独立范围抽样，再访
Independent Range Sampling, Revisited Again
摘要：
我们重新审视范围采样问题：输入是一组点，其中每个点都与实值权重相关联。我们的目标是将它们存储在一个结构中，使得给定查询范围和整数$k$，我们可以从查询范围内的点中提取$k$独立的随机样本，其中采样点的概率与其权重成比例。这项工作由胡，乔和陶于2014年发起，后来由Afshani和Wei跟进。第一行的工作主要是研究未加权，但动态版本的问题在一个维度，而第二个结果认为静态加权问题在一个维度，以及未加权的问题在3D半空间查询。我们提供了三个主要的结果和一些有趣的见解，错过了以前的工作：我们表明，它是可以建立有效的数据结构范围抽样查询，如果我们允许查询时间保持在预期（第一个结果），或获得有效的最坏情况下的查询边界，允许采样概率近似成比例的重量（第二个结果）。第三个结果是一个有条件的下界，它基本上表明需要前面两个让步中的一个。例如，对于3D范围采样查询，前两个结果给出了具有近线性空间和多对数查询时间的有效数据结构，而下界显示了具有近线性空间的最坏情况查询时间必须接近$n^{2/3}$，忽略多对数因子。据我们所知，这是第一个这样的范围搜索问题的预期和最坏情况下的查询时间之间的重大差距。
英文摘要：
We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer $k$, we can extract $k$ independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to $n^{2/3}$, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
