Relative Convex Hulls in Semi-dynamic Subdivisions

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) log n) 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 √ n polylog n) to O((m + n) polylog (mn)).