Dynamic Query Evaluation Over Structures with Low Degree

We consider the evaluation of first-order queries over classes of databases that have bounded degree and low degree. More precisely, given a query and a database, we want to efficiently test whether there is a solution, count how many solutions there are, or be able to enumerate the set of all solutions. Bounded and low degree are rather natural notions and both yield efficient algorithms. For example, Berkholz, Keppeler, and Schweikardt showed in 2017 that over databases of bounded degree, not only any first order query can efficiently be tested, counted and enumerated, but the data structure used can be updated when the database itself is updated. This paper extends existing results in two directions. First, we show that over classes of databases with low degree, there is a data structure that enables us to test, count and enumerate the solutions of first order queries. This data structure can also be efficiently recomputed when the database is updated. Secondly, for classes of databases with bounded degree we show that, without increasing the preprocessing time, we can compute a data structure that does not depend on the query but only on its quantifier rank. We can therefore perform a single preprocessing that can later be used for many queries.