关系代数

关系代数（Relational Algebra）与SQL查询语言类似，不过多用于理论，SQL查询语言用于现实的数据库查询。

Three types of operations/operators on relations：

• fundamental operations：基本运算

• additive operations：由基本运算组合而成的附加运算

• extended operations：扩展运算

Fundamental operations 部分

Select Operation

相当于SQL查询语言中的where

Select operation is defined as:

$$\Large \sigma_p(r)=\{t|t \in r \land p(t) \}$$

$p$ is called the selection predicate, connected by : $\land$ (and), $\lor$ (or), $eg$ (not) , op is one of: $=$, $e$, $>$, $<$, $\le$, $\ge$

Project Operation

相当于SQL查询语言中的select

Notation:

$$\Large \Pi_{A_1,A_2,\dots,A_k}(r)$$

where $A_i$ are attribute names and $r$ is a relation name

例子1：

$$\Large \Pi_{ID,name,salary}(instructor)$$

SELECT distinct ID, name, salary
FROM instructor;

例子2：

$$\Large \Pi_{name}(\sigma_{dept\_name='Physics'}(instructor))$$

SELECT distinct name
FROM instructor
WHERE dept_name='Physics';

Positional Notation（位置标记）

Use $1， $2, . . . refer to the first attribute, the second attribute, and so on.

例子：

如果表结构为instructor(ID, name, dept_name, salary)，那么：

$$\Large \Pi_{\$4} (\sigma_{\$4 < \$8} (instructor \times instructor ))$$

SELECT distinct i1.salary
FROM instructor i1 JOIN instructor i2
WHERE i1.salary < i2.salary;

即选出所有不是最高工资的工资。

Union Operation

Union operation on relation r and s is defined as:

$$\Large r \cup s=\{t|t \in r \lor t \in s \}$$

• r, s must have the same arity (same number of attributes)

• The attribute domains must be compatible（兼容的）

例子：

Based on “emp-dept” schema, find all employees working as 'CLERK' or having salary higher than 2000, or fulfill both conditions.

$$\Large \Pi_{name}(\sigma_{job='CLERK'}(emp)) \cup \Pi_{name}(\sigma_{salary>2000}(emp))$$

Set-Difference Operation

Set difference operation on relation r and s is defined as:

$$\Large r - s=\{t|t \in r \land t \notin s \}$$

• r, s must have the same arity

• The attribute domains must be compatible

例子：

Based on “emp-dept” schema, find all employees working as 'CLERK' ，but not having salary higher than 2000.

$$\Large \Pi_{name}(\sigma_{job='CLERK'}(emp)) - \Pi_{name}(\sigma_{salary>2000}(emp))$$

Cartesian-product operation

Cartesian-product operation on relation r and s is defined as:

$$\Large r \times s=\{t|t=t_1t_2 \land t_1 \in r \land t_2 \in s \}$$

例子1：

Give the relational algebra expression for the operations shown:

$$\Large \Pi_{A,B,C,D,E}(\sigma_{A=C}(r \times s))$$

例子2:

Based on “emp-dept” schema， find all employees’ name and their departments' name .

$$\Large \Pi_{emp.name, dept.name}(\sigma_{emp.deptno=dept.deptno}(emp \times dept))$$

Rename Operation

returns the result of expression E under the name x:

$$\Large \rho_{\text{x}}(E)$$

If a relational-algebra expression E has arity n then:

$$\Large \rho_{\text{x}(A_1,A_2,\dots,A_n)}(E)$$

Additive operations 部分

We define additional operations that do not add any power to the relational algebra, but that simplify common queries. An additional operation can be replaced/re-represented by basic operations.

