Relative Convex Hulls in Semi-Dynamic Arrangements

We present a data structure for maintaining the geodesic hull of a set of points (sites) in the presence of pairwise noncrossing line segments (barriers) that subdivide a bounding box into simply connected faces. For m barriers and n sites, our data structure has O((m+n)logn) size. It supports a mixed sequence of O(m) barrier insertions and O(n) site deletions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O((m+n) \operatorname{polylog}(mn))$\end{document} total time, and answers analogues of standard convex hull queries in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\operatorname{polylog}(mn))$\end{document} time.