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Staircase visibility and computation of kernels

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Abstract

Let\(\mathcal{O}\) be some set of orientations, that is,\(\mathcal{O} \subseteq [0^\circ ,360^\circ ]\). We consider the consequences of defining visibility based on curves that are monotone with respect to the orientations in\(\mathcal{O}\). We call such curves\(\mathcal{O}\)-staircases. Two points p andq in a polygonP are said to\(\mathcal{O}\)-see each other if an\(\mathcal{O}\)-staircase fromp toq exists that is completely contained inP. The\(\mathcal{O}\) -kernel of a polygonP is then the set of all points which\(\mathcal{O}\)-see all other points. The\(\mathcal{O}\)-kernel of a simple polygon can be obtained as the intersection of all {θ}-kernels, with θ∈\(\mathcal{O}\). With the help of this observation we are able to develop an\(O(n\log \left| \mathcal{O} \right|)\) algorithm to compute the\(\mathcal{O}\)-kernel of a simple polygon, for finite\(\mathcal{O}\). We also show how to compute theexternal \(\mathcal{O}\) -kernel of a polygon in optimal time\(O(n + \left| \mathcal{O} \right|)\). The two algorithms are combined to compute the (\(\mathcal{O}\)-kernel of a polygon with holes in time\(O(n^2 + n\left| \mathcal{O} \right|)\).

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Communicated by K. Mehlhorn.

This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and the Natural Sciences and Engineering Research Council of Canada and Information Technology Research Centre of Ontario.

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Schuierer, S., Wood, D. Staircase visibility and computation of kernels. Algorithmica 14, 1–26 (1995). https://doi.org/10.1007/BF01300371

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