Multidimensional Included and Excluded Sums

Abstract

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 $\Theta(d N)$ time, asymptotically beating the classical summed-area table (SAT) algorithm, which runs in $\Theta(2^d N)$ 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 $\Theta(d N)$ time, asymptotically beating the state-of-the-art corners algorithm by Demaine et al., which runs in $\Omega(2^d N)$ 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 $\Theta(d N)$ 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 $4$.