Relational Algebra

Relational Algebra Operators Expression Trees

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.

What is Relational Algebra? • An algebra whose operands are relations or variables that represent relations. • Operators are designed to do the most common things that we need to do with relations in a database. • The result is an algebra that can be used as a query language for relations.

What we will learn… • Core (or traditionally) relational algebra

Core Relational Algebra • Union, intersection, and difference. • Usual set operations, but require both operands have the same relation schema. • Selection: picking certain rows. • Projection: picking certain columns. • Products and joins: compositions of relations.

Operators

Set Operators • R U S: union, the set of elements that are in R or S or both. • R S: intersection, the set of elements that are in both R and S. • R - S: difference, the set of elements that are in R but no in S. • Required • R and S must have schema with identical set of attributes, and • Before calculation, the columns of R and S must be ordered.

RS RS A B C A B C a1 b1 c1 a1 b2 c2 a2 b2 c1 a1 b3 c2 a1 b2 c2 a2 b2 c1 R S Ａ　　　Ｂ　　　ＣＡ　　　Ｂ　　Ｃ a1 b1 c1 a1 b2 c2 a2 b2 c1 a1 b2 c2 a1 b3 c2 a2 b2 c1 R-S A B C a1 b1 c1

Projection • L (R) • L is a list of attributes from the schema of R. • The result is a new relation that has only some of R’s columns. • Eliminate duplicate tuples, if any. • Example title, year, length (Movie)

Selection • C (R) • C is a condition (as in “if” statements) that refers to attributes of R. • The result is a new relation with a subset of R’s tuples that satisfy C. • Example lentgh>=100 AND studioName = ‘Fox’(Movie)

Cartesian Product • or just product • R  S • Pair each tuple t1 of R with each tuple t2 of S. • Result: a new relation with new tuples, each of them concatenation a pair of t1t2, the attributes of R and S are in ordered. • But beware attribute A of the same name in R and S: use R.A and S.A.

R S Ａ　　　Ｂ　　　ＣＡ　　　Ｂ　　Ｃ a1 b1 c1 a1 b2 c2 a2 b2 c1 a1 b2 c2 a1 b3 c2 a2 b2 c1 RS A B C A B C a1 b1 c1 a1 b2 c2 a1 b1 c1 a1 b3 c2 a1 b1 c1 a2 b2 c1 a1 b2 c2 a1 b2 c2 a1 b2 c2 a1 b3 c2 a1 b2 c2 a2 b2 c1 a2 b2 c1 a1 b2 c2 a2 b2 c1 a1 b3 c2 a2 b2 c1 a2 b2 c1

Natural Join • A frequent type of join connects two relations by: • Equating attributes of the same name, and • Projecting out one copy of each pair of equated attributes. • Called natural join. • Denoted: R1 R2

Theta-Join • R C S • Take the product R x S. • Then apply C to the result. • C can be any boolean-valued condition. • Historic versions of this operator allowed only A theta B, where theta was =, <, etc.

RR.B=S.BS 等值连接 RC<ES R S A B C a1 b1 5 a1 b2 6 a2 b3 8 a2 b4 12 B E b1 3 b2 7 b3 10 b3 2 b5 2 A R.B C S.B E a1 b1 5 b2 7 a1 b1 5 b3 10 a1 b2 6 b2 7 a1 b2 6 b3 10 a2 b3 8 b3 10 A R.B C S.B E a1 b1 5 b1 3 a1 b2 6 b2 7 a2 b3 8 b3 10 a2 b3 8 b3 2 Ａ　Ｂ　　Ｃ　　Ｅ a1 b1 5 3 a1 b2 6 7 a2 b3 8 10 a2 b3 8 2 自然连接 RS

Outerjoin • Suppose we join RC S. • A tuple of R that has no tuple of S with which it joins is said to be dangling. • Similarly for a tuple of S. • Outerjoin preserves dangling tuples by padding them with a special NULL symbol in the result.

R OUTERJOIN S = A B C 1 2 3 4 5 NULL NULL 6 7 Example: Outerjoin R = A B S = B C 1 2 2 3 4 5 6 7 (1,2) joins with (2,3), but the other two tuples are dangling.

R S = R – (R – S) R C S = C (R x S) R S = L( C (R x S)) Dependent and Independent Operations

Combining Operations to Form Query • Algebras allow us to express sequences of operations in a natural way. • Example: in arithmetic --- (x + 4)*(y - 3). • Relational algebra allows the same. For example title, year( lentgh>=100 (Movie)studioName = ‘Fox’(Movie))

Expressions • Precedence of relational operators: • Unary operators --- select, project--- have highest precedence, bind first. • Then come products and joins. • Then intersection. • Finally, union and set difference bind last. • But you can always insert parentheses to force the order you desire.

Expression Trees • Leaves are operands --- either variables standing for relations or particular, constant relations. • Interior nodes are operators, applied to their child or children.

title, year studioName = ‘Fox’ lentgh >= 100 Movie Movie For example:

例：学生—课程数据库，包括Student，Course，SC三个关系例：学生—课程数据库，包括Student，Course，SC三个关系 Student Sno Sname Ssex Sage Sdept 95001 李勇　　　　男　　　　20CS 95002 刘晨　　　　女 19 IS 95003　　王敏　　　　女 18 MA 95004　张立　　　　男 19 IS Course SC • Cno Cname Cpqo Ccredit • 1 数据库 5 4 • 2　　　　数学 2 • 　　信息系统　　　 1 4 • 4　　　操作系统 6 3 • 5　　　数据结构 7 4 • 6　　　数据处理 2 • 7Pascal语言　　 6 4 Sno Cno Grade 95001 1 92 95001 2 85 95001 3 88 95002 2 90 95002 3 80

Sname（ Cpno = ‘5’ (Course) SC Sno,Sname(Student) ） Sname,Sdept(Student) Sdept = ‘IS’(Student) Sno(Cno = ‘1’ (SC)) Sno(Cno = ‘1’ or Cno=‘3’ (SC))

Reading Guide • Required: 4.1 • Recommended: 《数据库系统概论》第二章中的关系代数

练习 • 图书馆管理数据库 • 读者(读者编号, 姓名, 单位) • 图书(书号, 书名, 作者, 出版社, 单价, 类型) • 借阅记录(读者编号, 书号, 借阅日期, 应还日期) • 还书记录(读者编号, 书号, 归还日期) • 用关系代数描述以下查询要求 • 查询“人民邮电出版社”出版的所有图书的相关信息 • 查询单价在15元以上的书名和作者 • 查询8号读者2003年3月10日所借图书的相关信息 • 查询超期归还图书的读者姓名和单位 • 查询借阅过《天龙八部》的读者的信息 • 查询借阅过“金庸”所有著作的读者的姓名 • 查询没有借阅过任何图书的读者的姓名

Relational Algebra