Post on 14-Dec-2015
Tallahassee, Florida, 2014
COP4710
Database Systems
Relational Algebra
Fall 2014
Why Do We Learn This?
• Querying the database: specify what we want from our database– Find all the people who earn more than
$1,000,000 and pay taxes in Tallahassee• Could write in C++/Java, but a bad idea
• Instead use high-level query languages:
– Theoretical: Relational Algebra, Datalog
– Practical: SQL
– Relational algebra: a basic set of operations on relations that provide the basic principles
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What is an “Algebra”?
• Mathematical system consisting of:– Operands --- variables or values from which new
values can be constructed
– Operators --- symbols denoting procedures that construct new values from given values
• Examples– Arithmetic(Elementary) algebra, linear algebra,
Boolean algebra ……• What are operands?
• What are operators?
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What is Relational Algebra?
• An algebra – Whose operands are relations or variables that
represent relations
– Whose operators are designed to do common things that we need to do with relations in a database
• relations as input, new relation as output
– Can be used as a query language for relations
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Relational Operators at a Glance
• Five basic RA operations:– Basic Set Operations
• union, difference (no intersection, no complement)
– Selection: s– Projection: p – Cartesian Product: X
• When our relations have attribute names:– Renaming: r
• Derived operations:– Intersection, complement– Joins (natural join, equi-join, theta join, semi-join,
……)
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Set Operations
• Union: all tuples in R1 or R2, denoted as R1 U R2 – R1, R2 must have the same schema– R1 U R2 has the same schema as R1, R2– Example:
• Active-Employees U Retired-Employees
– If any, is duplicate elimination required?
• Difference: all tuples in R1 and not in R2, denoted as R1 – R2– R1, R2 must have the same schema– R1 - R2 has the same schema as R1, R2 – Example
• All-Employees - Retired-Employees
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Selection
• Returns all tuples which satisfy a condition, denoted as sc(R)– c is a condition: =, <, >, AND, OR, NOT– Output schema: same as input schema– Find all employees with salary more than
$40,000:• sSalary > 40000 (Employee)
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SSN Name Dept-ID Salary
111060000 Alex 1 30K
754320032 Bob 1 32K
983210129 Chris 2 45K
SSN Name Dept-ID Salary
983210129 Chris 2 45K
Projection
• Unary operation: returns certain columns, denoted as PA1,…,An (R)– Eliminates duplicate tuples !– Input schema R(B1, …, Bm)– Condition: {A1, …, An} {B1, …, Bm}– Output schema S(A1, …, An)
• Example: project social-security number and names:– P SSN, Name (Employee)
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SSN Name Dept-ID Salary
111060000 Alex 1 30K
754320032 Bob 1 32K
983210129 Chris 2 45K
SSN Name
111060000 Alex
754320032 Bob
983210129 Chris
Selection vs. Projection
• Think of relation as a table
– How are they similar?
– How are they different?
• Horizontal vs. vertical?
• Duplicate elimination for both? What about in real systems?
– Why do you need both?
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Cartesian Product
• Each tuple in R1 with each tuple in R2, denoted as R1 x R2– Input schemas R1(A1,…,An), R2(B1,…,Bm)
– Output schema is S(A1, …, An, B1, …, Bm)• Two relations are combined!
– Very rare in practice; but joins are very common
– Example: Employee x Dependent
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Example
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SSN Name
111060000 Alex
754320032 Brandy
Employee-SSN Dependent-Name
111060000 Chris
754320032 David
Employee Dependent
SSN Name Employee-SSN Dependent-Name
111060000 Alex 111060000 Chris
111060000 Alex 754320032 David
754320032 Brandy 111060000 Chris
754320032 Brandy 754320032 David
Employee x Dependent
Renaming
• Does not change the relational instance, denoted as Notation: r S(B1,…,Bn) (R)
• Changes the relational schema only– Input schema: R(A1, …, An)– Output schema: S(B1, …, Bn)
• Example:r
Soc-sec-num, firstname(Employee)
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SSN Name
111060000 Alex
754320032 Bob
983210129 Chris
Soc-sec-num firstname
111060000 Alex
754320032 Bob
983210129 Chris
Set Operations: Intersection
• Intersection: all tuples both in R1 and in R2, denoted as R1 R2– R1, R2 must have the same schema
– R1 R2 has the same schema as R1, R2
– Example• UnionizedEmployees RetiredEmployees
• Intersection is derived:– R1 R2 = R1 – (R1 – R2) why ?
