1. Relational Algebra

2. Relational Query Languages
Recall: Query = “Retrieval Program”
Language Examples:
Theoretical:
Relational Algebra
Relational Calculus
Tuple Relational Calculus (TRC)
Domain Relational Calculus (DRC)
Practical:
SQL (originally: SEQUEL from System R)
Quel (used in Ingres)
Datalog (Prolog-like – used in research lab systems)
Theoretical QLs give semantics to Practical QLs
CSCI1270, Lecture 2

3. Relational Algebra Basic Operators select ( σ ) project ( p )
union (  )
set difference ( – )
cartesian product (  )
rename ( ρ )
Closure Property
Relation
Relational
Operator
Relation
Relational
Operator
Relation
CSCI1270, Lecture 2

4. Select ( σ ) Notation: σpredicate (Relation)
Relation: Can be name of table or result of another query
Predicate:
Simple
attribute1 = attribute2
attribute = constant value (also: ≠, <, >, ≤, ≥)
2. Complex
predicate AND predicate
predicate OR predicate
NOT predicate
Idea:
Select rows from a relation based on a predicate
CSCI1270, Lecture 2

5. Bank Database Account bname acct_no balance Branch bname bcity assets
Downtown
Mianus
Perry
R.H.
Brighton
Redwood
A-101
A-215
A-102
A-305
A-201
A-222
A-217
500
700
400
350
900
750
Branch
bname
bcity
assets
Downtown
Redwood
Perry
Mianus
R.H.
Pownel
N. Town
Brighton
Brooklyn
Palo Alto
Horseneck
Bennington
Rye 9M
2.1M
1.7M
0.4M 8M
0.3M
3.7M
7.1M
Depositor
cname
acct_no
Johnson
Smith
Hayes
Turner
Jones
Lindsay
A-101
A-215
A-102
A-305
A-201
A-217
A-222
Borrower
cname
lno
Jones
Smith
Hayes
Jackson
Curry
Williams
Adams
L-17
L-23
L-15
L-14
L-93
L-11
L-16
Customer
cname
cstreet
ccity
Jones
Smith
Hayes
Curry
Lindsay
Turner
Williams
Adams
Johnson
Glenn
Brooks
Green
Main
North
Park
Putnam
Nassau
Spring
Alma
Sand Hill
Senator
Walnut
Harrison
Rye
Pittsfield
Stanford
Princeton
Palo Alto
Woodside
Brooklyn
Loan
bname
lno
amt
Downtown
Redwood
Perry
Mianus
R.H.
L-17
L-23
L-15
L-14
L-93
L-11
L-16
1000
2000
1500
500
900
1300
CSCI1270, Lecture 2

6. Select ( σ ) Notation: σpredicate (Relation)
bname
bcity
assets
Downtown
Brighton
Brooklyn 9M
7.1M
σ bcity = “Brooklyn” (branch) =
σ assets > $8M (σ bcity = “Brooklyn” (branch)) =
bname
bcity
assets
Downtown
Brooklyn 9M
CSCI1270, Lecture 2

7. Project ( p ) Notation: pA1, …, An (Relation)
Relation: name of a table or result of another query
Each Ai is an attribute
Idea: p selects columns (vs. σ which selects rows)
p cstreet, ccity (customer) =
cstreet
ccity
Main
North
Park
Putnam
Nassau
Spring
Alma
Sand Hill
Senator
Walnut
Harrison
Rye
Pittsfield
Stanford
Princeton
Palo Alto
Woodside
Brooklyn
CSCI1270, Lecture 2

8. Project ( p ) p bcity (σassets > 5M (branch)) = bcity Brooklyn
Horseneck
Question: Does the result of Project always have the same number
of tuples as its input?
CSCI1270, Lecture 2

9. Union (  ) Notation: Relation1  Relation2 Example:
R  S valid only if:
R, S have same number of columns (arity)
R, S corresponding columns have same name and domain (compatibility)
Example:
cname
Johnson
Smith
Hayes
Turner
Jones
Lindsay
Jackson
Curry
Williams
Adams
(p cname (depositor))  (p cname (borrower)) =
Schema:
Depositor
cname
acct_no
Borrower
cname
lno
CSCI1270, Lecture 2

