concepts of db(3) relational operation & sql

Spring, 2006 KUT

Concepts of DB(3)Relational Operation & SQL

한국기술대학교 인터넷미디어공학부

민준기

Spring, 2006 KUT

2

Relational Data Operation

• Procedural language– The user instructs the system to perform a sequence of

operations on the database to compute the desired result what & how

– Relational algebra• Nonprocedural language

– The user describes the information desired without a specific procedure for obtaining the information what

– Relational calculus• 1.Tuple Relational Calculus• 2.Domain Relational Calculus

• Relational Algebra and Relational Calculus have same expression/computing power

Spring, 2006 KUT

3

Relational Algebra• Relational Algebra consists of several groups of operations

– Relational Algebra Operations From Set Theory• UNION ( )• INTERSECTION ( )• DIFFERENCE (or MINUS, – )• CARTESIAN PRODUCT ( x )

– Relational Operations• Unary Relational Operations

– SELECT (symbol: (sigma))– PROJECT (symbol: (pi))– RENAME (symbol: (rho))

• Binary Relational Operations– JOIN (several variations of JOIN exist, )– DIVISION ( ÷ )

– Additional Relational Operations• OUTER JOINS, • OUTER UNION• AGGREGATE FUNCTIONS (These compute summary of information: for

example, SUM, COUNT, AVG, MIN, MAX)

• closure property– Operand and operation results are relation– Support nested expressions

Spring, 2006 KUT

UNION

DIFFERENCE

10

INTERSECT

a

b

c

x

y

a

a

b

b

c

c

x

y

x

y

x

y

PRODUCT RESTRICTION

a1

a2

a3

b1

b1

b2

b1

b2

b3

c1

c2

c3

a1

a2

a3

b1

b1

b2

c1

c1

c2

JOIN

DIVIDE

a

b

c

x

y

a

a

b

b

c

c

x

y

x

y

x

y

PROJECTION

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Relational Algebra Operations From Set Theoryⅰ. union,∪

R S ∪ ＝ { t | t R t S }∈ ∨ ∈ |R S| ≤ |R| + |S|∪

ⅱ. intersect,∩R∩S ＝ { t | t R t S }∈ ∧ ∈|R∩S| ≤ min{ |R|, |S| }

ⅲ. difference,- R S ＝ { t | t R t S }∈ ∧ |R S| ≤ |R|

ⅳ. Cartesian product,× R×S ＝ { r·s | r R s S }∈ ∧ ∈

· : concatenation |R×S| = |R|×|S|degree ＝ R’s degree + S’s degree

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▶Relational Operations • Relation : R(X) ＝ R(A1, ... , An)

• R’s tuple r : <a1, ... , an>

R={r | r ＝ <a1, ... , an> }

– ai : tuple r’s attribute Ai value

– ai ＝ r.Ai ＝ r[Ai]

• In general,– <r.A1 , r.A2 ,…, r.An > = < r[A1], r[A2], …, r[An] >

= r[A1, A2, … An] = r[X]

Spring, 2006 KUT

Slide 6- 7

Unary Relational Operations: SELECT• The SELECT operation (denoted by (sigma)) is used to select a

subset of the tuples from a relation based on a selection condition.– The selection condition acts as a filter– Keeps only those tuples that satisfy the qualifying condition– Tuples satisfying the condition are selected whereas the

other tuples are discarded (filtered out) horizontal subset

Av(R) ＝ { r | r R r.Aθv }∈ ∧ AB(R) ＝ { r | r R r.Aθr.B }∈ ∧

where, θ(theta) : <, >, ≤, ≥, ＝ , ≠ • Examples:

– Select the EMPLOYEE tuples whose department number is 4: DNO = 4 (EMPLOYEE)

– Select the employee tuples whose salary is greater than $30,000:

SALARY > 30,000 (EMPLOYEE)

Spring, 2006 KUT

Slide 6- 8

Unary Relational Operations: SELECT (contd.)

