Abstract
We present data structures for maintaining the relative convex hull of a set of points (sites) in the presence of pairwise non-crossing line segments (barriers) that subdivide a bounding box into simply connected faces. Our data structures have O((n + m)logn) size for n sites and m barriers. They support O(m) barrier insertions and O(n) site deletions in O((m + n) polylog (mn)) total time, and can answer analogues of standard convex hull queries in O(polylog(mn)) time.
Our data structures support a generalization of the sweep line technique, in which the sweep wavefront may have arbitrary polygonal shape, possibly bending around obstacles. We reduce the total time of m online updates of a polygonal sweep wavefront from \(O(m\sqrt{n}\,{\rm polylog} n)\) to O((m + n) polylog (mn)).
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Ishaque, M., Tóth, C.D. (2008). Relative Convex Hulls in Semi-dynamic Subdivisions. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_65
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DOI: https://doi.org/10.1007/978-3-540-87744-8_65
Publisher Name: Springer, Berlin, Heidelberg
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