Enumerating Answers to First-Order Queries over Databases of Low Degree

TL;DR

This paper extends the understanding of first-order query evaluation over low-degree databases, demonstrating efficient counting, testing, and enumeration of query answers with pseudo-linear preprocessing.

Contribution

It generalizes previous results by providing algorithms for counting, testing, and enumerating query answers in low-degree databases with pseudo-linear time complexity.

Findings

Counting answers can be done in pseudo-linear time.

Answer testing can be performed in constant time after preprocessing.

Enumeration of answers has constant delay after pseudo-linear preprocessing.

Abstract

A class of relational databases has low degree if for all $\delta>0$, all but finitely many databases in the class have degree at most $n^{\delta}$, where $n$ is the size of the database. Typical examples are databases of bounded degree or of degree bounded by $\log n$. It is known that over a class of databases having low degree, first-order boolean queries can be checked in pseudo-linear time, i.e.\ for all $\epsilon>0$ in time bounded by $n^{1+\epsilon}$. We generalize this result by considering query evaluation. We show that counting the number of answers to a query can be done in pseudo-linear time and that after a pseudo-linear time preprocessing we can test in constant time whether a given tuple is a solution to a query or enumerate the answers to a query with constant delay.