通过排序、扫描、Zip 和 Flat Map 对支配点进行聚合
Aggregating over Dominated Points by Sorting, Scanning, Zip and Flat Maps
AI总结：
本文证明常数维空间中支配点聚合可通过排序、zip、前缀聚合和 flat map 等 O(1) 个基本操作实现，从而表明前缀聚合足以表达其多维推广，并便于迁移到分布式架构。
摘要：
前缀聚合操作（也称为扫描），及其特例前缀求和，是一种重要的并行原语，在研究文献中受到广泛关注。它也被用作许多算法中的一个步骤。在 $\mathbb{R}^m$ 中对支配点进行聚合是前缀聚合的多维推广。无论是作为并行原语，还是作为计算几何、空间数据库和数据仓库中遇到的实际问题，它也得到深入研究。在本文中我们证明，对于常数维 $m$，在 $\mathbb{R}^m$ 中对支配点进行聚合可以通过 $O(1)$ 个基本操作来计算，这些操作包括对整个数据集进行排序、对元素的有序列表进行 zip、计算元素列表的前缀聚合以及 flat map，这些操作将数据规模从初始的 $n$ 扩展到 $n\log^{m-1}n$。由此我们确立：前缀聚合足以表达更高维中支配点上的聚合，尽管后者是前者影响深远的推广。许多已知可由支配点聚合表达的问题，也因此变得可由前缀聚合表达。我们依赖一小组原语操作，这保证了易于迁移到各种分布式架构，并保证实现具有某些期望的性质。
英文摘要：
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.
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