C20.0046: Database Management Systems Lecture #8

M.P. Johnson, DBMS, Stern/NYU, Sp2004

1

C20.0046: Database Management SystemsLecture #8

Matthew P. Johnson

Stern School of Business, NYU

Spring, 2004


2

Agenda Last time: Normalization This time:

1. 4NF

2. Relational Algebra Pep talk

OHs today, drop-ins (80809)


3

Normalization Review Q: What’s required for BCNF?

Q: What are the two types of violations?

Q: What’s the loophole for 3NF?

Q: How do we fix a non-BCNF relation?


4

Normalization Review Q: If AsBs violates BCNF, what do we do?

Q: In this case, could the decomposition be lossy?

Q: How do we combine two relations?

Q: Can BCNF decomp. lose FDs?

Q: Can 3NF decomp. lose FDs?


5

New topic: MVDs (3.7) Consider this relation

People ~ their jobs ~ their residences Person-address/city: many-many Person-job: many-many Address/city-job: independent

Chappaqua333 Some StreetFirst Lady456Hilary

Washington444 Embassy RowFirst Lady456Hilary

New York111 East 60th StreetCEO123Michael

London222 Brompton RoadCEO123Michael

444 Embassy Row

333 Some Street

444 Embassy Row

333 Some Street

222 Brompton Road

111 East 60th Street

Streets

Lawyer

Lawyer

Senator

Senator

Mayor

Mayor

Jobs

Washington456Hilary

Chappaqua789Hilary

Washington789Hilary

Chappaqua456Hilary

London123Michael

New York123Michael

CitysSSNName


6

Redundancy in BCNF

Lots of redundancy! Key? All fields

None determined by others! Non-trivial FDs? None! In BCNF? Yes!

Name Streets Citys Jobs

Michael 111 East 60th Street New York Mayor

Michael 222 Brompton Road London Mayor

Michael 111 East 60th Street New York CEO

Michael 222 Brompton Road London CEO

Hilary 333 Some Street Chappaqua Senator

Hilary 444 Embassy Row Washington Senator

Hilary 333 Some Street Chappaqua First Lady

Hilary 444 Embassy Row Washington First Lady

Hilary 333 Some Street Chappaqua Lawyer

Hilary 444 Embassy Row Washington Lawyer

Now what? New concept, leading

to another normal form: Multivalued

dependencies


7

As Bs if, when As are held fixedvalues in Bs are independent of

values in rest

More precisely: if t1 and t3 agree on As, we then can find t2 such that

t2, t2, t3 agree on As

t2, t1 agree of Bs

t2, t3 agree on Cs

MVD definition

As Bs Cst1

As Bs Cst2

As Bs Cst3

| |

| |

| |

| |


8

MVD example Claim: name streets,cities If true: can pick arbitrary t1, t3 and find a t2

We pick: first and last of Hilary’s tuples:

Now: if true, can find another Hilary row with street/address of t1 and job of t3

LawyerWashington444 Embassy RowHilary

JobsCitysStreetsName

SenatorChappaqua333 Some StreetHilaryt1

t3

LawyerChappaqua333 Some StreetHilaryt2


9

MVD example Now: if true, can find another Hilary row with

street/address of t1 and job of t3

Sure enough:

Hilary 333 Some Street Chappaqua Lawyert2












t2


10

MVD rules No splitting rule:

In the example, name streets,cities Do we have name streets?

No: 444 Embassy Row doesn’t go with Chappaqua

NB: City doesn’t determine street – could have >1 house But city, street aren’t independent




t1

t3


11

MVD rules Trivial dependencies:

As Bs iff As BsAi

Transitive rule: As Bs, Bs Cs As Cs

Complementation rule: As Bs As rest Intuition: if each value in Bs is assoc’ed w/each value in

rest, then each value of rest is assoc’ed w/each value in BsName Streets Citys Jobs






12

MVDs and FDs MVD is a generalization of FD Every FD is an MVD Pf: Suppose As Bs

Pick t1, t3 that agree on As.

Must find a t2. Let t2 be t3.

Then1) t2 agrees on As with both

2) t2 agrees on Bs with t1 (why?)

3) t2 agrees on rest with t3 (why?)

QED


13

Fourth Normal Form 4NF: like BCNF, but with MVDs not FDs An MVD As Bs is nontrivial if

No Bs are As Some attributes left over (why?)

