R ELATIONAL A LGEBRA M ORE POINTERS FROM T UESDAY.

RELATIONAL ALGEBRA

MORE POINTERS FROM TUESDAY

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RELATIONAL ALGEBRA - DIVISION

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Queries that give the same result as division are not replacements for division

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DIVISION EXAMPLE

Student TaskFred Database1Fred Database2Fred Compiler1Eugene Database1Eugene Compiler1Sara Database1Sara Database2

Completed DBProjectTaskDatabase1Database2

Want to perform Completed/DBProject to find all students who completed all projects.

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DIVISION EXAMPLE

Step 1) Project Completed onto it’s unique attributes

πStudent(Completed)


Completed

DBProjectTaskDatabase1Database2

AStudentFredEugeneSara

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DIVISION EXAMPLE

Step 2) Perform Cartesian product with DBProject

πStudent(Completed) × DBProject


Completed


Student TaskFred Database1Fred Database2Eugene Database1Eugene Database2Sara Database1Sara Database2

Every student is combined with every task.

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DIVISION EXAMPLE

Step 3) Subtract Completed

πStudent(Completed) × DBProject-Completed


Completed


We have the possible combinations that "could have" been, but weren't.

Student TaskFred Database1Fred Database2Eugene Database1Eugene Database2Sara Database1Sara Database2


Student TaskEugene Database2

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DIVISION EXAMPLE

Step 4) Project onto the unique attributes of Completed

πStudent( πStudent(Completed) × DBProject-Completed)


Completed


All students who have not completed all assignments.



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DIVISION EXAMPLE

Step 5) Subtract from Completed

πStudent(Completed) - πStudent( πStudent(Completed)

× DBProject-Completed)


Completed



AStudentFredEugeneSara


Completed

StudentFredSara

Completed/DBProject=

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

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):

))Re)''

((( servesBoatsredcolorsid

Tempred

sname Tempred Tempgreen Sailors(( ) )

))Re)''

((( servesBoatsgreencolorsid

Tempgreen

EXAMPLE QUERIES

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Find the names of sailors who’ve reserved all boats.

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

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

Book has lots of examples.

))(/)Re,

(( Boatsbid

servesbidsid

Tempsids

sname Tempsids Sailors( )

)''

(/ BoatsInterlakebnamebid

EXAMPLE QUERIES

DATABASE SYSTEMS I

QUERY OPTIMIZATION

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PRINCIPLES OF QUERY OPTIMIZATION

1. Display the minimum number of fields in a query.

2. Use primary key or indexes wherever possible.

3. Use numeric rather than text primary keys.

4. Use non blank unique fields.

5. Avoid domain aggregate functions such as Dlookup().

6. Use between and equal to , rather than > or <. It will speed up the queries.

7. Use count(*) rather than count(column).

8. Short table and field names run faster than long name.

9. Normalize the tables.

10. Avoid the use of distinct row queries.

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QUERY OPTIMIZATION A user of a commercial DBMS formulates SQL

queries The query optimizer translates this query into

an equivalent RA query, i.e. an RA query with the same result

In order to optimize the efficiency of query processing, the query optimizer can re-order the individual operations within the RA query

Re-ordering has to preserve the query semantics and is based on RA equivalences

Just like Math operations can be reordered, so can Relational Algebra operations

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QUERY OPTIMIZATION Why can re-ordering improve the efficiency? Different orders can imply different sizes of

the intermediate results The smaller the intermediate results, the

more efficient Example:

much (!) more efficient than

Why?

))Re)''

((( SailorsservesBoatsredcolor

))((Re''

BoatsSailorsservesredcolor

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RELATIONAL ALGEBRA EQUIVALENCES

The most important RA equivalences are commutative and associative laws.

A commutative law about some operation states that the order of (two) arguments does not matter.

An associative law about some (binary) operation states that (more than two) arguments can be grouped either from the left or from the right.

If an operation is both commutative and associative, then any number of arguments can be (re-)ordered in an arbitrary manner.

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RELATIONAL ALGEBRA EQUIVALENCES The following (binary) RA operations are

commutative and associative: For example, we have:

Proof method: show that each tuple produced by the expression on the left is also produced by the expression on the right and vice versa.

(R S) (S R) (Commutative)

R (S T) (R S) T (Associative)

>< Ç

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Selections are crucial from the point of view of query optimization, because they typically reduce the size of intermediate results by a significant factor.

Laws for selections only:

RR cnccnANDANDc ...1...1

c c c cR R1 2 2 1

(Splitting)

(Commutative)

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Laws for the combination of selections and other operations:

if R has all attributes mentioned in c

if S has all attributes mentioned in c

The above laws can be applied to “push selections down” as much as possible in an expression, i.e. performing selections as early as possible.

SRSR cc )()(

)()( SRSR cc

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A projection commutes with a selection that only uses attributes retained by the projection.

Selection between attributes of the two arguments of a Cartesian product converts Cartesian product to a join.

Similarly, if a projection follows a join R S, we can ‘push’ it by retaining only attributes of R (and S) that are needed for the join or are kept by the projection.

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SUMMARY Several ways of expressing a given query; a

query optimizer chooses the most efficient version.

Query optimization exploits RA equivalencies to re-order the operations within an RA expression.

Optimization criterion is to minimize the size of intermediate relations.

DATABASE SYSTEMS I

RELATIONAL CALCULUS

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RELATIONAL CALCULUS Nonprocedural

Describes the set of answers without saying how they should be computed

Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC).

Calculus has variables, constants, comparison ops, logical connectives and quantifiers.

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TUPLE RELATIONAL CALCULUS Query has the form: {T | p(T)}

p(T) denotes a formula in which tuple variable T appears.

Answer is the set of all tuples T for which the formula p(T) evaluates to true.

Formula is recursively defined: start with simple atomic formulas (get tuples

from relations or make comparisons of values) build bigger and better formulas using the logical

connectives.

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DOMAIN RELATIONAL CALCULUS Query has the form:

Answer includes all tuples that

make the formula be true. Formula is recursively defined, starting with

simple atomic formulas (getting tuples from relations or making comparisons of values), and building bigger and better formulas using the logical connectives.

x x xn p x x xn1 2 1 2, ,..., | , ,...,

x x xn1 2, ,...,

p x x xn1 2, ,...,

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TRC FORMULAS

An Atomic formula is one of the following: R Rel (R is a tuple in relation Rel) R.a op S.b (comparing two fields) R.a op constant (comparing field to constant)

op is one of

A formula can be: an atomic formula where p and q are formulas where variable R is a tuple variable where variable R is a tuple variable

, , , , ,

p p q p q, ,))(( RpR))(( RpR

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SELECTION AND PROJECTION

Find all sailors with rating above 7{S |S Sailors S.rating > 7}

I N T A I N T A Sailors T, , , | , , ,

7

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JOINS

Find sailors rated > 7 who’ve reserved boat #103

{S | SSailors S.rating > 7 R(RReserves R.sid = S.sid

R.bid = 103)}

Note the use of to find a tuple in Reserves that ‘joins with’ the Sailors tuple under consideration.

I N T A I N T A Sailors T, , , | , , ,

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Ir Br D Ir Br D serves Ir I Br, , , , Re 103

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UNSAFE QUERIES, EXPRESSIVE POWER

It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe. e.g., S S Sailors|

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SUMMARY

Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)

Algebra and safe calculus have same expressive power, leading to the notion of relational completeness.

R ELATIONAL A LGEBRA M ORE POINTERS FROM T UESDAY.

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