Towards Approximate Query Enumeration with Sublinear Preprocessing Time

TL;DR

This paper presents new algorithms for approximate query enumeration on sparse databases that significantly reduce preprocessing time to polylogarithmic levels, with guarantees on accuracy and completeness under certain conditions.

Contribution

It introduces a novel model for approximate enumeration with sublinear preprocessing, extending prior work to achieve faster preprocessing with probabilistic guarantees.

Findings

Constant delay enumeration after sublinear preprocessing on bounded degree databases.

High-probability guarantees that enumerated tuples are close to actual query answers.

Speed-up in preprocessing from linear to polylogarithmic time, with controllable accuracy.

Abstract

This paper aims at providing extremely efficient algorithms for approximate query enumeration on sparse databases, that come with performance and accuracy guarantees. We introduce a new model for approximate query enumeration on classes of relational databases of bounded degree. We first prove that on databases of bounded degree any local first-order definable query can be enumerated approximately with constant delay after a constant time preprocessing phase. We extend this, showing that on databases of bounded tree-width and bounded degree, every query that is expressible in first-order logic can be enumerated approximately with constant delay after a sublinear (more precisely, polylogarithmic) time preprocessing phase. Durand and Grandjean (ACM Transactions on Computational Logic 2007) proved that exact enumeration of first-order queries on databases of bounded degree can be done with constant delay after a linear time preprocessing phase. Hence we achieve a significant speed-up in the preprocessing phase. Since sublinear running time does not allow reading the whole input database even once, sacrificing some accuracy is inevitable for our speed-up. Nevertheless, our enumeration algorithms come with guarantees: With high probability, (1) only tuples are enumerated that are answers to the query or `close' to being answers to the query, and (2) if the proportion of tuples that are answers to the query is sufficiently large, then all answers will be enumerated. Here the notion of `closeness' is a tuple edit distance in the input database. For local first-order queries, only actual answers are enumerated, strengthening (1). Moreover, both the `closeness' and the proportion required in (2) are controllable. We combine methods from property testing of bounded degree graphs with logic and query enumeration, which we believe can inspire further research.