发表机构
Boston University; Max Planck Institute for Software Systems; TU Wien; University of Washington（波士顿大学; 马克斯·普朗克软件系统研究所; 维也纳工业大学; 华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究卷积和积查询（CSPQ）的评估，将最坏情况最优连接、树分解等技术适配到CSPQ，定义三种宽度度量，证明方法可应用于任意域，解决了现有技术渐近次优的问题。
AI 中文摘要
我们研究和积查询（SPQ）扩展形式的查询评估，该扩展允许包含变量线性组合的原子（例如$A(2X+3Y-Z, Y+Z)$），我们将其称为卷积和积查询（CSPQ）。这些查询既出现在实际场景（如图像处理工作负载）中，也出现在理论场景（如$(\text{min},+)$-卷积和$k$-SUM猜想）中，且捕获了带有附加线性等式约束的SPQ。尽管先前的工作已考虑线性等式和不等式约束对查询评估的影响，但该场景下开发的技术对于CSPQ而言渐近次优。为解决此问题，我们描述了几种利用线性代数技术（如秩分析、商空间和变量替换）的CSPQ评估算法。首先，我们将最坏情况最优连接适配到CSPQ，实现了与合取查询类似的运行时间。然后，我们将树分解（TD）的定义扩展到CSPQ，并描述了一种因式分解执行方式，在此扩展中，线性组合是一等公民，发挥着与传统TD中变量相同的作用。我们定义了三种宽度度量，分别针对域值大小、活跃域大小以及关系支撑大小来限定此执行的复杂度。最后，我们展示了这些方法如何应用于有理数之外的任意域。
英文摘要
We study query evaluation for an extension of sum-product queries (SPQ) that allows atoms with linear combinations of variables (e.g. $A(2X+3Y-Z, Y+Z)$), which we call convolution sum-product queries (CSPQs). These queries arise in both practical settings (e.g. image processing workloads) and theoretical ones (e.g. the $(\min,+)$-convolution and $k$-SUM conjectures), and capture SPQs with added linear equality constraints. While prior work has considered the impact of linear in- and dis-equality constraints on query evaluation, the techniques developed in that setting are asymptotically sub-optimal for CSPQs. To address this, we describe several evaluation algorithms for CSPQs that leverage linear algebra techniques like rank analysis, quotient spaces, and variable substitution. First, we adapt Worst-Case Optimal Joins to CSPQs, achieving a runtime similar in spirit to that for conjunctive queries. Then, we extend the definition of tree decompositions (TDs) to CSPQs, and describe a factorized execution. In this extension, linear combinations are first-class citizens and play the same role as variables in traditional TDs. We define three width measures that bound the complexity of this execution with respect to the size of the domain values, of the active domain, and of the relation's support. Lastly, we show how these methods can be applied to arbitrary fields beyond the rationals.