• SELECT Operation Properties– The SELECT operation <selection condition>(R) produces a relation S that

has the same schema (same attributes) as R

– SELECT is commutative: <condition1>( < condition2> (R)) = <condition2> ( < condition1> (R))

– Because of commutativity property, a cascade (sequence) of SELECT operations may be applied in any order:

<cond1>(<cond2> (<cond3> (R)) = <cond2> (<cond3> (<cond1> ( R)))

– A cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions:

<cond1>(< cond2> (<cond3>(R)) = <cond1> AND < cond2> AND < cond3>(R)))

– The number of tuples in the result of a SELECT is less than (or equal to) the number of tuples in the input relation R

– Selectivity

Spring, 2006 KUT

Join

• 세타조인 (theta-join)R(X), S(Y), A X, B Y ∈ ∈ 에 대하여R AθB S ＝ { r · s | r R s S ( r.Aθs.B) }∈ ∧ ∈ ∧– A, B : joining attribute– 결과 차수 = R 의 차수 + S 의 차수

• example– 학생 학번 = 학번 등록

• 동일조인 (equi-join)세타조인에서 θ 가 " ＝ " 인 경우R A=BS ＝ { r·s | r R s S ( r.A∈ ∧ ∈ ∧ ＝ s.B ) }

Spring, 2006 KUT

Slide 6- 10

Unary Relational Operations: PROJECT• PROJECT Operation is denoted by (pi) • In Relation R(X),

if Y X and⊆ Y={B1,B2, … ,Bm},

Y(R) ＝ { <r.B1, ... , r.Bm> | r R }∈ vertical subset• This operation keeps certain columns (attributes)

from a relation and discards the other columns.– PROJECT creates a vertical partitioning

• The list of specified columns (attributes) is kept in each tuple• The other attributes in each tuple are discarded

• Example: To list each employee’s first and last name and salary, the following is used:

LNAME, FNAME,SALARY(EMPLOYEE)

Spring, 2006 KUT

Slide 6- 11

Binary Relational Operations: JOIN• JOIN Operation (denoted by )

– The sequence of CARTESIAN PRODECT followed by SELECT is used quite commonly to identify and select related tuples from two relations

– A special operation, called JOIN combines this sequence into a single operation

– This operation is very important for any relational database with more than a single relation, because it allows us combine related tuples from various relations

– The general form of a join operation on two relations R(A1, A2, . . ., An) and S(B1, B2, . . ., Bm) is:

R <join condition>S– where R and S can be any relations that result from

general relational algebra expressions.

Spring, 2006 KUT

Slide 6- 12

Some properties of JOIN

• Consider the following JOIN operation:– R(A1, A2, . . ., An) S(B1, B2, . . ., Bm)

R.Ai=S.Bj

– Result is a relation Q with degree n + m attributes:• Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.

– The resulting relation state has one tuple for each combination of tuples—r from R and s from S, but only if they satisfy the join condition r[Ai]=s[Bj]

– Hence, if R has nR tuples, and S has nS tuples, then the join result will generally have less than nR * nS tuples.

– Only related tuples (based on the join condition) will appear in the result

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Theta JOIN

For R(X), S(Y), A X, B Y ,∈ ∈R AθB S ＝ { r · s | r R s S ( r.Aθs.B) }∈ ∧ ∈ ∧

– A, B : join attribute– θ can be any general boolean expression on the

attributes of R and S; for example:• R.Ai<S.Bj AND (R.Ak=S.Bl OR R.Ap<S.Bq)

Spring, 2006 KUT

Slide 6- 14

NATURAL JOIN Operation

• NATURAL JOIN Operation – Another variation of JOIN called NATURAL JOIN — denoted by N— was

created to get rid of the second (superfluous) attribute in an EQUIJOIN condition.

• because one of each pair of attributes with identical values is superfluous

• natural join: N)

If R(X), S(Y)’s join attribute is Z(=X∩Y)

R NS

= {<r · s>[X Y] | r R s S r[Z]∪ ∈ ∧ ∈ ∧ ＝ s[Z] }

= X Y∪ (Z=Z(R×S))

= X Y∪ (R Z=ZS)

– The standard definition of natural join requires that the two join attributes, or each pair of corresponding join attributes, have the same name in both relations

Spring, 2006 KUT

• example: Q R(A,B,C,D) N S(C,D,E)

– The implicit join condition includes each pair of attributes with

the same name, “AND”ed together:

• R.C=S.C AND R.D.S.D

– Result keeps only one attribute of each such pair:

• Q(A,B,C,D,E)

NATURAL JOIN Operation

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DIVISION: ÷ (1)• DIVISION Operation

– The division operation is applied to two relations R(X), S(Y) – R÷S ＝ { t | t ∈ D(R) t · s R for all s S }, where Y∧ ∈ ∈ X.

