The VC-Dimension of Queries and Selectivity Estimation Through Sampling

TL;DR

This paper introduces a new method for estimating SQL query selectivity using VC-dimension theory, enabling accurate, sample-based predictions that are independent of database size and query count, with practical validation.

Contribution

The work provides an explicit VC-dimension bound for query outcome spaces and develops a sampling method for accurate selectivity estimation applicable to multiple queries simultaneously.

Findings

The VC-dimension depends on query predicate complexity, not database size.

A small, representative sample can accurately estimate query selectivity.

The method outperforms existing techniques like PostgreSQL and SQL Server in experiments.

Abstract

We develop a novel method, based on the statistical concept of the Vapnik-Chervonenkis dimension, to evaluate the selectivity (output cardinality) of SQL queries - a crucial step in optimizing the execution of large scale database and data-mining operations. The major theoretical contribution of this work, which is of independent interest, is an explicit bound to the VC-dimension of a range space defined by all possible outcomes of a collection (class) of queries. We prove that the VC-dimension is a function of the maximum number of Boolean operations in the selection predicate and of the maximum number of select and join operations in any individual query in the collection, but it is neither a function of the number of queries in the collection nor of the size (number of tuples) of the database. We leverage on this result and develop a method that, given a class of queries, builds a concise random sample of a database, such that with high probability the execution of any query in the class on the sample provides an accurate estimate for the selectivity of the query on the original large database. The error probability holds simultaneously for the selectivity estimates of all queries in the collection, thus the same sample can be used to evaluate the selectivity of multiple queries, and the sample needs to be refreshed only following major changes in the database. The sample representation computed by our method is typically sufficiently small to be stored in main memory. We present extensive experimental results, validating our theoretical analysis and demonstrating the advantage of our technique when compared to complex selectivity estimation techniques used in PostgreSQL and the Microsoft SQL Server.