4NF: for every nontrivial MVD

As Bs, As is a superkey In example name streets,cities, but

name isn’t a superkeyName Streets Citys Jobs




14

Decomposition to 4NF Again, analogous to BCNF If we can find As Bs for R where As isn’t

a superkey, replace R with R1(As,Bs) and R2(As,rest)

Running example: name streets,cities People(name,streets,cities,jobs) becomes

Residences(name,street,city) and Employment(name,job)


15

4NF: another construal In nontrivial As Bs, As must be superkey After df of 4NF, text says: “That is, … every

nontrivial MVD is really a FD with a superkey on the left” (p123).

We know: FDs are* MVDs but not vice versa So: Why does this follow? Is it true? Yes. As is a superkey As everything As Bs the MVD is an FD Two kinds of MVDs: FDs and “true” MVDs 4NF eliminates exactly the true ones

* The typo swapping these was fixed.


16

Summary of normal forms

Guaranteed to 3NF BCFN 4NF

Eliminate FD redundancy

Mostly Yes Yes

Eliminate MVD redundancy

No No Yes

Preserve FDs Yes No No

Preserve MVDs No No No


17

Combined isa/weak example Exercise 3.3.1

Convert from E/R to R, by E/R, OO and nulls

courses

Lab-courses

Depts

Computer-allocation

room

number

givenBy

name chair

isa


18

Next topic: relational algebra (5.1-2) Set operations: union, intersection, difference Projection, selection Cartesian Product Joins: natural joins, theta joins Combining operations to form queries Dependent and independent operations


19

What is relational algebra? An algebra for relations “High-school” algebra is an algebra for numbers Formalism for constructing expressions

Operations Operands: Variables, Constants, expressions

Expressions: Vars & constants Operators applied to expressions

Algebra Vars/consts Operators

High-school Numbers + * - / etc.

Relational Relations (=sets of tupes)

union, intersection, join, etc.


20

Why do we care about relational algebra? Why construct expressions on relations? The exprs are the form questions about the

take The relations these exprs cash out to are the

answers to our questions First proof of RDBMS/RA concept: System R

(1979) Modern implementation of RA: SQL


21

Relation operators Five basic operators:

Union: Intersection: Difference: - Selection: Projection: Cartesian Product:

Derived/auxiliary operators: Intersection, complement Joins (natural, equijoin, theta join, semijoin) Renaming:


22

Operators Relations are sets have set-theoretic ops

Venn diagrams

Union: R1 R2 Example:

ActiveEmployees RetiredEmployees

Difference: R1 – R2 Example:

AllEmployees – RetiredEmployees = ActiveEmployees


23

Set operations - exampleName Address Gender Birthdate

Fisher 123 Maple F 9/9/99

Hamill 456 Oak M 8/8/88

Name Address Gender Birthdate


Ford 345 Palm M 7/7/77

R:

S:





R S:


24







R:

S:

R - S: Name Address Gender Birthdate



25

Operators Intersection: R1 R2 Example:

UnionizedEmployees RetiredEmployees

Intersection can be derived from and – R1 R2 = R1 – (R1 – R2) R1 R2 = -(-R1 -R2) (allowed?)


26







R:

S:

R S: Name Address Gender Birthdate



27

Operators Selection Selects all tuples satisfying a condition Notation: c(R)

Examples salary > 100000(Employee) name = “Smith”(Employee)

The condition c can have comparison ops:=, <, , >, , <> boolean ops: and, or


28

Selection example

Select the movies at Angelica: Theater=“Angelica”(Showings)

City of GodVillageFilm Forum

Village

Village

N’hood

Fog of War

City of God

Title

Angelica

Angelica

Theater

Village

Village

N’hood

Fog of War

City of God

Title

Angelica

Angelica

Theater


29

Operators Projection: op we used for decomposition

Eliminates columns, then removes duplicates

Notation: A1,…,An(R)


30

Operators Cartesian Product

Cross product Each tuple in R1 combines w/each tuple in R2

Notation: R1 R2

If R1, R2 fields overlap, include both and disambiguate: R1.A, R2.A

Fairly rare in practice used to express joins

Q: Where does the name come from? Q: If R1 has n1 rows and R2 has n2, how

large is R1 x R2?


