Answering FO+MOD queries under updates on bounded degree databases

TL;DR

This paper presents a dynamic data structure for efficiently evaluating FO and FO+MOD queries on bounded degree databases, enabling immediate updates and enumeration with constant delay.

Contribution

It extends static bounded degree query evaluation techniques to dynamic settings, allowing constant-time query result updates after each database modification.

Findings

Constant-time query result reporting after updates

Linear time data structure construction

Immediate enumeration with constant delay

Abstract

We investigate the query evaluation problem for fixed queries over fully dynamic databases, where tuples can be inserted or deleted. The task is to design a dynamic algorithm that immediately reports the new result of a fixed query after every database update. We consider queries in first-order logic (FO) and its extension with modulo-counting quantifiers (FO+MOD), and show that they can be efficiently evaluated under updates, provided that the dynamic database does not exceed a certain degree bound. In particular, we construct a data structure that allows to answer a Boolean FO+MOD query and to compute the size of the result of a non-Boolean query within constant time after every database update. Furthermore, after every update we are able to immediately enumerate the new query result with constant delay between the output tuples. The time needed to build the data structure is linear in the size of the database. Our results extend earlier work on the evaluation of first-order queries on static databases of bounded degree and rely on an effective Hanf normal form for FO+MOD recently obtained by Heimberg, Kuske, and Schweikardt (LICS 2016).