间接覆盖子树问题的一个优化算法
An Optimal Algorithm for the Indirect Covering Subtree Problem
摘要：
我们考虑间接覆盖子树问题（Kim等人，1996年）。输入是边加权树图以及位于节点处的客户。每个客户都与半径和惩罚相关联。我们的目标是找到一个树形设施，使设置和惩罚成本的总和最小化。设置成本等于设施占用的边长之和，惩罚成本是到设施的距离超过其半径的所有客户的惩罚之和。间接覆盖子树问题推广了树上的单个最大覆盖定位问题，其中设施是节点而不是子树。间接覆盖子树可以在$O(n\log^2 n)$时间内求解（Kim等人，1996年）。已经提供了一种运行时间为$O(n\log^2n/\log\log n)$的单个最大覆盖位置的稍快算法（Spoerhase和Wirth，2009）。我们实现了间接覆盖子树的时间$O(n\log n)$，从而为这两个问题提供了最快的已知算法。我们的结果也意味着更快的算法，如$(1,X)$ -medianoid和$(1,p)$ -质心的树木竞争的位置问题。我们补充我们的结果由一个下界的$Ω(n\log n)$为单一的最大覆盖范围的位置和$(1,X)$ -medianoid上的实数RAM模型表明，我们的算法是最佳的运行时间。
英文摘要：
We consider the indirect covering subtree problem (Kim et al., 1996). The input is an edge weighted tree graph along with customers located at the nodes. Each customer is associated with a radius and a penalty. The goal is to locate a tree-shaped facility such that the sum of setup and penalty cost is minimized. The setup cost equals the sum of edge lengths taken by the facility and the penalty cost is the sum of penalties of all customers whose distance to the facility exceeds their radius. The indirect covering subtree problem generalizes the single maximum coverage location problem on trees where the facility is a node rather than a subtree. Indirect covering subtree can be solved in $O(n\log^2 n)$ time (Kim et al., 1996). A slightly faster algorithm for single maximum coverage location with a running time of $O(n\log^2n/\log\log n)$ has been provided (Spoerhase and Wirth, 2009). We achieve time $O(n\log n)$ for indirect covering subtree thereby providing the fastest known algorithm for both problems. Our result implies also faster algorithms for competitive location problems such as $(1,X)$-medianoid and $(1,p)$-centroid on trees. We complement our result by a lower bound of $Ω(n\log n)$ for single maximum coverage location and $(1,X)$-medianoid on a real-number RAM model showing that our algorithm is optimal in running time.
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