Set-Intersection Operation

For relation r and s, set-intersection operation on r and s is defined as:

$$\Large r \cap s=\{t|t \in r \land t \in s \}=r-(r-s)$$

例子：

Based on “emp-dept” schema, find all employees working as 'CLERK' and having salary higher than 2000.

$$\Large \Pi_{name}(\sigma_{job='CLERK'}(emp)) \cap \Pi_{name}(\sigma_{salary>2000}(emp))$$

Natural-Join Operation

Let r and s be relations on schemas R and S respectively, and assuming $R\cap S = {A_1， A_2， \dots， A_n }$, natural-join operation is defined as:

$$\Large r \bowtie s = \sigma_{r.A_1=s.A_1 \land \dots \land r.A_n=s.A_n}(r \times s)$$

• Natural join is associative(结合律)

  $$\Large (A \bowtie B) \bowtie C = A \bowtie (B \bowtie C)$$
• Natural join is commutative (交换律)

  $$\Large A \bowtie B = B \bowtie A$$

Theta Join

The theta join operation is defined as:

$$\Large r \bowtie_\theta s = \sigma_\theta(r \times s)$$

where $\theta$ is a predicate on attributes in $R \cup S$

Division

记不住公式没问题，看例子理解。

$$\Large R \div S = \Pi_{R-S}(r) - \Pi_{R-S}((\Pi_{R-S}(r) \times S) - \Pi_{R-S,S}(r))$$

例子1：

Find all students who have taken all courses offered in the Biology department.

$$\Large \begin{align} &\Pi_{name,course\_id}(\rho_{S}(student) \bowtie_{S.ID=T.ID} \rho_{T}(takes)) \\ \div &\Pi_{course\_id}(\sigma_{dept\_name = 'Biology'}(course)) \end{align}$$

select distinct S.name
from student as S
where not exists (
    (
        select course_id
        from course
        where dept_name = 'Biology'
	)
    except
    (
        select T.course_id
        from takes as T
        where S.ID = T.ID
    ) 
);

例子2：

Based on “emp-dept” schema, find the jobs provided by all the departments having employees.

首先，确定目标的是job，条件是deptno

$$\Large \Pi_{job,deptno}(emp) \div \Pi_{deptno}(dept)$$

例子3：

找到提供了工资在3000到4000之间所有职位的部门，给出部门名称。

首先确定目标是deptno，条件是job

\Large \Pi_{dname}(dept \bowtie \Pi_{deptno,job}(emp)\div\Pi_{job}(\sigma_{sal\ge 3000 \and sal <4000}(emp)))

Assignment Operation

The assignment operation ($\leftarrow$) provides a convenient way to express complex queries. 相当于with ... as...

Outer Join

Outer join is an extension of the join operation that avoids loss of information.

left outer join operation is defined as:

$$\Large (r \bowtie s) \cup (r- \Pi_R(r \bowtie s) \times \{ (\text{null},\dots, \text{null})\})$$

Extended operations部分

Generalized Projection

Generalized project（广义投影） extends the projection operation by allowing arithmetic functions to be used in the projection list.

$$\Large \Pi_{F_1,F_2,\dots,F_n}(E)$$

each of $F_1,F_2,\dots,F_n$ are arithmetic expressions involving constants and attributes in the schema of E

Aggregation functions

Aggregate operation（聚集运算） in relational algebra is defined as:

$$\Large _{G_1,G_2,\dots,G_n} \mathcal{G}_{F_1(A_1),F_2(A_2),\dots,F_n(A_n)}(E)$$

• $E$ is any relational-algebra expression
• $G_1,G_2,\dots,G_n$is a list of attributes on which to group (can be empty)
• Each $F_i$ is an aggregate function
• Each $A_i$ is an attribute name

例子1：

Based on “emp-dept” schema,Find the average salary in each department

$$\Large _{deptno} \mathcal{G}_{avg(sal)}(emp)$$

例子2：

获得工资比所在部门平均工资高的员工姓名、工资以及所在部门的平均工资。

$$\Large \Pi_{ename,sal,avgsal}(\sigma_{sal>avgsal}(emp \bowtie(_{deptno} \mathcal{G}_{avg(sal) \, as\, avgsal}(emp))))$$