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Theta Join
• A join that involves a predicate , q denoted as R1 q
R2 – Input schemas: R1(A1,…,An), R2(B1,…,Bm)
– Output schema: S(A1,…,An,B1,…,Bm)
– Derived operator:
R1 q R2 = s q (R1 x R2)
1. Take the product R1 x R2
2. Then apply SELECTC to the result
– As for SELECT, C can be any Boolean-valued condition
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Theta Join: Example
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Name AddressAJ's 1800 Tennessee
Michael's Pub 513 Gaines
Bar Beer Price
AJ’s Bud 2.5
AJ’s Miller 2.75
Michael’s Pub Bud 2.5
Michael’s Pub Corona 3.0
Bar Sells
BarInfo := Sells Sells.Bar=Bar.Name Bar
Bar Beer Price Name Address
AJ’s Bud 2.5 AJ's 1800 Tennessee
AJ’s Miller 2.75 AJ's 1800 Tennessee
Michael’s Pub Bud 2.5 Michael's Pub 513 Gaines
Michael’s Pub Corona 3.0 Michael's Pub 513 Gaines
Natural Join
• Notation: R1 R2• Input Schema: R1(A1, …, An), R2(B1, …, Bm)• Output Schema: S(C1,…,Cp)
– Where {C1, …, Cp} = {A1, …, An} U{B1, …, Bm}
• Meaning: combine all pairs of tuples in R1 and R2 that agree on the attributes:– {A1,…,An} {B1,…, Bm} (called the join
attributes)
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Natural Join: Examples
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SSN Name
111060000 Alex
754320032 Brandy
SSN Dependent-Name
111060000 Chris
754320032 David
Employee Dependent
SSN Name Dependent-Name
111060000 Alex Chris
754320032 Brandy David
Employee Dependent =P SSN, Name, Dependent-Name(sEmployee.SSN=Dependent.SSN(Employee x Dependent)
Natural Join: Examples
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A BX YX ZY ZZ V
B C
Z U
V W
Z V
A B C
X Z U
X Z V
Y Z U
Y Z V
Z V W
R S
R S
Equi-join
• Special case of theta join: condition c contains only conjunction of equalities– Result schema is the same as that of Cartesian
product
– May have fewer tuples than Cartesian product
– Most frequently used in practice:
R1 =A B R2
– Natural join is a particular case of equi-join
– A lot of research on how to do it efficiently
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A Joke About Join
A join query walks up to two tables
in a restaurant and asks :“Mind if I join you?”
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Division
• A/B for A(x,y) and B(y)– Contains all tuples (x) such that for every y tuple
in B, there is an xy tuple in A
– Useful for expressing “for all” queries– 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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),( AyxByxBA
1. Disqualified x values: )))((( ABAxx
2. A/B: )(Ax Disqualified x values
Division: Example
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sno pno s1 p1 s1 p2 s1 p3 s1 p4 s2 p1 s2 p2 s3 p2 s4 p2 s4 p4
pnop2
pnop2p4
pnop1p2p4
snos1s2s3s4
snos1s4
snos1
A
B1B2
B3
A/B1 A/B2 A/B3
Building Complex Expressions
• Algebras allow us to express sequences of operations in a natural way– Example
• In arithmetic algebra: (x + 4)*(y - 3)
– Relational algebra allows the same
• Three notations, just as in arithmetic:1. Sequences of assignment statements2. Expressions with several operators3. Expression trees
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Sequences of Assignments
• Create temporary relation names• Renaming can be implied by giving relations a list of
attributes• Example: R3 := R1 JOINC R2 can be written:
R4 := R1 x R2R3 := SELECTC (R4)
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Expressions with Several Operators
• Example: the theta-join R3 := R1 JOINC R2 can be
written: R3 := SELECTC (R1 x R2)
• Precedence of relational operators:1. Unary operators --- select, project, rename --- have highest
precedence, bind first
2. Then come products and joins
3. Then intersection
4. Finally, union and set difference bind last
• But you can always insert parentheses to force the order you desire
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Expression Trees
• Leaves are operands – either variables standing for relations or
particular constant relations
• Interior nodes are operators, applied to their child or children
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Expression Tree: Examples
Given Bars(name, addr), Sells(bar, beer, price), find the names of all the bars that are either on Tennessee St. or sell Bud for less than $3
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Bars Sells
SELECTaddr = “Tennessee St.” SELECT price<3 AND beer=“Bud”
PROJECTname
RENAMER(name)
PROJECTbar
UNION
Summary of Relational Algebra
• Why bother ? Can write any RA expression directly in C++/Java, seems easy– Two reasons:
• Each operator admits sophisticated implementations (think of and s C)
• Expressions in relational algebra can be rewritten: optimized
s(age >= 30 AND age <= 35)(Employees)– Method 1: scan the file, test each employee– Method 2: use an index on age
Employees Relatives– Iterate over Employees, then over Relatives? Or iterate over
Relatives, then over Employees?– Sort Employees, Relatives, do “merge-join”– “hash-join”– etc.
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