Multidimensional Included and Excluded Sums
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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 ⊕ on the elements. The included-sum problem requires that the elements within overlapping boxes cornered at each element within the array be reduced using ⊕. The excluded-sum problem reduces the elements outside each box. The weak versions of these problems assume that the operator ⊕ has an inverse ⊖, 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×. The 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× 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 ⊖, BDBSC can beat box-complement by up to a factor of 4.
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