$d$-dimensional (for $d>1$) efficient range-summability ($d$D-ERS) of ideally i.i.d. random variables (RVs) is a fundamental algorithmic problem that has applications to two important families of database problems, namely, fast approximate wavelet tracking (FAWT) on data streams and approximately answering range-sum queries over a data cube. Whether there are efficient solutions to the $d$D-ERS problem, or to the latter database problem, have been two long-standing open problems. Both are solved in this work. Specifically, we propose a novel solution framework to $d$D-ERS on RVs that have Gaussian or Poisson distribution. Our $d$D-ERS solutions are the first ones that have polylogarithmic time complexities. Furthermore, we develop a novel $k$-wise independence theory that allows our $d$D-ERS solutions to have both high computational efficiencies and strong provable independence guarantees. Finally, we show that under a sufficient and likely necessary condition, certain existing solutions for 1D-ERS can be generalized to higher dimensions.

翻译：i.d.随机变量(RV)是一个基本的算法问题,适用于两个数据库问题的重要组,即数据流快速近似波子跟踪(FAWT)和对数据立方体的测距查询。对于美元-ERS问题,或对于后一个数据库问题,是否有有效的解决办法,这是两个长期未解决的问题。在这项工作中,这两个问题都得到了解决。具体地说,我们提议了一个具有高山或普瓦森分布的RV上美元-ERS的新解决方案框架。我们的美元-ERS解决方案是第一个具有多孔时间复杂性的解决方案。此外,我们开发了一个新颖的美元-明智的独立理论,使我们的美元-ERS解决方案既具有高计算效率,又具有强有力的独立保障。最后,我们表明,在足够和可能必要的条件下,现有的1D-ERS解决方案可以推广到更高的层面。