Size and Treewidth Bounds for Conjunctive Queries

This paper provides new worst-case bounds for the size and treewith of the result Q(D) of a conjunctive query Q to a database D. We derive bounds for the result size |Q(D) | in terms of structural properties of Q, both in the absence and in the presence of keys and functional dependencies. These bounds are based on a novel “coloring ” of the query variables that associates a coloring number C(Q) to each query Q. Using this coloring number, we derive tight bounds for the size of Q(D) in case (i) no functional dependencies or keys are specified, and (ii) simple (one-attribute) keys are given. These results generalize recent size-bounds for join queries obtained by Atserias, Grohe, and Marx (FOCS 2008). An extension of our coloring technique also gives a lower bound for |Q(D) | in the general setting of a query with arbitrary functional dependencies. Our new coloring scheme also allows us to precisely characterize (both in the absence of keys and with simple keys) the treewidth-preserving queries— the queries for which the output treewidth is bounded by a function of the input treewidth. Finally we characterize the queries that preserve the sparsity of the input in the general setting with arbitrary functional dependencies

Decision Problems in Information Theory

Constraints on entropies are considered to be the laws of information theory. Even though the pursuit of their discovery has been a central theme of research in information theory, the algorithmic aspects of constraints on entropies remain largely unexplored. Here, we initiate an investigation of decision problems about constraints on entropies by placing several different such problems into levels of the arithmetical hierarchy. We establish the following results on checking the validity over all almost-entropic functions: first, validity of a Boolean information constraint arising from a monotone Boolean formula is co-recursively enumerable; second, validity of "tight" conditional information constraints is in ???. Furthermore, under some restrictions, validity of conditional information constraints "with slack" is in ???, and validity of information inequality constraints involving max is Turing equivalent to validity of information inequality constraints (with no max involved). We also prove that the classical implication problem for conditional independence statements is co-recursively enumerable

Join Cardinality Estimation Graphs: Analyzing Pessimistic and Optimistic Estimators Through a Common Lens

Join cardinality estimation is a fundamental problem that is solved in the query optimizers of database management systems when generating efficient query plans. This problem arises both in systems that manage relational data as well those that manage graph-structured data where systems need to estimate the cardinalities of subgraphs in their input graphs. We focus on graph-structured data in this thesis. A popular class of join cardinality estimators uses statistics about sizes of small size queries to make estimates for larger queries. Statistics-based estimators can be broadly divided into two groups: (i) optimistic estimators that use statistics in formulas that make degree regularity and conditional independence assumptions; and (ii) the recent pessimistic estimators that estimate the sizes of queries using a set of upper bounds derived from linear programs, such as the AGM bound, or tighter bounds, such as the MOLP bound that are based on information theoretic bounds. In this thesis, we introduce a new framework that we call cardinality estimation graph (CEG) that can represent the estimates of both optimistic and pessimistic estimators. We observe that there is generally more than one way to generate optimistic estimates for a query, and the choice has either been ad-hoc or unspecified in previous work. We empirically show that choosing the largest candidate yields much higher accuracy than pessimistic estimators across different datasets and query workloads, and it is an effective heuristic to combat underestimations, which optimistic estimators are known to suffer from. To further improve the accuracy, we demonstrate how hash partitioning, an optimization technique designed to improve pessimistic estimators' accuracy, can be applied to optimistic estimators, and we evaluate the effectiveness. CEGs can also be used to obtain insights of pessimistic estimators. We show MOLP estimator is at least as tight as the pessimistic estimator and are identical on acyclic queries over binary relations, and the MOLP CEG offers an intuitive combinatorial proof that the MOLP bound is tighter than the DBPLP bound