10. Set Difference ( – ) Notation: Relation1 - Relation2 Example:
R - S valid only if:
R, S have same number of columns (arity)
R, S corresponding columns have same domain (compatibility)
Example:
(p bname (σamount ≥ 1000 (loan))) –
(p bname (σ balance < 800 (account))) =
bname
lno
amount
Downtown
Redwood
Perry
L-17
L-23
L-15
L-14
L-16
1000
2000
1500
500
300
bname
acct_no
balance
Mianus
Brighton
Redwood
A-215
A-201
A-222
A-217
700
900
850
bname
Downtown
Perry
CSCI1270, Lecture 2

11. What About Intersection?
Remember: R ⋂ S = R – (R – S)
R – S R S

12. Cartesian Product (  ) Notation: Relation1  Relation2 Example:
R  S like cross product for mathematical relations:
every tuple of R appended to every tuple of S
flattened!!!
Example:
depositor  borrower =
depositor.
cname
acct_no
borrower.
lno
Johnson
Smith …
A-101
A-215
Jones
Hayes
Jackson
Curry
Williams
Adams
L-17
L-23
L-15
L-14
L-93
L-11
L-16
How many tuples in the result?
A: 56
CSCI1270, Lecture 2

13. Rename ( ρ ) Notation: r identifier (Relation)
renames a relation, or
Notation: r identifier0 (identifier1, …, identifiern) (Relation)
renames relation and columns of n-column relation
Use:
massage relations to make , – valid, or  more readable
CSCI1270, Lecture 2

14. Rename ( ρ )
Notation: r identifier0 (identifier1, …, identifiern) (Relation)
Example:
r result (dcname, acctno, bcname, lno) (depositor  borrower) =
result
dccname
acctno
bcname
lno
Johnson
Smith …
A-101
A-215
Jones
Hayes
Jackson
Curry
Williams
Adams
L-17
L-23
L-15
L-14
L-93
L-11
L-16
CSCI1270, Lecture 2

15. Example Query in RA
Determine lno for loans that are for an amount that is larger than the amount of some other loan. (i.e. lno for all non-minimal loans)
Can do in steps:
Temp1  …
Temp2  … Temp1 … …
CSCI1270, Fall 2008, Lecture 2

16. Example Query in RA 1. Find the base data we need
lno
amt
L-17
L-23
L-15
L-14
L-93
L-11
L-16
1000
2000
1500
500
900
1300
Temp1  p lno,amt (loan)
2. Make a copy of (1)
Temp2  ρ Temp2 (lno2,amt2) (Temp1)
lno2
amt2
L-17
L-23
L-15
L-14
L-93
L-11
L-16
1000
2000
1500
500
900
1300
CSCI1270, Lecture 2

17. Example Query in RA 3. Take the cartesian product of 1 and 2
Temp3  Temp1  Temp2
lno
amt
lno2
amt2
L-17 …
L-23
1000
2000
L-16
1300
CSCI1270, Lecture 2

18. Example Query in RA p lno (σamt > amt2 (p lno,amt (loan)  (
4. Select non-minimal loans
Temp4  σamt > amt2 (Temp3)
5. Project on lno
Result  p lno (Temp4)
… or, if you prefer…
p lno (σamt > amt2 (p lno,amt (loan)  (
ρTemp2 (lno2,amt2) (p lno,amt (loan)))))
CSCI1270, Fall 2008, Lecture 2

19. Review Theoretical Query Languages
Relational Algebra
SELECT ( σ )
PROJECT ( π )
UNION (  )
SET DIFFERENCE ( – )
CARTESIAN PRODUCT (  )
RENAME ( ρ )
Relational algebra gives semantics to practical query languages
Above set: minimal relational algebra
 will now look at some redundant (but useful!) operators
CSCI1270, Fall 2008, Lecture 2

