We investigate trade-offs in static and dynamic evaluation of hierarchical queries with arbitrary free variables. In the static setting, the trade-off is between the time to partially compute the query result and the delay needed to enumerate its tuples. In the dynamic setting, we additionally consider the time needed to update the query result under single-tuple inserts or deletes to the database. Our approach observes the degree of values in the database and uses different computation and maintenance strategies for high-degree (heavy) and low-degree (light) values. For the latter it partially computes the result, while for the former it computes enough information to allow for on-the-fly enumeration. We define the preprocessing time, the update time, and the enumeration delay as functions of the light/heavy threshold. By appropriately choosing this threshold, our approach recovers a number of prior results when restricted to hierarchical queries. We show that for a restricted class of hierarchical queries, our approach achieves worst-case optimal update time and enumeration delay conditioned on the Online Matrix-Vector Multiplication Conjecture.

翻译：我们研究了具有任意自由变量的分层查询在静态与动态评估中的权衡问题。在静态设定下，权衡体现在部分计算结果所需的时间与枚举元组的延迟之间；在动态设定下，我们还额外考虑了数据库单条元组插入或删除操作下更新查询结果所需的时间。本方法通过观测数据库中值的度数，对高度数（重值）和低度数（轻值）采用不同的计算与维护策略：针对低度数值部分计算结果，针对高度数值则预先计算足够的信息以实现即时枚举。我们将预处理时间、更新时间和枚举延迟定义为轻/重阈值的函数。通过适当选择该阈值，本方法在限制为分层查询时能够复现若干先前成果。我们证明，对于受限的分层查询类别，在基于在线矩阵向量乘法猜想的前提下，本方法能够实现最坏情况最优的更新时间和枚举延迟。