在更新情况下回答带有聚合的合取查询
Answering Conjunctive Queries with Aggregations under Updates
- Nanyang Technological University（南洋理工大学）
- University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究在更新情况下带聚合的合取查询，任意更新下维护难度与布尔半环相当，仅插入更新时边界变化，新类强连接出现。通过适配CROWN框架给出上下界，得到查询和半环参数化二分法，明确聚合增加的难度。
AI中文摘要：
动态查询处理在插入和删除操作期间保持查询答案的最新状态。对于集合语义下的合取查询（CQs），可维护类是确切已知的：在任意更新下是q - 分层CQs，在仅插入更新下扩展到自由连接CQs。但现代分析聚合，包括袋计数、SUM/COUNT、出处、访问控制和最短路径等，都通过在正交换半环上评估CQ来捕获。我们研究聚合是否改变了能有效维护的内容，以及如果改变，在何时改变。在任意更新下，没有改变：维护至少与在布尔半环上一样困难。在仅插入更新下，有改变：边界从自由连接退化为一个我们称为强连接的新类，其中q - 分层$\subsetneq$强连接$\subsetneq$自由连接$\subsetneq$无环。对于每个带有合适单调序列的有序半环（如和积和热带），在OuMv和OMv猜想下，没有自由连接但非强连接的CQ能在$O(|D|^{1/2 - \epsilon})$时间内维护。我们在组合k - 团和广义OuMv猜想下进一步将此下界强化为一个由查询的高度和维度参数化的族；这些量化了随着查询规模增加注释难度增长的程度。在算法方面，一个单一框架通过将CROWN适配到注释关系来匹配这些边界。它在仅插入更新下以$O(1)$摊销时间维护每个强连接CQ，无论底层半环如何。此外，在任意更新下，如果半环具有$O(1)$可删除聚合，则它以$O(1)$摊销时间维护每个q - 分层CQ。上下界一起给出了查询和半环参数化的二分法，恢复了布尔情况并精确指出了聚合增加的难度。
英文摘要：
Dynamic query processing keeps query answers up to date during insertions and deletions. For conjunctive queries (CQs) under set semantics, the classes maintainable in constant amortized time are known exactly: the $q$-hierarchical CQs under arbitrary updates, and the free-connex CQs under insertion-only updates. Many analytics tasks, including \textsf{SUM}/\textsf{COUNT} aggregations, provenance, and access control, are captured by evaluating a CQ over a positive commutative semiring. We thus ask whether aggregation changes what can be maintained efficiently, and if so, when. Under \emph{insertion-only} updates, it does: the boundary retreats from free-connex to a new class we call \emph{strong-connex}, with $q\text{-hierarchical} \subsetneq \text{strong-connex} \subsetneq \text{free-connex} \subsetneq \text{acyclic}$. For every \emph{strictly monotone} semiring, including the sum-product and tropical semirings, no free-connex but non-strong-connex CQ is maintainable in $O(|D|^{1/2-ε})$ time under the OuMv and OMv conjectures, whereas every strong-connex CQ is maintainable in $O(1)$ amortized time over every semiring. Under \emph{arbitrary} updates, the boundary stays at the $q$-hierarchical CQs for every semiring with $O(1)$-deletable aggregates, and maintenance over any semiring is at least as hard as over the Boolean semiring. We further strengthen the lower bounds to semirings that fall outside the class and to query with different \emph{height} and \emph{dimension}, under the combinatorial $k$-clique and generalized OuMv conjectures. All upper bounds come from a single framework, obtained by adapting CROWN to annotated relations; together with the lower bounds, they yield dichotomies parameterized by both the query and the semiring, recovering the Boolean results as a special case.
