Query Decomposition and Data Localization

Transcription

1 Query Decomposition and Data Localization Query Decomposition and Data Localization Query decomposition and data localization consists of two steps: Mapping of calculus query (SQL) to algebra operations (select, project, join, rename) [Query Decomposition] Query Decomposition Applying data distribution information to the algebra operations [Data Localication] Data Localization Spring, 2006 Arturas Mazeika Page 2 Query Decomposition Query Decomposition (Normalization) Normalization Manipulate query quantifiers and qualification Analysis Detect and reject incorrect queries Simplification Eliminate redundant predicates Restructuring Use transformation rules to optimize query Lexical and syntactic analysis Check validity (similar to compilers) Check for attributes and relations Type checking on the qualification Put into normal form Conjunctive normal form Disjunctive normal form (p 11 p 12 p 1n ) (p m1 p m2 p mn ) (p 11 p 12 p 1n ) (p m1 p m2 p mn ) OR s mapped into union AND s mapped into join or selection Spring, 2006 Arturas Mazeika Page 3 Spring, 2006 Arturas Mazeika Page 4

2 Query Decomposition (Analysis 1/3) Query Decomposition (Analysis Example 2/3) Type incorrect Checks whether the attributes and relation names of a query are defined in the global schema Checks whether the operations on attributes do not conflict with the types of the attributes Semantically incorrect Checks whether the components contribute in any way to the generation of the result Only a subset of relational calculus queries can be tested for correctness, i.e those that do not contain disjunction and negation Typical data structures used to detect the semantically incorrect queries are the following: Connection graph (query graph) Join graph Consider a query: SELECT ENAME,RESP, ASG, PROJ WHERE EMP.ENO = ASG.ENO AND ASG.PNO = PROJ.PNO AND PNAME = "CAD/CAM" AND DUR 36 AND TITLE = "Programmer" Spring, 2006 Arturas Mazeika Page 5 Query Decomposition (Analysis Example 3/3) Spring, 2006 Arturas Mazeika Page 6 Query Decomposition (Simplification 1/2) Consider a query: SELECT ENAME,RESP, ASG, PROJ WHERE EMP.ENO = ASG.ENO AND PNAME = "CAD/CAM" AND DUR 36 AND TITLE = "Programmer" If the graph is not connected the query is wrong Transformation rules are used to simplify queries p 1 p 1 false p 1 (p 1 p 2 ) p 1 p 1 false p 1 p 1 true p 1 Spring, 2006 Arturas Mazeika Page 7 Spring, 2006 Arturas Mazeika Page 8

3 Query Decomposition (Simplification Example 2/2) Consider the following query: SELECT TITLE WHERE EMP.ENAME = "J. Doe" OR (NOT(EMP.TITLE = "Programmer") AND (EMP.TITLE = "Elect. Eng." OR EMP.TITLE = "Elect. Eng.") AND NOT(EMP.TITLE = "Programmer")) Simplified query: SELECT TITLE WHERE EMP.ENAME = "J. Doe" Spring, 2006 Arturas Mazeika Page 9 Query Decomposition (Restructuring, Transformation Rules 2/5) Query Decomposition (Restructuring 1/5) Convert relational calculus to relational algebra. Finds an efficient expression. Example: Find the names of employees other than J. Doe who worked on the CAD/CAM project for either 1 or 2 years. SELECT ENAME, ASG, PROJ WHERE EMP.ENO = ASG.ENO AND ASG.PNO = PROJ.PNO AND ENAME "J. Doe" AND PNAME = "CAD/CAM" AND (DUR = 12 OR DUR = 24) A query tree is typically build for the transformation Relations are leaves (FROM clause) Attributes of the result is root (SELECT clause) Intermediate leaves should give a result from the leaves to the root Spring, 2006 Arturas Mazeika Page 10 Query Decomposition (Restructuring, Transformation Rules 3/5) Commutativity of binary operations R S = S R R S = S R R S = S R Associativity of binary operations R S) T = R (S T ) (R S) T = R (S T ) Idempotence of unary operations Π A (Π A (R)) = Π A (R) σ p1(a1) (σ p2(a2) (R)) = σ p1(a1) p2(a2) (R) Commuting selection with binary operations σ p(a) (R S) (σ p(a) (R)) S σ p(a1 )(R (A2,B 2 ) S) (σ p(a1 )(R) (A2,B 1 ) S σ p(a) (R T ) σ p(a) (R) σ p(a) (T ) A belongs to R and T Commuting projection with binary operations Π C (R S) Π A (R) Π B (S) Π C (R (A,B) S) Π A (R) (A,B) Π B (S) Π C (R S) Π C (R) Π C (S) C = A B, A A, B B Spring, 2006 Arturas Mazeika Page 11 Spring, 2006 Arturas Mazeika Page 12