Let D = X-Y(and hence X =D Y); that is, let D be the set of attributes of R that are not attributes of S.

– The result of DIVISION is a relation T(Y) that includes a tuple t if tuples tR appear in R with tR [Y] = t, and with

• tR [X] = ts for every tuple ts in S.

– For a tuple t to appear in the result T of the DIVISION, the values in t must appear in R in combination with every tuple in S.

• Note : ((R ÷ S) × S) R⊆

Spring, 2006 KUT

학번(SNO)

과목번호(CNO)

100 C413100 E412

200 C123

300 C312

300 C324

300 C413400 C312400 C324

400 C413

400 E412

500 C312

과목번호(CNO)C413

과목번호(CNO)C312C413

과목번호(CNO)C312C413

E412

학번(SNO)100

300

400

학번(SNO)300

400

학번(SNO)400

학과목 (SC) 과목 1(C1) 과목 2(C2) 과목 3(C3)

SC ÷ C1 SC ÷ C2 SC ÷ C3

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Extension of Relational Algebra(1)

ⅰ. semijoin: – R S: R’s tuples that can be natual join with S

• Let R(X), S(Y)’s join attribute be Z(=X∩Y), R S ＝ R N( Z(S)) ＝ X(R NS)

• Property– R S ≠ S R

– R NS ＝ (R S) NS ＝ (S R) NR

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Natual Join & SemiJoin

R S R S

R S X∩Y(S)

( 자연 조인 )

( 세미 조인 )

A B C

a1a2a3a4

b1b1b1b2

c1c1c2c3

B C D

b1b1b2

c1c1c3

d1d2d3

B C

b1b2

c1c3

A B C D

a1a1a2a2a4

b1b1b1b1b2

c1c1c1c1c3

d1d2d1d2d3

A B C

a1a2a4

b1b1b2

c1c1c3

NN N

N

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• The OUTER JOIN Operation– In NATURAL JOIN and EQUIJOIN, tuples without a

matching (or related) tuple are eliminated from the join result

• Tuples with null in the join attributes are also eliminated• This amounts to loss of information.

– A set of operations, called OUTER joins, can be used when we want to keep all the tuples in R, or all those in S, or all those in both relations in the result of the join, regardless of whether or not they have matching tuples in the other relation.

ⅱ. outerjoin, +

Spring, 2006 KUT

• The left outer join operation keeps every tuple in the first or left relation R in R S; if no matching tuple is found in S, then the attributes of S in the join result are filled or “padded” with null values.

• A similar operation, right outer join, keeps every tuple in the second or right relation S in the result of R S.

• A third operation, full outer join, denoted by

or + , keeps all tuples in both the left and the right relations when no matching tuples are found, padding them with null values as needed.


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Natural Join and Outer JoinR S

(Outer Join) (Natural Join)

R S

+R S

A B C

a1a2a3a4

b1b1b1b2

c1c1c2c3

B C D

b1b1b2b3

c1c1c3c3

d1d2d3d3

A B C D

a1a1a2a2a3a4

b1b1b1b1b1b2b3

c1c1c1c1c2c3c3

d1d2d1d2

d3d3

A B C D

a1a1a2a2a4

b1b1b1b1b2

c1c1c1c1c3

d1d2d1d2d3

N

+

N

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ⅲ. outer-union, ∪+

– The outer union operation was developed to take the union of tuples from two relations if the relations are not type compatible.

– This operation will take the union of tuples in two relations R(X, Y) and S(X, Z) that are partially compatible, meaning that only some of their attributes, say X, are type compatible.

– The attributes that are type compatible are represented only once in the result, and those attributes that are not type compatible from either relation are also kept in the result relation T(X, Y, Z).

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Outer UnionR S

A B C

a1a2a3a4

b1b1b1b2

c1c1c2c3

B C D

b1b1b2

c1c1c2

d1d2d3

A B C D

a1a2a3a4

b1b1b1b2b1b1b2

c1c1c2c3c1c1c2

d1d2d3

∪+

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• A type of request that cannot be expressed in the basic relational algebra is to specify mathematical aggregate functions on collections of values from the database.

• Examples of such functions include retrieving the average or total salary of all employees or the total number of employee tuples.

– These functions are used in simple statistical queries that summarize information from the database tuples.

• Common functions applied to collections of numeric values include– SUM, AVERAGE, MAXIMUM, and MINIMUM.

