We investigate the fine-grained complexity of dynamically maintaining the result of fixed self-join free conjunctive queries under single-tuple updates. Prior work shows that free-connex queries can be maintained in update time $O(|D|^δ)$ for some $δ\in [0.5, 1]$, where $|D|$ is the size of the current database. However, a gap remains between the best known upper bound of $O(|D|)$ and lower bounds of $Ω(|D|^{0.5-ε})$ for any $ε\ge 0$. We narrow this gap by introducing two structural parameters to quantify the dynamic complexity of a conjunctive query: the height $k$ and the dimension $d$. We establish new fine-grained lower bounds showing that any algorithm maintaining a query with these parameters must incur update time $Ω(|D|^{1-1/\max(k,d)-ε})$, unless widely believed conjectures fail. These yield the first super-$\sqrt{|D|}$ lower bounds for maintaining free-connex queries, and suggest the tightness of current algorithms when considering arbitrarily large $k$ and~$d$. Complementing our lower bounds, we identify a data-dependent parameter, the generalized $H$-index $h(D)$, which is upper bounded by $|D|^{1/d}$, and design an efficient algorithm for maintaining star queries, a common class of height 2 free-connex queries. The algorithm achieves an instance-specific update time $O(h(D)^{d-1})$ with linear space $O(|D|)$. This matches our parameterized lower bound and provides instance-specific performance in favorable cases.

翻译：我们研究了在单元组更新下动态维护固定无自连接合取查询结果的细粒度复杂度。先前工作表明，自由连接查询可在更新时间为 $O(|D|^δ)$（其中 $δ\in [0.5, 1]$，$|D|$ 表示当前数据库规模）的条件下进行维护。然而，现有最佳上界 $O(|D|)$ 与任意 $ε\ge 0$ 对应的下界 $Ω(|D|^{0.5-ε})$ 之间仍存在差距。为缩小这一差距，我们引入两个量化合取查询动态复杂度的结构参数：高度 $k$ 与维度 $d$。通过建立新的细粒度下界，我们证明除非广泛接受的猜想不成立，否则维护具有这些参数的查询的任何算法必须承受 $Ω(|D|^{1-1/\max(k,d)-ε})$ 的更新时间。这首次为维护自由连接查询提出了超 $\sqrt{|D|}$ 级下界，并表明当考虑任意大的 $k$ 和 $d$ 时，现有算法可能已达最优。作为下界的补充，我们定义了一个数据依赖型参数——广义 $H$ 指数 $h(D)$（其上界为 $|D|^{1/d}$），并针对星型查询（一类常见的高度为 2 的自由连接查询）设计了高效算法。该算法在 $O(|D|)$ 线性空间复杂度下实现了实例特定的更新时间 $O(h(D)^{d-1})$，既符合我们的参数化下界，又在有利情况下提供了实例特定的性能表现。