Relational Algebra

Chapter 4, Part A

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Relational Query Languages

• Query languages: Allow manipulation and retrieval of data from a database.
• Relational model supports simple, powerful QLs:
• Strong formal foundation based on logic.
• Allows for much optimization.
• Query Languages != programming languages!
• QLs not expected to be “Turing complete”.
• QLs not intended to be used for complex calculations.
• QLs support easy, efficient access to large data sets.

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Formal Relational Query Languages

Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation:

• Relational Algebra: More operational, very useful for representing execution plans.
• Relational Calculus: Lets users describe what they want, rather than how to compute it. (Non-operational, declarative.)

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• Understanding Algebra & Calculus is key to
• understanding SQL, query processing!

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Preliminaries

• A query is applied to relation instances, and the result of a query is also a relation instance.
• Schemas of input relations for a query are fixed (but query will run regardless of instance!)
• The schema for the result of a given query is also fixed! Determined by definition of query language constructs.
• Positional vs. named-field notation:
• Positional notation easier for formal definitions, named-field notation more readable.
• Both used in SQL

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Example Instances

• “Sailors” and “Reserves” relations for our examples. See schemas for relations in text
• We’ll use positional or named field notation, assume that names of fields in query results are `inherited’ from names of fields in query input relations.

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R1

S1

S2

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Relational Algebra

• Basic operations:
• Selection ( ) Selects a subset of rows from relation.
• Projection ( ) Deletes unwanted columns from relation.
• Cross-product ( ) Allows us to combine two relations.
• Set-difference ( ) Tuples in reln. 1, but not in reln. 2.
• Union ( ) Tuples in reln. 1 and in reln. 2.
• Additional operations:
• Intersection, join, division, renaming: Not essential, but (very!) useful.
• Since each operation returns a relation, operations can be composed! (Algebra is “closed”.)

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Projection

• Deletes attributes that are not in projection list.
• Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation.
• Projection operator has to eliminate duplicates! (Why??)
• Note: real systems typically don’t do duplicate elimination unless the user explicitly asks for it. (Why not?)

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Selection

• Selects rows that satisfy selection condition.
• No duplicates in result! (Why?)
• Schema of result identical to schema of (only) input relation.
• Result relation can be the input for another relational algebra operation! (Operator composition.)

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Union, Intersection, Set-Difference

• All of these operations take two input relations, which must be union-compatible:
• Same number of fields.
• `Corresponding’ fields have the same type.
• What is the schema of result?

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Cross-Product

• Each row of S1 is paired with each row of R1.
• Result schema has one field per field of S1 and R1, with field names `inherited’ if possible.
• Conflict: Both S1 and R1 have a field called sid.

• Renaming operator:

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Joins

• Condition Join:

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• Result schema same as that of cross-product.
• Fewer tuples than cross-product, might be able to compute more efficiently
• Sometimes called a theta-join.

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Joins

• Equi-Join: A special case of condition join where the condition c contains only equalities.

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• Result schema similar to cross-product, but only one copy of fields for which equality is specified.
• Natural Join: Equijoin on all common fields.

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Division

• Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats.
• Let A have 2 fields, x and y; B have only field y:
• A/B =
• i.e., A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A.
• Or: If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B.
• In general, x and y can be any lists of fields; y is the list of fields in B, and x y is the list of fields of A.

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Examples of Division A/B

A

B1

B2

B3

A/B1

A/B2

A/B3

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Expressing A/B Using Basic Operators

• Division is not essential op; just a useful shorthand.
• (Also true of joins, but joins are so common that systems implement joins specially.)
• Idea: For A/B, compute all x values that are not `disqualified’ by some y value in B.
• x value is disqualified if by attaching y value from B, we obtain an xy tuple that is not in A.

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Disqualified x values:

A/B:

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Example tables

Sailors(sid: integer, sname: string, rating: integer, age: real)

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Boats(bid: integer, bname: string, color: string)

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Reserves(sid: integer, bid: integer, day: date)

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If the key for the Reserves relation contained only

the attributes sid and bid, how would the semantics differ?

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Database Management Systems, R. Ramakrishnan and J. Gehrke

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Find names of sailors who’ve reserved boat #103

• Solution 1:

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Find names of sailors who’ve reserved a red boat

• Information about boat color only available in Boats; so need an extra join:

• A query optimizer can find this given the first solution!

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Find sailors who’ve reserved a red or a green boat

• Can identify all red or green boats, then find sailors who’ve reserved one of these boats:

• Can also define Tempboats using union! (How?)

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Find sailors who’ve reserved a red and a green boat

• Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors):

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Find the names of sailors who’ve reserved all boats

• Uses division; schemas of the input relations to / must be carefully chosen:

• To find sailors who’ve reserved all ‘Interlake’ boats:

.....

Database Management Systems, R. Ramakrishnan and J. Gehrke

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Summary

• The relational model has rigorously defined query languages that are simple and powerful.
• Relational algebra is more operational; useful as internal representation for query evaluation plans.
• Several ways of expressing a given query; a query optimizer should choose the most efficient version.

Database Management Systems, R. Ramakrishnan and J. Gehrke

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