• The COUNT function is used for counting tuples or values.– AVGgrade(enroll)

• retrieves the average score value from the enroll relation– GROUPyear(student)

• Grouping student into subgroups with respect to year– GROUPcnoAVGscore(enroll)

• Retrieve the average score of each cno group of enroll– General Form : GAFB(E)

• E : Relational Algebra expression• F : aggregation fuction ( SUM, AVG, MAX, MIN, COUNT)• B : Aggregate attribute• G : GROUP Function• A :Group attribute

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▶ Algebra Expression

• Retrieve all students’ name and dept

sname,dept (student)

• Retrieve name and score of a studuent who register C413

course

sname,score(cno='C413' (student NRegister))

• Retrieve name of a professor who teaches ‘database’

profname(cname=‘database'(course))

Spring, 2006 KUT

SQL

• SQL– A standard language for RDB– Based on relational algebra and calculus

• Features of SQL– No-procedural language

• SQL language : handling a set of data satisfying the conditions

– Interactive or embedded

Spring, 2006 KUT

Sample Table

학번(SNO)

이름(SNANE)

학년(YEAR)

학과(DEPT)

100 나 연 묵 4 컴퓨터200 이 찬 영 3 전기300 정 기 태 1 컴퓨터400 송 병 호 4 컴퓨터500 박 종 화 2 산공

학생(STUDENT)

과목번호(CNO)

과목이름(CNANE)

학점(CREDIT)

학과(DEPT)

C123 프로그래밍 3 컴퓨터C312 자료 구조 3 컴퓨터C324 파일 처리 3 컴퓨터C413 데이타 베이스 3 컴퓨터C412 반 도 체 3 전자

담당교수(PRNAME)

김성기황수찬이규철이석호홍봉희

과목(COURSE)

Spring, 2006 KUT

Basic of SQL

• SELECT statement– Retrieve data from table– SELECT clause and FROM clause are mandatory

in SELECT statement– WHERE clause is optional to describe the

condition

SELECT SNAME

FROM STUDENT학번(SNO)

이름(SNANE)

학년(YEAR)

학과(DEPT)


학생(STUDENT)

Spring, 2006 KUT

SQL 의 기초SELECT SNAME

FROM STUDENT

WHERE YEAR =4

학번(SNO)

이름(SNANE)

학년(YEAR)

학과(DEPT)


학생(STUDENT)

Spring, 2006 KUT

SQL• UNION :

– SQL ex : SELECT a FROM R UNION SELECT b FROM S;

• INTERSECT : – SQL ex : SELECT a FROM R INTERSECT SELECT b FROM S;

• DIFFERENCE : – SQL ex : SELECT a FROM R MINUS SELECT b FROM S;

• PRODUCT : – SQL ex : SELECT a, b FROM R, S;

• RESTRICTION(SELECTION) : – SQL ex : SELECT * FROM R WHERE r.A=10;

• PROJECTION :– SQL ex : SELECT r.A1, r.A2 FROM R;

• JOIN– SQL ex : SELECT r.A, r.B FROM R, S WHERE r.A = s.B;

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Query Processing

• Index– Random data(tuple) access inefficient

– Additional data structure

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keyaddressK1K2K3

Index file Data File

▶Index method• indexed file consists of

– index file– data file

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Traditional Index Structure

• B-Tree• B+-Tree

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index : (1) B-tree• B-tree (degree = m)

– m-way search tree1. Except root and leaf, the number of subtrees of internal

node is at least ⌈m/2 , at most, m⌉1. at most, the number of key is ⌈m/2 -1⌉

2. if root is not a leaf, root has two subtree ats least.3. all leaf is same level

• balanced tree

Note: degree is the maximum number of subtrees

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3-nary B-tree

18 20 30 36 42 50 58 62 65 70 110 120 130 136 140 1507 15

19

16 26 60 100 132 145

43 128 138

69 ^

^ 40 ^ ^ ^ ^

a

b c

d e f g h i

j k l m n o p q r s t u v

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▶ Operation(1)• B-tree

– random access : branch by search key– sequential access : in order traversal– Insert/delete : keep balance

• split : by node overflow

• merge : by node underflow

• Insert– Insert done at leaf node

• has free space : simple insertion

• overflow(no free space) there m keys in a leaf node

1) split

2) m/2 th key insert parent node

3) remains left, right

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Example • 59 insert

• 57 insert

50 50

o o’

