R ELATIONAL A LGEBRA Joe Meehean 1. T ABLES FOR T ODAY ’ S L ECTURE 2 StdNoLast Name First Name...

1

RELATIONAL ALGEBRA

Joe Meehean

2

TABLES FOR TODAY’S LECTURE

StdNo Last Name

First Name

Major GPA

732 Arnett Will Theatre 3.74967 Crumb Robert Art 2.3764 Vogelzan

gAnna Music 3.5

568 Park Phil CS 3.6587 Park Anna CS 3.9

OfferingID StdNo CreditsAA7 732 4BB8 732 3

OfferingID CourseName

AA7 Calculus IBB8 Art History

Students

Enrolled Offering

3

RELATIONAL ALGEBRA Algebra for tables Transform 1 or more tables into another table Based on a small set of operators

4

ALTERNATIVE TERMINOLOGYTable-Oriented(Database)

Set-Oriented (Mathematics)

Table RelationRow TupleColumn AttributeAND ΛOR V

5

SELECTION OPERATOR Represented by σ symbol AKA restrict operator Produces a subset of the input relation

(table) subset of the tuples (rows)

Uses condition to indicate which rows to retain condition can include: =, ≠, <, >, ≥, ≤, V, Λ

e.g., for relation Students(stdno, last_name, first_name, major, gpa) σgpa > 3.5(Students) gives the students that have gpa greater than

3.5

6

SELECTION OPERATOR Conceptual Picture

StdNo Last Name

First Name

Major GPA


gAnna Music 3.5


7

SELECTION OPERATOR Conceptual Picture

StdNo Last Name

First Name

Major GPA


gAnna Music 3.5


8

PROJECTION OPERATOR Represented by π Produces a subset of the input table

subset of the columns e.g., πlast_name,major Student

gives the Student relation without the first_name and stdnr attributes

Also removes duplicates projection removes attributes, so duplicate rows

can occur these duplicates are removed real database systems often do not remove

duplicates

9

PROJECTION OPERATOR Conceptual Picture

StdNo Last Name

First Name

Major GPA


gAnna Music 3.5


10


StdNo Last Name

First Name

Major GPA


gAnna Music 3.5


11


StdNo Last Name

First Name

Major GPA


gAnna Music 3.5


12

PRODUCT OPERATOR Represented by × AKA extended cross product operator Produces all possible combination of tuples

(rows) between 2 relations (tables) # tuples (rows) in output

# tuples in input1 * # tuples in input2 # attributes (columns) in output

# attributes in input1 + #attributes in input2

13

PRODUCT EXAMPLE Student × EnrolledStdNo Last

NameFirst Name

Major GPA

732 Arnett Will Theatre 3.74764 Vogelzan

gAnna Music 3.5

OfferingID StdNo CreditsAA7 732 4BB8 732 3StdNo Last

NameFirst Name

Major GPA OfferingID

StdNo Credits

732 Arnett Will Theatre 3.74 AA7 732 4732 Arnett Will Theatre 3.74 BB8 732 3764 Vogelzan

gAnna Music 3.5 AA7 732 4

764 Vogelzang

Anna Music 3.5 BB8 732 3

14

JOIN OPERATOR Represented by Also combines 2 tables Requires a condition between tuples

all conditions from selection are OK Join is derived from product and selection

R c S = σc(R × S)

15

JOIN OPERATOR Equi-join

matching condition is that 2 rows have same value for some attribute(s)

c = d

Natural join two relations have same attribute values of attributes must match for tuple to

be included duplicate attribute is not included in

resulting relation no need to include condition for natural join

16

JOIN OPERATOR EXAMPLE Student Enrolled StdNo Last

NameFirst Name

Major GPA

732 Arnett Will Theatre 3.74764 Vogelzan

gAnna Music 3.5

OfferingID StdNo Credits

AA7 732 4BB8 732 3StdNo Last

NameFirst Name

Major GPA OfferingID

Credits

732 Arnett Will Theatre 3.74 AA7 4732 Arnett Will Theatre 3.74 BB8 3

QUESTIONS?

17

18

QUICK REVIEW Select operator (σ)

removes tuples (rows) from relation (table) Project operator (π)

removes attributes (columns) from relation (table)

Cross product operator (×) combines attributes from 2 relations matches every tuple from both input relations

Join operator ( ) combines attributes from 2 relations matches tuples based on condition often same value for an attribute

19

RENAME OPERATOR Represented by ρ Renames relations or attributes (or both) Rename relation

ρ(Z, A) relation A is now called Z

Rename attributes ρ((old1 → new1, old2 → new2, old3 → new3), A) A’s attributes old1, old2, and old3 are renamed

20

RENAME OPERATOR Rename both attributes and relation

ρ(Z, (old1 → new1, old2 → new2, old3 → new3), A)

Simplify expressions ρ( music_majors, σmajor = music student) πlast_name, first_name(σgpa > 3.5 music_majors)

21

UNION OPERATOR Represented by U Combines two relations Result contains all tuples in both relations e.g., orange and green parts

σmajor = music students U σ major = theatre students

22

INTERSECTION OPERATOR Represented by ∩ Combines two relations Results contains only tuples that are in both

input relations e.g., music majors with good gpa

σmajor = music students ∩ σ gpa > 3.5 students

23

DIFFERENCE OPERATOR Represented by − Combines two relations Tuples in the first table, but not in second e.g. music majors that don’t have a good gpa

σmajor = music students − σ gpa > 3.5 students

24

UNION COMPATIBLE Relations are union compatible if

they have the same number of attributes AND corresponding attributes have the same

data type (ints, chars, etc…) usually means relations have same attributes

Union, intersect, and difference require union compatible relations

e.g. red parts supplied by XYZ ρ(XYZparts, parts (catalog

σsname=XYZsupplier)) σcolor = red parts ∩ πpname, pid, color XYZparts

25

DIVISION OPERATOR Represented by / Combines two relations, A & B

A has two attributes (x,y) B has one attribute (y)

A/B is all x’s such that (x,y) exists in A for all y’s in B

Result has only a singe atribute (x)

26

DIVISION OPERATOR For each x value in A

consider the set of y values x maps to in A if this set contains all y values in B then x is in A/B

e.g. student enrolled in all offerings ρ(super_student_ids,

πstd_no,offering_id enrolled / πoffering_idofferings) πlastname, firstname(students super_student_ids)

27

DIVISION OPERATOR Derive A/B

think about what doesn’t belong in result get set of all x,y mappings (πx A) × B remove those mappings that appear in A ((πx A) × B) − A what’s left must be x’s that don’t map to all y’s πx ( ((πx A) × B) − A ) remove them from the set of x’s πx A − πx ( ((πx A) × B) − A )

QUESTIONS?

28

R ELATIONAL A LGEBRA Joe Meehean 1. T ABLES FOR T ODAY ’ S L ECTURE 2 StdNoLast Name First Name...

Documents

Transcript of R ELATIONAL A LGEBRA Joe Meehean 1. T ABLES FOR T ODAY ’ S L ECTURE 2 StdNoLast Name First Name...