We consider the problem of computing the \emph{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 [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 $O(n\log h)$ time, where $h$ is the number of points in the skyline of $P$. We also show that the decision problem can be solved in $O(n\log k)$ time and the optimization problem can be solved in $O(n \log k + n \operatorname{loglog} n)$ time. This improves previous algorithms and is optimal for a large range of values of $k$.

中文翻译：

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基于更快距离的代表天际线和沿帕累托前沿的$ k $中心

我们考虑在飞机中计算\ emph {基于距离的代表天际线}的问题，这是由Tao，Ding，Lin和Pei提出的问题。第25届IEEE数据工程国际会议（ICDE），2009年]由Dupin，Nielsen和Talbi [在第三目标国际会议上，优化与学习– OLA 2020]在多目标优化的背景下独立审议。给定平面中的一组$ P $的$ n $点和一个参数$ k $，任务是选择由$ P $定义的天际线的$ k $点（也称为$ P $的帕累托锋）最小化从天际线的点到选定点的最大距离。我们证明可以在$ O（n \ log h）$时间内解决问题，其中$ h $是$ P $天际线中的点数。我们还表明，决策问题可以在$ O（n \ log k）$时间内解决，而优化问题可以在$ O（n \ log k + n \ operatorname {loglog} n）$时间内解决。这改进了先前的算法，并且对于$ k $的较大值范围而言是最佳的。