多维包含和排除和
Multidimensional Included and Excluded Sums
摘要：
本文介绍了包含和排除和问题的科学计算应用程序，如快速多极子方法的算法。这些问题被定义为一个$d$维数组的$N$元素和一个二元结合算子~ $\oplus$的元素。包含和问题要求使用$\oplus$减少数组中每个元素的重叠框内的元素。异和问题减少了每个框外的元素。这些问题的弱版本假设算子$\oplus$有逆$\ominus$，而强版本不需要这个假设。除了研究现有的算法来解决这些问题，我们介绍了三个新的算法。双向盒和（BDBS）算法在$Θ(d N)$时间内解决了强包含和问题，渐进地击败了经典的面积和表（SAT）算法，后者在$Θ(2^d N)$中运行并且仅解决了问题的弱版本。根据经验，BDBS算法在更高维度上优于SAT算法达$17.1\times$。\defn{box-complement}算法可以在$Θ(d N)$时间内解决强异和问题，渐进地击败Demaine等人的最先进的角点算法，其在$Ω(2^d N)$时间内运行。在3维中，在给定类似的空间量的情况下，盒补算法凭经验优于角算法约$1.4\times$。弱异和问题可以通过双向盒和求补（BDBSC）算法在$Θ(d N)$时间内求解，该算法是BDBS算法的一个简单扩展.给定算子逆$\ominus$，BDBSC可以击败盒补，最多可达$4$的因子。
英文摘要：
This paper presents algorithms for the included-sums and excluded-sums problems used by scientific computing applications such as the fast multipole method. These problems are defined in terms of a $d$-dimensional array of $N$ elements and a binary associative operator~$\oplus$ on the elements. The included-sum problem requires that the elements within overlapping boxes cornered at each element within the array be reduced using $\oplus$. The excluded-sum problem reduces the elements outside each box. The weak versions of these problems assume that the operator $\oplus$ has an inverse $\ominus$, whereas the strong versions do not require this assumption. In addition to studying existing algorithms to solve these problems, we introduce three new algorithms. The bidirectional box-sum (BDBS) algorithm solves the strong included-sums problem in $Θ(d N)$ time, asymptotically beating the classical summed-area table (SAT) algorithm, which runs in $Θ(2^d N)$ and which only solves the weak version of the problem. Empirically, the BDBS algorithm outperforms the SAT algorithm in higher dimensions by up to $17.1\times$. The \defn{box-complement} algorithm can solve the strong excluded-sums problem in $Θ(d N)$ time, asymptotically beating the state-of-the-art corners algorithm by Demaine et al., which runs in $Ω(2^d N)$ time. In 3 dimensions the box-complement algorithm empirically outperforms the corners algorithm by about $1.4\times$ given similar amounts of space. The weak excluded-sums problem can be solved in $Θ(d N)$ time by the bidirectional box-sum complement (BDBSC) algorithm, which is a trivial extension of the BDBS algorithm. Given an operator inverse $\ominus$, BDBSC can beat box-complement by up to a factor of $4$.
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