《数据库系统概念 Database System Concepts》原书教学资源（第五版，PPT课件讲稿，英文版）Chapter 14 Query Optimization

Transformation of Relational Expressions Two relational algebra expressions are said to be equivalent if the two expressions generate the same set of tuples on every legal database instance Note:order of tuples is irrelevant In SQL,inputs and outputs are multisets of tuples Two expressions in the multiset version of the relational algebra are said to be equivalent if the two expressions generate the same multiset of tuples on every legal database instance. An equivalence rule says that expressions of two forms are equivalent Can replace expression of first form by second,or vice versa Database System Concepts-5th Edition,Oct 5,2006. 14.7 @Silberschatz,Korth and Sudarshan

Database System Concepts - 5 14.7 ©Silberschatz, Korth and Sudarshan th Edition, Oct 5, 2006. Transformation of Relational Expressions Two relational algebra expressions are said to be equivalent if the two expressions generate the same set of tuples on every legal database instance Note: order of tuples is irrelevant In SQL, inputs and outputs are multisets of tuples Two expressions in the multiset version of the relational algebra are said to be equivalent if the two expressions generate the same multiset of tuples on every legal database instance. An equivalence rule says that expressions of two forms are equivalent Can replace expression of first form by second, or vice versa

Equivalence Rules 1.Conjunctive selection operations can be deconstructed into a sequence of individual selections. eno,(E)=Oo(Oo,(E)) 2.Selection operations are commutative. 06(o4,(E)=o6,(o6(E)》 3.Only the last in a sequence of projection operations is needed,the others can be omitted. Πz(Π，(.（Πn(E).…》=Π(E) 4. Selections can be combined with Cartesian products and theta joins. ao(E1XE2)=E1☒9E2 b.o01(E1凶2E2)=E1凶91A92E2 Database System Concepts-5th Edition,Oct 5,2006. 14.8 ©Silberschat乜，Korth and Sudarshan

Database System Concepts - 5 14.8 ©Silberschatz, Korth and Sudarshan th Edition, Oct 5, 2006. Equivalence Rules 1. Conjunctive selection operations can be deconstructed into a sequence of individual selections. 2. Selection operations are commutative. 3. Only the last in a sequence of projection operations is needed, the others can be omitted. 4. Selections can be combined with Cartesian products and theta joins. a.  (E1X E2 ) = E1  E2 b. 1 (E1 2 E2 ) = E1 1 2 E2 ( ( )) ( ( )) 1 2 2 1     E =    E ( ) ( ( )) 1 2 1 2    E =    E ( ( ( ( )) )) ( ) 1 2 1 L L  Ln E  = L E

Equivalence Rules (Cont.) 5.Theta-join operations(and natural joins)are commutative. E1凶gE2=E2☒gE1 6.(a)Natural join operations are associative: (E凶E2)凶E3=E凶(E2E3) (b)Theta joins are associative in the following manner: (E1凶91E2)☒203E3=E1☒01A93(E2凶2E3) where 02 involves attributes from only E2 and E3. Database System Concepts-5th Edition,Oct 5,2006. 14.9 @Silberschatz,Korth and Sudarshan

Database System Concepts - 5 14.9 ©Silberschatz, Korth and Sudarshan th Edition, Oct 5, 2006. Equivalence Rules (Cont.) 5. Theta-join operations (and natural joins) are commutative. E1  E2 = E2  E1 6. (a) Natural join operations are associative: (E1 E2 ) E3 = E1 (E2 E3 ) (b) Theta joins are associative in the following manner: (E1 1 E2 ) 2 3 E3 = E1 1 3 (E2 2 E3 ) where 2 involves attributes from only E2 and E3

Pictorial Depiction of Equivalence Rules Rule 5 E1 E2 E2 E1 Rule 6a ☒ E3 E1 E1 E2 E2 E3 68 Rule 7a If 0 only has ☒ attributes from El 60 E2 E1 E2 E1 Database System Concepts-5th Edition,Oct 5,2006. 14.10 @Silberschatz,Korth and Sudarshan

Database System Concepts - 5 14.10 ©Silberschatz, Korth and Sudarshan th Edition, Oct 5, 2006. Pictorial Depiction of Equivalence Rules

Equivalence Rules (Cont.) 7.The selection operation distributes over the theta join operation under the following two conditions: (a)When all the attributes in 0o involve only the attributes of one of the expressions (E1)being joined. 0(E1E2)=(o0(E1)凶gE2 (b)When 0 1 involves only the attributes of E1 and 02 involves only the attributes of E2. o1N2(E1凶E2)=(o1(E1)Xg(o2(E2) Database System Concepts-5th Edition,Oct 5,2006. 14.11 ©Silberschat乜，Korth and Sudarshan

Database System Concepts - 5 14.11 ©Silberschatz, Korth and Sudarshan th Edition, Oct 5, 2006. Equivalence Rules (Cont.) 7. The selection operation distributes over the theta join operation under the following two conditions: (a) When all the attributes in 0 involve only the attributes of one of the expressions (E1 ) being joined. 0 (E1  E2 ) = (0 (E1 ))  E2 (b) When  1 involves only the attributes of E1 and 2 involves only the attributes of E2 . 1 (E1  E2 ) = (1 (E1 ))  ( (E2 ))