BG (Base Graph)

Base Graph (BG) is a graph-based representation of a database schema that serves as the basis for query optimization in a database management system (DBMS). It was developed in the early 1980s as part of the System R project at IBM Research and has since become a standard approach for query optimization in many commercial and open-source DBMSs.

At a high level, the BG consists of two parts: a directed acyclic graph (DAG) that represents the logical structure of the database schema, and a set of transformation rules that describe how to transform a query expressed in terms of the DAG into an equivalent query that can be executed efficiently by the DBMS.

The DAG represents the schema as a set of nodes, each of which corresponds to a relation or table in the database. The nodes are connected by directed edges that represent foreign key relationships between the tables. In other words, if there is a foreign key from table A to table B, there is a directed edge from node A to node B in the DAG. The DAG is acyclic because cycles in the schema can lead to problems during query optimization, such as infinite recursion or redundant subqueries.

The DAG provides a way to reason about the structure of the schema and the relationships between tables, which is essential for optimizing queries. For example, if a query involves joining two tables that are connected by a foreign key relationship in the DAG, the optimizer can use this information to determine the most efficient join order and join algorithm to use.

The transformation rules are a set of rules that describe how to transform a query expressed in terms of the DAG into an equivalent query that can be executed efficiently by the DBMS. There are many different transformation rules, each of which represents a different optimization technique. Some common optimization techniques that can be implemented using transformation rules include join reordering, projection pushdown, predicate pushdown, and index selection.

Join reordering is a technique that involves changing the order in which tables are joined in a query to minimize the amount of data that needs to be processed. For example, if a query involves joining three tables A, B, and C, the optimizer can consider different join orders such as (A join B) join C, A join (B join C), or (A join C) join B, and choose the one that minimizes the number of intermediate results that need to be generated.

Projection pushdown is a technique that involves pushing projection operations (i.e., selecting only certain columns from a table) as close to the source data as possible. This can reduce the amount of data that needs to be processed and can improve query performance.

Predicate pushdown is a technique that involves pushing filter operations (i.e., selecting only certain rows from a table based on some condition) as close to the source data as possible. This can also reduce the amount of data that needs to be processed and can improve query performance.

Index selection is a technique that involves choosing the most appropriate index to use for a given query based on the selectivity of the query and the characteristics of the index. For example, if a query involves selecting rows from a table based on a certain column value, the optimizer can choose to use an index on that column to speed up the query.

The transformation rules are typically applied in a sequence of passes, with each pass applying a subset of the rules in a specific order. The order in which the rules are applied can have a significant impact on the quality of the optimized query. Therefore, DBMSs typically use heuristics or cost-based optimization techniques to determine the best order in which to apply the rules.

In summary, the Base Graph (BG) is a graph-based representation of a database schema that provides a way to reason about the structure of the schema and the relationships between tables, which is essential for optimizing queries. The BG consists of a DAG that represents the logical structure of the schema and a set of transformation rules that describe how to transform a query expressed in terms of the DAG into an equivalent query that can be executed efficiently by the DBMS.

The BG approach to query optimization has several advantages over other approaches. First, it is highly modular and extensible, meaning that new optimization techniques can be added to the set of transformation rules as needed. This makes it easy to adapt to new query patterns or changing workload characteristics. Second, it is based on a formal, mathematical model of the schema and the query, which makes it possible to reason about the correctness and completeness of the optimization process. Third, it can be implemented efficiently, even for large and complex databases, because the DAG can be constructed and manipulated using standard graph algorithms.

However, there are also some limitations to the BG approach. One limitation is that it relies heavily on the quality of the schema design. If the schema is poorly designed, with many complex or redundant relationships between tables, the BG may not be able to optimize queries as effectively. Another limitation is that it can be difficult to optimize queries that involve complex operations such as subqueries or aggregation functions. In these cases, additional optimization techniques or heuristics may be needed to achieve good query performance.

Despite these limitations, the BG remains a widely used and effective approach to query optimization in modern DBMSs. Many commercial and open-source DBMSs, including IBM DB2, Oracle, PostgreSQL, and MySQL, use some variant of the BG approach to optimize queries. In addition, research continues to explore new optimization techniques and extensions to the BG model, such as support for distributed databases, multi-objective optimization, and machine learning-based optimization.

In conclusion, the Base Graph (BG) is a graph-based representation of a database schema that provides a way to reason about the structure of the schema and the relationships between tables, which is essential for optimizing queries. The BG approach to query optimization is highly modular and extensible, based on a formal, mathematical model of the schema and the query, and can be implemented efficiently for large and complex databases. While there are some limitations to the BG approach, it remains a widely used and effective approach to query optimization in modern DBMSs.