Worst-case Optimal Join Algorithms

TL;DR

This paper introduces a new algorithm for natural join queries that is proven to be worst-case optimal, improving over traditional methods and providing constructive proofs of key geometric inequalities.

Contribution

It presents the first worst-case optimal join algorithm for all natural join queries, bridging theoretical bounds with practical algorithm design.

Findings

The algorithm achieves worst-case optimality for all natural join queries.

It provides a constructive proof of the fractional cover bound without Shearer's inequality.

The algorithm also proves geometric inequalities like Loomis-Whitney and Bollobás-Thomason.

Abstract

Efficient join processing is one of the most fundamental and well-studied tasks in database research. In this work, we examine algorithms for natural join queries over many relations and describe a novel algorithm to process these queries optimally in terms of worst-case data complexity. Our result builds on recent work by Atserias, Grohe, and Marx, who gave bounds on the size of a full conjunctive query in terms of the sizes of the individual relations in the body of the query. These bounds, however, are not constructive: they rely on Shearer's entropy inequality which is information-theoretic. Thus, the previous results leave open the question of whether there exist algorithms whose running time achieve these optimal bounds. An answer to this question may be interesting to database practice, as it is known that any algorithm based on the traditional select-project-join style plans typically employed in an RDBMS are asymptotically slower than the optimal for some queries. We construct an algorithm whose running time is worst-case optimal for all natural join queries. Our result may be of independent interest, as our algorithm also yields a constructive proof of the general fractional cover bound by Atserias, Grohe, and Marx without using Shearer's inequality. This bound implies two famous inequalities in geometry: the Loomis-Whitney inequality and the Bollob\'as-Thomason inequality. Hence, our results algorithmically prove these inequalities as well. Finally, we discuss how our algorithm can be used to compute a relaxed notion of joins.