Using Color Refinement to Boost Enumeration and Counting for Acyclic CQs of Binary Schemas

TL;DR

This paper introduces the color-index, an index structure based on color refinement, to efficiently evaluate and count acyclic conjunctive queries over binary schemas, often outperforming the database size.

Contribution

The paper proposes the color-index, leveraging color refinement for fast enumeration and counting of acyclic conjunctive queries, a novel approach in database query evaluation.

Findings

Preprocessing time is linear in the size of the color-index.

Enumeration and counting can be performed with constant delay and time proportional to the index size.

The index size can be significantly smaller than the database, enabling more efficient query processing.

Abstract

We present an index structure, called the color-index, to boost the evaluation of acyclic conjunctive queries (ACQs) over binary schemas. The color-index is based on the color refinement algorithm, a widely used subroutine for graph isomorphism testing algorithms. Given a database $D$, we use a suitable version of the color refinement algorithm to produce a stable coloring of $D$, an assignment from the active domain of $D$ to a set of colors $C_D$. The main ingredient of the color-index is a particular database $D_c$ whose active domain is $C_D$ and whose size is at most $|D|$. Using the color-index, we can evaluate any free-connex ACQ $Q$ over $D$ with preprocessing time $O(|Q| \cdot |D_c|)$ and constant delay enumeration. Furthermore, we can also count the number of results of $Q$ over $D$ in time $O(|Q| \cdot |D_c|)$. Given that $|D_c|$ could be much smaller than $|D|$ (even constant-size for some families of databases), the color-index is the first index structure for evaluating free-connex ACQs that allows efficient enumeration and counting with performance that may be strictly smaller than the database size.