In this paper, we investigate space-time tradeoffs for answering Boolean conjunctive queries. The goal is to create a data structure in an initial preprocessing phase and use it for answering (multiple) queries. Previous work has developed data structures that trade off space usage for answering time and has proved conditional space lower bounds for queries of practical interest such as the path and triangle query. However, most of these results cater to only those queries, lack a comprehensive framework, and are not generalizable. The isolated treatment of these queries also fails to utilize the connections with extensive research on related problems within the database community. The key insight in this work is to exploit the formalism of relational algebra by casting the problems as answering join queries over a relational database. Using the notion of boolean {\em adorned queries} and {\em access patterns}, we propose a unified framework that captures several widely studied algorithmic problems. Our main contribution is three-fold. First, we present an algorithm that recovers existing space-time tradeoffs for several problems. The algorithm is based on an application of the {\em join size bound} to capture the space usage of our data structure. We combine our data structure with {\em query decomposition} techniques to further improve the tradeoffs and show that it is readily extensible to queries with negation. Second, we falsify two proposed conjectures in the existing literature related to the space-time lower bound for path queries and triangle detection for which we show unexpectedly better algorithms. This result opens a new avenue for improving several algorithmic results that have so far been assumed to be (conditionally) optimal. Finally, we prove new conditional space-time lower bounds for star and path queries.

翻译：本文研究了布尔合取查询的时空权衡问题。目标是在初始预处理阶段创建数据结构，并利用它来回答（多个）查询。先前的工作已开发出权衡存储空间以换取查询回答时间的数据结构，并对路径查询和三角形查询等具有实际意义的查询证明了条件性空间下界。然而，这些结果大多仅针对特定查询，缺乏统一框架，且不具有普适性。对这些查询的孤立处理也未能利用与数据库领域相关问题的广泛研究之间的联系。本文的关键洞察在于利用关系代数的形式化方法，将问题转化为在关系数据库上回答连接查询。通过引入布尔"带修饰查询"和"访问模式"的概念，我们提出了一个统一框架，能够涵盖多个被广泛研究的算法问题。我们的主要贡献包含三个方面。首先，我们提出了一种算法，可恢复多个问题的现有时空权衡。该算法基于应用"连接大小界"来捕捉数据结构的空间使用量。我们将数据结构与"查询分解"技术相结合，以进一步改善权衡，并证明其易于扩展到带否定操作的查询。其次，我们否证了现有文献中两个关于路径查询和三角形检测时空下界提出的猜想，并为此展示了出乎意料更优的算法。这一结果为改进先前被认为是（条件性）最优的若干算法结果开辟了新途径。最后，我们证明了星型查询和路径查询的新条件性时空下界。