Fast vertex guarding for polygons with and without holes
Computational Geometry: Theory and Applications
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Many geometric problems are intrinsically linked to the issue of splitting or classifying points. We investigate two such families of problems in two separate branches of research. Guarding problems are motivated by the issue of guarding a region with security cameras or illuminating it with lights. Such problems have been studied for decades, but there are two significant guarding problems whose complexity is not completely understood. First, we investigate the problem of guarding simple polygons; this problem is known to be NP-complete but its approximability is not known. Second, we investigate the complexity of guarding monotone chains, also known as 1.5-dimensional terrains. Understanding the interaction of 'visibility polygons' and how they separate point sets is crucial for the investigation of such problems. We resolve a significant open problem by proving strong NP-completeness for terrain guarding. We also present an approximation algorithm for guarding simple polygons with perimeter guards; this new algorithm improves the state of the art.A geometric split tree is a data structure for storing point sets that recursively splits the space, and in turn the data, in some random way. Understanding the distribution of such a tree's structure is a matter of understanding the distribution of the splits. We investigate the distributions associated with several natural splitting methods. We make new connections between an important problem in discrete geometry and natural probability distributions. With the goal of analyzing geometric split trees based on their splits, we introduce a random tree model that is general while still allowing powerful comparisons with random trees from more restricted models.