Two approximate solutions to the Art Gallery Problem
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In this proposal, a dense mesh of potential observers is placed in the open space that surrounds/immerses the obstacles in the gallery (Figure 1a) which basically discretises the open space. This aspect highlights the central assumption in this approach that the open space of the art gallery is covered with dense arrangement of observers. Each observer i v has a visual connectivity expressed as a set called isovist ai where ai = {vi,vj,...,vn}; 1≥i,j,...,n≤N, vj,...,vn are observers visible from vi and N is the set of all observers in the gallery. |ai| is referred as the rank of the observer. With this premise, the following algorithms are proposed to solve the art gallery problem.
[1] Michael Batty,et al. Visualising the Structure of Architectural Open Spaces Based on Shape Analysis , 2004, ArXiv.
[2] V. Chvátal. A combinatorial theorem in plane geometry , 1975 .