propose an exact algorithm to solve the Orthogonal Art Gallery problem in which guards can only be placed on the vertices of the polygon P representing the gallery. Our approach is based on a discretization of P into a finite set of points in its interior. The algorithm repeatedly solves an instance of the Set Cover problem obtaining a minimum set Z of vertices of P that can view all points in the current discretization. Whenever P is completely visible from Z, the algorithm halts; otherwise, the discretization is refined and another iteration takes place. We establish that the algorithm always converges to an optimal solution by presenting a worst case analysis of the number of iterations that could be effected. Even though these could theoretically reach O(n^4), our computational experiments reveal that, in practice, they are linear in n and, for n \leq 200, they actually remain less than three in almost all instances. <p>Furthermore, the low number of points in the initial discretization, O(n^2), compared to the possible O(n^4) atomic visibility polygons, renders much shorter total execution times. Optimal solutions found for different classes of instances of polygons with up to 200 vertices are also described.</p>
M. C. Couto, P. J. Rezende and C. C. Souza, "An Exact and Efficient Algorithm for the Orthogonal Art Gallery Problem," XX Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI 2007)(SIBGRAPI), Belo Horizonte, Minas Gerais, Brazil, 2007, pp. 87-94.