20. Review Express the following query in the RA:
Find the names of customers who have both accounts and loans
T1  ρT1 (cname2, lno) (borrower)
T2  depositor  T1
T3  σcname = cname2 (T2)
Result  π cname (T3)
Above sequence of operators (ρ, , σ) very common.
Motivates additional (redundant) RA operators.
CSCI1270, Lecture 2

21. Relational Algebra Additional Operators
Natural Join ( )
2. Division ( )
Generalized Projection (π)
Aggregation
5. Outer Joins ( )
6. Update (  ) (we’ve already been using this)
1&2 Redundant: Can be expressed in terms of minimal RA
e.g. depositor borrower =
π …(σ…(depositor  ρ…(borrower)))
3 – 6 Added for extra power
CSCI1270, Lecture 2

22. Natural Join Relation1 Notation: Relation2 Idea: combines ρ, , σ r s3 α Β β + - 10 20
‘a’
‘b’
‘c’ A B C D 1 2 3 α β + - 10 20 E B D
‘a’
‘b’
‘c’ α β 10 20 = r s
πcname,acct_no,lno (σcname=cname2 (depositor  ρt(cname2,lno) (borrower))) ≡
depositor
borrower
CSCI1270, Lecture 2

23. Division Notation: Relation1 Relation2
Idea: expresses “for all” queries r A B α β γ δ 1 23 3 4 6 2 B 1 2 s A α δ =
Query: Find values for A in r which have corresponding B values for all B values in s
CSCI1270, Lecture 2

24. Division ¸ ´ Another way to look at it: and Relational Division
17 ¸ 3 = 5
The largest value of i such that: i ´ 3 ≤ 17
Relational Division r A B α β γ δ 1 23 3 4 6 2 s B 1 2 t A α δ =
The largest value of t such that:
( t  s Í r )
CSCI1270, Lecture 2

25. Division A More Complex Example r t s ? A B C D E α β γ a b 1 3 A B C11 = ?
CSCI1270, Lecture 2

26. Division Adds No Power
Definition in terms of the basic algebra operation Let r(R) and s(S) be relations, and let S  R
r  s = R-S (r) – R-S ( (R-S (r) x s) – R-S,S(r))
To see why
R-S,S(r) simply reorders attributes of r
R-S(R-S (r) x s) – R-S,S(r)) gives those tuples t in R-S (r) such that for some tuple u  s, tu  r.

27. Generalized Projection
Notation:
p e1,…,en (Relation)
e1,…,en can include arithmetic expressions – not just attributes
Example
cname
limit
balance
Jones
Turner
5000
3000
2000
2500
credit =
Then…
cname
limit-balance
Jones
Turner
3000
500
π cname, limit - balance (credit) =
CSCI1270, Lecture 2

28. Aggregate Functions and Operations
An aggregate function takes a collection of values and returns a single value as a result.
avg: average value min: minimum value max: maximum value sum: sum of values count: number of values
Aggregate operation in relational algebra
G1, G2, …, Gn g F1( A1), F2( A2),…, Fn( An) (E )
E is any relational-algebra expression
G1, G2 …, Gn is a list of attributes on which to group
(can be empty)
Each Fi is an aggregate function
Each Ai is an attribute name
CSCI1270, Lecture 2

29. Aggregate Operation – ExampleB C
Relation r:     7 3 10
sum-C
g sum(c) (r)
No grouping 27
CSCI1270, Lecture 2

30. Aggregate Operation – Example
Relation account grouped by branch-name:
branch-name
account-number
balance
Perryridge
Brighton
Redwood
A-102
A-201
A-217
A-215
A-222
400
900
750
700
branch-name g sum(balance) (account)
branch-name
sum(balance)
Perryridge
Brighton
Redwood
1300
1500
700
CSCI1270, Lecture 2

31. Aggregate Functions (Cont.)
Result of aggregation does not have a name
Can use rename operation to give it a name
For convenience, we permit renaming as part of aggregate operation
branch-name g sum(balance) as sum-balance (account)
CSCI1270, Lecture 2