Abstract
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 adorned queries and 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 join size bound to capture the space usage of our data structure. We combine our data structure with 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.
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Notes
- 1.
Here we extend the valuation to mean \(v((a_1, \dots , a_n)) = (v(a_1), \dots , v(a_n))\).
- 2.
We will omit the parameter \(\textbf{u}\) from the notation of \(\alpha \) whenever it is clear from the context.
- 3.
For all results in this paper, S includes the space requirement of the input as well. If we are interested in only the space requirement of the constructed data structure, then the |D| term in the space requirement of Theorem 1 can be removed.
- 4.
All answering times \(T > \sqrt{|E|}\) are trivial to achieve using linear space by using the data structure for \(T' = \sqrt{E}\) and holding the result back until time T has passed.
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Deep, S., Hu, X., Koutris, P. (2023). General Space-Time Tradeoffs via Relational Queries. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_21
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