31

Cartesian product example

Street City

333 Some Street Chappaqua

444 Embassy Row Washington

333 Some Street Chappaqua

Hillary-addresses

Job

Senator

First Lady

Lawyer

Hillary-jobs

Street City Job

333 Some Street Chappaqua Senator

444 Embassy Row Washington Senator

333 Some Street Chappaqua First Lady

444 Embassy Row Washington First Lady

333 Some Street Chappaqua Lawyer

444 Embassy Row Washington Lawyer

Hillary-addresses x Hillary-jobs


32

Operators Natural join: our join up to now

But always merging shared attributes Notation: R1 ⋈ R2 Meaning:

R1 ⋈ R2 = every att once(shared atts =(R1 R2)) I.e., first compute the cross product R1 x R2

Next, select the rows in which shared fields agree

Finally, project onto the union of R1 and R2’s fields (remove duplicates)


33

Natural join example

Name Street City

Hilary 333 Some Street Chappaqua

Hilary 444 Embassy Row Washington

Hilary 333 Some Street Chappaqua

Addresses

Name Job

Hilary Senator

Hilary First Lady

Hilary Lawyer

Jobs

Addresses ⋈ JobsName Street City Job








34

Natural Join R S

R ⋈ S= ?

Unpaired tuples called dangling

A B

X Y

X Z

Y Z

Z V

B C

Z U

V W

Z V


35

Natural Join Given the schemas R(A, B, C, D), S(A, C, E),

what is the schema of R ⋈ S ?

Given R(A, B, C), S(D, E), what is R ⋈ S?

Given R(A, B), S(A, B), what is R ⋈ S?


36

Theta Join Like natural join, but

includes only rows that satisfy arbitrary condition Does not project away shared attributes

R1 ⋈ R2 = (R1 R2)

Here can be any condition If condition is always satisfies, then theta join

becomes natural join


37

Theta-join exampleA B C

1 2 3

6 7 8

9 7 8

B C D

2 3 4

2 3 5

7 8 10

A U.B U.C V.B V.C D

1 2 3 2 3 4

1 2 3 2 3 5

1 2 3 7 8 10

6 7 8 7 8 10

9 7 8 7 8 10

U V

U V

A<D


38

Equijoin A theta join where is an equality R1 ⋈A=B R2 = A=B(R1 R2) = lower-case sigma Example:

Employee ⋈SSN=SSN Dependents

Most useful join in practice


39

Semijoin R ⋉ S = {atts of R}(R ⋈ S) Q: What does this mean?

Natural join of R and S; Then project onto R’s atts

A: The rows of R for which >1 row in S agree on shared atts


40

Semijoin example

SSN Name

. . . . . .

DSSN Dname SSN

. . . . . .

EmployeeDependents

network

Employee ⋉ Dependents =

{employees who have dependents}

Employee ⋉ Dependents =

{employees who have dependents}


41

Renaming Changes the schema, not the instance Notation: B1,…,Bn(R) is spelled “rho”, pronounced “row” Example:

Employee(ssn,name) social, name)(Employee)

Or just: (Employee)


42

Complex RA Expressions Q: How long was Star Wars (1977)?

Strategy: find the row with Star Wars; then project the length field

Title Year Length inColor Studio Prdcr#

Star Wars 1977 124 True Fox 12345

M.Ducks 1991 104 True Disney 67890

W.World 1992 95 True Paramount 99999


43

Combining operations Schema: Movies (Title, year, length, filmType, studioName)

Query: select titles and years of movies by Fox that are at least 100 minutes long.

Title Year Length Filmtype StudioStar wars 1977 124 Color Fox

Mighty ducks 1991 104 Color Disney

Wayne’s world 1992 85 Color Paramount


44

Complex RA Expressions Reps(ssn, name, etc.) Clients(ssn, name, rssn) Q: Find George’s client names Clients.name(Reps.name=George(Reps.ssn=rssn(

Reps x Clients))) Or: Clients.name(Reps.name=George and Reps.ssn=rssn(Reps x

Clients)) Or: Clients.name(Reps.name=George(Reps x Clients)

Reps.ssn=rssn(Reps x Clients))


45

For next time Finish chapter 5 Come to office hours!

Documents

C20.0046: Database Management Systems Lecture #8