57

60

58

f f

o

50 58

^p

60

50

o

b

59

o’p

b

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Example• splite by insert 54

• in Parent node f, insert 54

60 58 60 54

f f f’

o o’ p o o’’ o’ p

50 50

o o’ o’’

57 57 54 goes to parent node f

58 goes to parent node b

^ ^

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Example• parent node b, insert 58

• parent node a, insert 43

69

a

b c

43 69

a

b b’ c

^

5819 43

19

b b b’

d e f d e f f’

43 goesto parent a

^ ^

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▶ Example (2)

• Delete– Delete is done at leaf node– Deletion key is not in leaf node

• swap with following key • deletion

– if # of key < m/2 -1, underflow⌈ ⌉1. redistribution

– sibling node having keys whose number >= m/2 ⌈ ⌉ (parent node key → underflow node key)

(sibling node key →parent node key)

merge– can not redistribution

(sibling node + parent node + underflow node)

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Example• delete 60

• delete 20

60 62

50 6562 50 6560

b b

62

50 65

b

f f f

o o op p p5050

26

20 3630

b

e

l m

40

42n

30

26 36

b

e

l m

40

42n

Spring, 2006 KUT

B-Tree insertion (m =5)

• Insert 77 72 84

74 75 76 78 2 7 40 89 90 91

72 76 84

74 75 2 7 40 89 90 91 77 78

Split

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B-Tree Deletion (m = 5)

• Delete 84 72 76 84

74 75 2 7 40 89 90 91 77 78

72 76 89

74 75 2 7 40 84 90 91 77 78

Swap & Delete

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B-Tree Deletion

• Delete 74 72 76 89

74 75 2 7 40 90 91 77 78

72 76 89

74 75 2 7 40 90 91 77 78

Underflow 발생

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B-Tree Deletion Redistribution Using A Adjacent Sibling whose number of key greater than or

equal to ceiling(m/2)

72 76 89

74 75 2 7 40 90 91 77 78

40 76 89

72 75 2 7 90 91 77 78

{2,7, 40, 72, 75} is redistributed , [m/2] th value ( 즉 , 40) go to parent node

Spring, 2006 KUT

B-Tree Deleton

• Delete 40 40 76 89

72 75 2 7 90 91 77 78

72 76 89

40 75 2 7 90 91 77 78

Swapcannot Redistribution

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B-Tree Deletion

72 76 89

40 75 2 7 90 91 77 78

72 89

2 7 90 91 75 76 77 78

merge with right sibling and parent

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(2) B+-Tree• B+-tree consists of index set and sequence set

1. index set– consists of internal node

– support access path to leaf nodes

– support direct access

2. sequence set– consists of leaf nodes

– leaf nodes store whole keys support sequential access

– leaf node and internal node has different structures

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▶ B+-Tree(2)• B+-tree with degree m

– node structure ＜ n, P0, K1, P1, K2, P2, … , Pn-1, Kn, Pn>

• n : # of keys ( 1≤n<m )

• P0, …, Pn :pointer to subtree

• K1, …, Kn : key value

– root has 0, 2~m subtrees– Except root and leaf, internal node has ⌈m/2 ~⌉ m

subtrees– All leaf nodes are same level– key values in nodes is ascending order

Spring, 2006 KUT

Slide 14- 51

Difference between B-tree and B+-tree

• In a B-tree, pointers to data records exist at all levels of the tree

• In a B+-tree, all pointers to data records exists at the leaf-level nodes

• A B+-tree can have less levels (or higher capacity of search values) than the corresponding B-tree

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▶ B+-tree operation• search

– B+-tree index set : m-nary search tree– Record is obtained at leaf node

• Insert– similar to B-tree

• Delete– done at leaf (when redistribution/merge)

• key in index set is not deleted∵ it act as seperator access path

– redristribution: change key in index set– merge : delete key in index set

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B+-tree with degree 3

Index

set

sequence set

69

20 11043

2015 4035 43 6955 9070 110 125120

a

b c

d e f g h

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Example• B+-Tree , delete 43 43 in index set is not removed

• delete 125 (underflow redistribution)

69

20 11043

2015 4035 6955 9070 110 125120

69

20 90

2015 4035 6955 9070 120110

43

Spring, 2006 KUT

55

Exapme• delete 55(under flow merge)

69

20 90

2015 4035 9070 11069 120

concepts of db(3) relational operation & sql

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