4 Query Decomposition (Restructuring, Example 4/5) Query Decomposition (Restructuring, Example 5/5) Spring, 2006 Arturas Mazeika Page 13 Data Localization Spring, 2006 Arturas Mazeika Page 14 Data Localization (Example, Brute Force 1/3) Input: algebraic query on global conceptual schema Purpose: determine which fragments are involved Substitute global query with data manipulation queries on fragments Optimize the global query Assume EMP is fragmented into EMP1, EMP2, EMP3 as follows: EMP1=ENO E3 (EMP) EMP2= E3 <ENO E6 (EMP) EMP3=ENO> E6 (EMP) ASG fragmented into ASG1 and ASG2 as follows: ASG1=ENO E3 (ASG) ASG2=ENO> E3 (ASG) Replace EMP by (EMP1 EMP2 EMP3) and ASG by (ASG1 ASG2) in all queries Spring, 2006 Arturas Mazeika Page 15 Spring, 2006 Arturas Mazeika Page 16

5 Data Localization (Example, Provides Parallelism 2/3) Data Localization (Example, Eliminates Unnecessary Work 3/3) Spring, 2006 Arturas Mazeika Page 17 Data Localizations Issues Spring, 2006 Arturas Mazeika Page 18 Data Localizations Issues (Reduction of HF with Selection 1/4) Consider relation R with horizontal fragmentation F = {R 1, R 2,..., R k }, where R i = σ pi (R) Rule can be formalized by the following: Reduction of horizontal fragmentation (HF) Reduction with selection Reduction with join Reduction of vertical fragmentation (VF) Find empty relations σ pj (R i ) = p i (x) p j (x) = false, x R i Beneficial when fragmentation predicate is inconsistent with the query selection predicate (returns empty relations) Example: SELECT * WHERE ENO="E5" Spring, 2006 Arturas Mazeika Page 19 Spring, 2006 Arturas Mazeika Page 20

6 Data Localizations Issues (Reduction of HF with Join 2/4) Data Localizations Issues (Reduction of HF with Join 3/4) Possible if fragmentation is done on the join attribute Distribute join over union (R 1 R 2 ) S (R 1 S) (R 2 S) Rule can be formalized by the following. Given R i = σ pi (R) and R j = σ pj (R), j = 1, 2,..., k R i R j = p i (x) p j (x) = false, x R j R i pi,p 1 pi,p 2 pi,p 3 Fragmentation: EMP1=ENO E3 (EMP) EMP2= E3 <ENO E6 (EMP) EMP3=ENO> E6 (EMP) ASG1=ENO E3 (ASG) ASG2=ENO> E3 (ASG) Query: SELECT *, ASG WHERE EMP.ENO=ASG.ENO R 1 R 2 R 3 R i R 1 R i R 2 R i R 3 Spring, 2006 Arturas Mazeika Page 21 Spring, 2006 Arturas Mazeika Page 22 Data Localizations Issues (Derived Horizontal Fragmentation 4/4) Data Localizations Issues (Reduction of VF) Find empty relations Assume a good fragmentation of EMP is EMP1 = σ TITLE= Prgrammer (EMP) EMP2 = σ TITLE Prgrammer (EMP) To achieve efficient joins one can also fragment ASG: ASG1= ASG < ENO EMP1 ASG2= ASG < ENO EMP2 Let R be fragmented into R 1, R 2,..., R k, R i = Π Ai (R), A i A. Then Π D (R i ) is empty iff D is not in A i. For example EMP 1 = Π ENO,ENAME (EMP ) EMP 2 = Π ENO,T IT LE (EMP ) Query: SELECT ENAME Spring, 2006 Arturas Mazeika Page 23 Spring, 2006 Arturas Mazeika Page 24

7 Conclusion Query decomposition and data localization maps calculus query into algebra operations and applys data distribution information to the algebra operations Query decomposition consists of normalization, analysis, simplification, and restructuring Data localization reduces horizontal fragmentation with join and selection, and vertical fragmentation with joins, and aims to find empty relations Spring, 2006 Arturas Mazeika Page 25