Online k-color spanning disk problems

Abstract

Inspired by the applications in on-demand manufacturing, we introduce the online k-color spanning disk problem, the first online model for color spanning problems to the best of our knowledge. Given a set P of n colored points in a plane, with each color chosen from a set C of $m \le n$ colors, the online k-color spanning disk problem determines the location of the center that minimizes the accumulated radius of the minimum spanning disks for a sequence of color sets, denoted by $\delta =\langle C_1,C_2,\ldots ,C_T\rangle$, $C_t\subseteq C$, $|C_t| \ge k$, $t\in \{1, 2, \ldots , T\}$, as they are presented online. Here, a minimum spanning disk for a color set means a disk contains at least one point of each color. We construct a special instance to establish a lower bound on the performance of any online algorithms. Then, an $O(nm\log n)$-time Voronoi-diagram-based algorithm is designed such that its competitive ratio matches the problem’s lower bound. This implies our algorithm is theoretically the best possible in terms of the competitive ratio. We also introduce and study a variant, named the online balanced k-color spanning disk problem, for which a non-trivial lower bound and a best possible algorithm are presented.