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.

受按需制造应用的启发，我们引入了在线k颜色跨越盘问题，这是我们所知的第一个颜色跨越问题的在线模型。给定平面上n个彩色点的集合P ，每种颜色从\(m \le n\)种颜色的集合C中选择，在线k色跨盘问题确定最小化累积半径的中心位置颜色集序列的最小跨盘数，表示为\(\delta =\langle C_1,C_2,\ldots ,C_T\rangle \) , \(C_t\subseteq C\) , \(|C_t| \ge k\) ,\(t\in \{1, 2, \ldots , T\}\)，因为它们在线呈现。这里，颜色集的最小跨度圆盘意味着圆盘包含每种颜色的至少一个点。我们构造一个特殊的实例来建立任何在线算法的性能下限。然后，设计一个基于O(nmlog n))时间的基于Voronoi图的算法，使其竞争比与问题的下界相匹配。这意味着我们的算法理论上在竞争比方面是最好的。我们还介绍并研究了一种变体，称为在线平衡k色跨盘问题，为此提出了一个不平凡的下界和最佳可能算法。