Beyond Worst-case Analysis for Joins with Minesweeper

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 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 β-acyclic queries in time linear in the certificate plus the output size, while for any β-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. We present em-pirical results that certificates can be much smaller than the input size, which suggests that ideas in minesweeper might lead to faster algorithms in practice