Max–min dispersion on a line

Given a set P of n locations on which facilities can be placed and an integer k, we want to place k facilities on some locations so that a designated objective function is maximized. The problem is called the k-dispersion problem. For instance it is desirable to locate fire departments far away each other. In this paper we give a simple O((2k2)kn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O((2k^2)^k n)$$\end{document} time algorithm to solve the max–min version of the k-dispersion problem if P is a set of points on a line. If k is a constant then this is an O(n) time algorithm. This is the first O(n) time algorithm to solve the max–min k-dispersion problem for the set of “unsorted” points on a line. If P is a set of sorted points on a line, and the input is given as an array in which the coordinates of the points are stored in the sorted order, then by slightly modifying the algorithm above one can solve the dispersion problem in O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log n)$$\end{document} time. This is the first sublinear time algorithm to solve the max–min k-dispersion problem for the set of sorted points on a line.