Aggregating over Dominated Points by Sorting, Scanning, Zip and Flat Maps

Abstract: Prefix aggregation operation (also called scan), and its particular case, prefix summation, is an important parallel primitive and enjoys a lot of attention in the research literature. It is also used in many algorithms as one of the steps. Aggregation over dominated points in $\mathbb{R}^m$ is a multidimensional generalisation of prefix aggregation. It is also intensively researched, both as a parallel primitive and as a practical problem, encountered in computational geometry, spatial databases and data warehouses. In this paper we show that, for a constant dimension $m$, aggregation over dominated points in $\mathbb{R}^m$ can be computed by $O(1)$ basic operations that include sorting the whole dataset, zipping sorted lists of elements, computing prefix aggregations of lists of elements and flat maps, which expand the data size from initial $n$ to $n\log^{m-1}n$. Thereby we establish that prefix aggregation suffices to express aggregation over dominated points in more dimensions, even though the latter is a far-reaching generalisation of the former. Many problems known to be expressible by aggregation over dominated points become expressible by prefix aggregation, too. We rely on a small set of primitive operations which guarantee an easy transfer to various distributed architectures and some desired properties of the implementation.