Faster distance-based representative skyline and k-center along pareto front in the plane

We consider the problem of computing the distance-based representative skyline in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Mathematics; Optimization and Learning - Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set P of n points in the plane and a parameter k, the task is to select k points of the skyline defined by P (also known as Pareto front for P) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in O(nlogh)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\log h)$$\end{document} time, where h is the number of points in the skyline of P. We also show that the decision problem can be solved in O(nlogk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\log k)$$\end{document} time and the optimization problem can be solved in O(nlogk+nloglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n \log k + n {{\,\textrm{loglog}\,}}n)$$\end{document} time. This improves previous algorithms and is optimal for a large range of values of k.