Answering Conjunctive Queries under Updates

Abstract

We consider the task of enumerating and counting answers to k-ary conjunctive queries against relational databases that may be updated by inserting or deleting tuples.We exhibit a new notion of q-hierarchical conjunctive queries and show that these can be maintained efficiently in the following sense. During a linear time preprocessing phase, we can build a data structure that enables constant delay enumeration of the query results; and when the database is updated, we can update the data structure and restart the enumeration phase within constant time. For the special case of self-join free conjunctive queries we obtain a dichotomy: if a query is not q-hierarchical, then query enumeration with sublinear * delay and sublinear update time (and arbitrary preprocessing time) is impossible.For answering Boolean conjunctive queries and for the more general problem of counting the number of solutions of k-ary queries we obtain complete dichotomies: if the query's homomorphic core is q-hierarchical, then size of the the query result can be computed in linear time and maintained with constant update time. Otherwise, the size of the query result cannot be maintained with sublinear update time.All our lower bounds rely on the OMv-conjecture, a conjecture on the hardness of online matrix-vector multiplication that has recently emerged in the field of fine-grained complexity to characterise the hardness of dynamic problems. The lower bound for the counting problem additionally relies on the orthogonal vectors conjecture, which in turn is implied by the strong exponential time hypothesis. * ) By sublinear we mean O(n 1−ε ) for some ε > 0, where n is the size of the active domain of the current database. * This is the full version of the conference contribution [6].

On the Enumeration Complexity of Unions of Conjunctive Queries

“…R . An enumeration algorithm A for E R is a 3 RAM that solves E R without repetitions. We say that A enumerates E R with delay d(|x |) if the time before the first output, the time between any two consecutive outputs, and the time between the last output and termination are each bound by d(|x |).…”

“…The reduction encodes the hard problem to the relations of a self-join free CQ, such that the answers of the CQ correspond to an answer of this problem [2,3,5,6]. However, using such an encoding for CQs within a union does not always work.…”

General dynamic Yannakakis: conjunctive queries with theta joins under updates

“…Dynamic Yannakakis (Dyn) is an algorithm to efficiently evaluate free-connex acyclic aggregate-equijoin queries under updates [23]. This algorithm matches two important theoretical lower bounds (for q-hierarchical CQs [7] and free-connex acyclic CQs [5]), and is highly efficient in practice. In this section we present a generalization of Dyn, called GDyn, to dynamically process free-connex acyclic GCQs.…”

Incremental View Maintenance with Triple Lock Factorization Benefits

“…Our approach with the relational payload ring strictly subsumes previous work on factorized IVM for acyclic joins [21] as it can support cyclic joins (see Section 6). The so-called q-hierarchical join queries (such as the Housing query in our experiments) are exactly those self-join-free conjunctive queries that admit constant time update [8]. Recent work on in-database maintenance of linear regression models shows how to compute such models using previously computed models over distinct sets of features [19].…”