Beyond Worst-Case Analysis for Joins with Minesweeper

TL;DR

This paper introduces Minesweeper, a new join algorithm with stronger runtime guarantees based on certificate complexity, outperforming traditional worst-case bounds for certain classes of database queries.

Contribution

Develops a framework for measuring stronger runtime guarantees called certificate complexity and establishes a dichotomy theorem for its application to join query evaluation.

Findings

Minesweeper evaluates β-acyclic queries in time linear in certificate plus output size.

For β-cyclic queries, some instances require superlinear time in the certificate.

Extends analysis to queries with bounded treewidth and the triangle query.

Abstract

We describe a new algorithm, Minesweeper, that is able to satisfy stronger runtime guarantees than previous join algorithms (colloquially, `beyond worst-case guarantees') for data in indexed search trees. Our first contribution is developing a framework to measure this stronger notion of complexity, which we call {\it certificate complexity}, that extends notions of Barbay et al. and Demaine et al.; a certificate is a set of propositional formulae that certifies that the output is correct. This notion captures a natural class of join algorithms. In addition, the certificate allows us to define a strictly stronger notion of runtime complexity than traditional worst-case guarantees. Our second contribution is to develop a dichotomy theorem for the certificate-based notion of complexity. Roughly, we show that Minesweeper evaluates $\beta$-acyclic queries in time linear in the certificate plus the output size, while for any $\beta$-cyclic query there is some instance that takes superlinear time in the certificate (and for which the output is no larger than the certificate size). We also extend our certificate-complexity analysis to queries with bounded treewidth and the triangle query.