The trichotomy of HAVING queries on a probabilistic database

Abstract

We study the evaluation of positive conjunctive queries with Boolean aggregate tests (similar to HAVING in SQL) on probabilistic databases. More precisely, we study conjunctive queries with predicate aggregates on probabilistic databases where the aggregation function is one of MIN, MAX, EXISTS, COUNT, SUM, AVG, or COUNT(DISTINCT) and the comparison function is one of =, , ≥, >, ≤, or < . The complexity of evaluating a HAVING query depends on the aggregation function, α, and the comparison function, θ. In this paper, we establish a set of trichotomy results for conjunctive queries with HAVING predicates parametrized by (α, θ). For such queries (without self joins), one of the following three statements is true: (1) The exact evaluation problem has P-time data complexity. In this case, we call the query safe. (2) The exact evaluation problem is P-hard, but the approximate evaluation problem has (randomized) P-time data complexity. More precisely, there exists an  for the query. In this case, we call the query apx-safe. (3) The exact evaluation problem is P-hard, and the approximate evaluation problem is also hard. We call these queries hazardous. The precise definition of each class depends on the aggregate considered and the comparison function. Thus, we have queries that are (MAX, ≥)-safe, (COUNT, ≤)-apx-safe, (SUM, =)-hazardous, etc. Our trichotomy result is a signifi- cant extension of a previous dichotomy result for Boolean conjunctive queries into safe and not safe. For each of the three classes we present novel techniques. For safe queries, we describe an evaluation algorithm that uses random variables over semirings. For apx-safe queries, we describe an  that relies on a novel algorithm for generating a random possible world satisfying a given condition. Finally, for hazardous queries we give novel proofs of hardness of approximation. The results for safe queries were previously announced [43], but all other results are new.

Anytime approximation in probabilistic databases

“…[12,40]. Beyond these restrictions, there is work on tractability of positive queries [10], on queries with aggregates [17,48] and with negation [20,58]. The approximate probability computation algorithm described in this article is used by the SPROUT query engine [18] to complement its optimized exact algorithm for tractable conjunctive queries without self-joins [40].…”

Querying and Learning in Probabilistic Databases

“…Motivated by queries whose probability computations entail #P-hard [16,15] instances, the query evaluation problem in PDBs has been studied very intensively [16,15,17,38,42,53,59]. Except for two works [50,51], which we are aware of, each of these approaches aims for computing all answers along with their probabilities.…”

Querying uncertain data with aggregate constraints

“…We further assume that the constraints actually disqualify 99.9999% possible worlds, out of the total 10 477,121 possible worlds. In fact, the problem is known to be #P-hard [23]. Next, we illustrate two observations that are important for designing an efficient sampling strategy.…”