[Paper Review] Entropy bounds for conjunctive queries with functional dependencies

Tomasz Gogacz, Szymon Toruńczyk|arXiv (Cornell University)|Dec 6, 2015

Complexity and Algorithms in Graphs8 references4 citations

TL;DR

This paper provides a precise characterization of the worst-case size bound for conjunctive queries under functional dependencies, showing that the entropy-based upper bound from Gottlob et al. (2012) is tight. It establishes that the bound can be exactly captured using entropy vectors and finite group constructions, resolving an open question and linking query size bounds to deep problems in information theory.

ABSTRACT

We study the problem of finding the worst-case bound for the size of the result $Q(\mathbb{ D})$ of a fixed conjunctive query $Q$ applied to a database $\mathbb{ D}$ satisfying given functional dependencies. We provide a precise characterization of this bound in terms of entropy vectors, and in terms of finite groups. In particular, we show that an upper bound provided by Gottlob, Lee, Valiant and Valiant is tight, answering a question from their paper. Our result generalizes the bound due to Atserias, Grohe and Marx, who consider the case without functional dependencies. Our result shows that the problem of computing the worst-case size bound, in the general case, is closely related to difficult problems from information theory.

Motivation & Objective

• To characterize the worst-case size of the result of a conjunctive query under functional dependencies.
• To determine whether the entropy-based upper bound proposed by Gottlob et al. (2012) is tight in the presence of functional dependencies.
• To establish a connection between query size bounds and information-theoretic entropy cones and finite group constructions.
• To generalize the AGM bound (Atserias, Grohé, Marx) to the case with functional dependencies.
• To explore the algorithmic complexity of evaluating conjunctive queries under functional dependencies.

Proposed method

• The paper uses entropy vectors indexed by subsets of query variables to represent the joint entropy of variable groups in a database instance.
• It proves that the worst-case query size exponent α(Q) equals the minimal entropy value over all entropy vectors satisfying the functional dependencies and query constraints.
• A finite group construction is used to achieve a matching lower bound, demonstrating tightness of the entropy bound.
• The method relies on translating functional dependencies into entropy constraints and using the entropy cone to characterize feasible entropy vectors.
• The analysis involves defining a notion of iterative width for functional dependencies, which controls the complexity of query evaluation algorithms.
• The paper applies results from information theory, particularly the structure of the entropy cone and non-Shannon inequalities, to analyze the problem.

Experimental results

Research questions

• RQ1 Is the entropy-based upper bound for conjunctive query result size tight when functional dependencies are present?
• RQ2 Can the worst-case query size exponent α(Q) be exactly characterized using entropy vectors under functional dependencies?
• RQ3 What is the relationship between finite group constructions and entropy-based bounds for conjunctive queries?
• RQ4 How does the presence of functional dependencies affect the applicability of the AGM bound (fractional edge covering)?
• RQ5 Can the problem of computing α(Q) be reduced to optimization over the entropy cone, and what are the computational challenges involved?

Key findings

• The entropy-based upper bound for query result size, previously proposed by Gottlob et al. (2012), is proven to be tight, answering their open question affirmatively.
• The worst-case size exponent α(Q) is exactly equal to the minimum entropy value over all entropy vectors satisfying the functional dependencies and query structure.
• A finite group construction achieves the entropy lower bound, proving that the entropy characterization is not only an upper bound but also tight.
• The problem of computing α(Q) is shown to be deeply connected to difficult problems in information theory, particularly optimization over the entropy cone.
• The paper generalizes the AGM bound to the case with functional dependencies, showing that the AGM bound remains valid only when no functional dependencies are present.
• An algorithm is presented that computes the query result in time O(|D|^α(Q) + m), where m is the iterative width of the functional dependencies, generalizing the result of Atserias et al. (2013).