Tight Fine-Grained Bounds for Direct Access on Join Queries

TL;DR

This paper establishes tight bounds on preprocessing time for lexicographic direct access to join query answers, using a decomposition algorithm and complexity-theoretic lower bounds based on the Zero-Clique Conjecture.

Contribution

It introduces a decomposition-based algorithm for direct access on join queries and proves its optimality through lower bounds derived from fine-grained complexity assumptions.

Findings

The algorithm achieves tight bounds for all join queries and orders.

Lower bounds are based on the hardness of the Set-Disjointness problem.

Self-joins do not affect the complexity of direct access.

Abstract

We consider the task of lexicographic direct access to query answers. That is, we want to simulate an array containing the answers of a join query sorted in a lexicographic order chosen by the user. A recent dichotomy showed for which queries and orders this task can be done in polylogarithmic access time after quasilinear preprocessing, but this dichotomy does not tell us how much time is required in the cases classified as hard. We determine the preprocessing time needed to achieve polylogarithmic access time for all join queries and all lexicographical orders. To this end, we propose a decomposition-based general algorithm for direct access on join queries. We then explore its optimality by proving lower bounds for the preprocessing time based on the hardness of a certain online Set-Disjointness problem, which shows that our algorithm's bounds are tight for all lexicographic orders on join queries. Then, we prove the hardness of Set-Disjointness based on the Zero-Clique Conjecture which is an established conjecture from fine-grained complexity theory. Interestingly, while proving our lower bound, we show that self-joins do not affect the complexity of direct access (up to logarithmic factors). Our algorithm can also be used to solve queries with projections and relaxed order requirements, though in these cases, its running time not necessarily optimal. We also show that similar techniques to those used in our lower bounds can be used to prove that, for enumerating answers to Loomis-Whitney joins, it is not possible to significantly improve upon trivially computing all answers at preprocessing. This, in turn, gives further evidence (based on the Zero-Clique Conjecture) to the enumeration hardness of self-join free cyclic joins with respect to linear preprocessing and constant delay.