Query equivalence and optimization

Transcription

1 Query equivalence and optimization Jens Grünberg Institute for Theoretical Computer Science Faculty of Computer Science, TU Dresden Supervisor: Janis Voigtlaender 27. Januar / 33

2 1 Data Model Axes Queries 2 3 Motivation Reverse Axis Anorther approach 2/ 33

3 Motivation Speed needed in huge databases of biological, physical or chemical projects needed for saving energy and resources Efficient Stream based processing 3/ 33

4 Motivation Speed needed in huge databases of biological, physical or chemical projects needed for saving energy and resources Efficient Stream based processing 3/ 33

5 Motivation Speed needed in huge databases of biological, physical or chemical projects needed for saving energy and resources Efficient Stream based processing 3/ 33

6 Tree View Data Model Axes Queries 4/ 33

7 Labeling Data Model Axes Queries Definition Let Σ be a finite set of labels. Lab L for L Σ is a unary relation representing the set of nodes labeled with L 5/ 33

8 Labeling Data Model Axes Queries journal editor1 editor2 anna bob Nodes = {, journal, editor1, editor2, anna, bob } 6/ 33

9 Labeling Data Model Axes Queries journal Labjournal "journal" editor1 editor2 "anna" Labanna anna bob Labbob "bob" Nodes = {, journal, editor1, editor2, anna, bob } Labjournal = { journal } Labanna = { anna } Labbob = { bob } 7/ 33

10 Labeling Data Model Axes Queries journal Labjournal "journal" editor1 editor2 Labeditor "editor" "anna" Labanna anna bob Labbob "bob" Nodes = {, journal, editor1, editor2, anna, bob } Labjournal = { journal } Labanna = { anna } Labbob = { bob } Labeditor = { editor1, editor2 } 8/ 33

11 Child and next-sibling relations Data Model Axes Queries Definition R child (x,y) is a binary relation where y is a child of x 9/ 33

12 Child and next-sibling relations Data Model Axes Queries Definition R child (x,y) is a binary relation where y is a child of x Nodes = {, journal, editor1, editor2, anna, bob } Rchild = { (, journal), (journal, editor1), (editor1, anna), (journal, editor2), (editor2, bob) } journal editor1 editor2 anna bob 10/ 33

13 Child and next-sibling relations Data Model Axes Queries Definition R child (x,y) is a binary relation where y is a child of x Nodes = {, journal, editor1, editor2, anna, bob } Rchild = { (, journal), (journal, editor1), (editor1, anna), (journal, editor2), (editor2, bob) } journal editor1 editor2 anna bob 11/ 33

14 Child and next-sibling relations Data Model Axes Queries Definition R child (x,y) is a binary relation where y is a child of x Nodes = {, journal, editor1, editor2, anna, bob } Rchild = { (, journal), (journal, editor1), (editor1, anna), (journal, editor2), (editor2, bob) } journal editor1 editor2 anna bob 12/ 33

15 Child and next-sibling relations Data Model Axes Queries Definition R next sibling (x,y) is a binary relation where y is a immediate right-sibling of x 13/ 33

16 Child and next-sibling relations Data Model Axes Queries Definition R next sibling (x,y) is a binary relation where y is a immediate right-sibling of x Nodes = {, journal, editor1, editor2, anna, bob } Rnext-sibling = { (editor1, editor2) } journal editor1 editor2 anna bob 14/ 33

17 Navigational Structure Data Model Axes Queries Definition σ nav = (Lab L,R child,r next sibling ) 15/ 33

18 Axes Data Model Axes Queries R child = {(, A),(A, B),(B, C),(C,D),(C,E),...} C B D E A Node Selected Node Context Node child descendant R descendant = {..., (B,C),(B,D),(B, E),...} R descendant = {..., (, A),(A,B), (, B),...} R ancestor = R 1 descendant = {..., (B,A),(B, ),...} ancestor 16/ 33

19 Axes Data Model Axes Queries R child = {(, A),(A, B),(B, C),(C,D),(C,E),...} C B D E A Node Selected Node Context Node child descendant R descendant = {..., (B,C),(B,D),(B, E),...} R descendant = {..., (, A),(A,B), (, B),...} R ancestor = R 1 descendant = {..., (B,A),(B, ),...} ancestor 16/ 33

20 Axes Data Model Axes Queries R child = {(, A),(A, B),(B, C),(C,D),(C,E),...} C B D E A Node Selected Node Context Node child descendant R descendant = {..., (B,C),(B,D),(B, E),...} R descendant = {..., (, A),(A,B), (, B),...} R ancestor = R 1 descendant = {..., (B,A),(B, ),...} ancestor 16/ 33

21 Queries - Syntax Data Model Axes Queries Definition p ::= p p cxp (1) cxp ::= /lp lp (2) lp ::= lp/lp axis :: test[q] (3) q ::= q q q q q cxp true (4) 17/ 33

22 Data Model Axes Queries Semantic of Queries - Node Testing Definition Definition T( ) = All Nodes T(L) = Lab L for every L Σ axis :: test[q] NodeSet (n) := {n T(test) (n,n ) R axis q Boolean (n )} 18/ 33

23 Data Model Axes Queries Semantic of Queries - Node Testing Definition Definition T( ) = All Nodes T(L) = Lab L for every L Σ axis :: test[q] NodeSet (n) := {n T(test) (n,n ) R axis q Boolean (n )} 18/ 33

24 Queries - Semantic Data Model Axes Queries Definition p 1 p 2 NodeSet (n) ::= p 1 NodeSet (n) p 2 NodeSet (n) (5) /p NodeSet (n) ::= p NodeSet (n 0 ) (6) p 1 /p 2 NodeSet (n) ::= p 2 NodeSet (n ) (7) n p 1 NodeSet (n) q 1 q 2 Boolean (n) ::= q 1 Boolean (n) q 2 Boolean (n) (8) q 1 q 2 Boolean (n) ::= q 1 Boolean (n) q 2 Boolean (n) (9) q Boolean (n) ::= q Boolean (n) (10) true Boolean (n) ::= true (11) p Boolean (n) ::= p NodeSet (n) (12) 19/ 33

25 How to define on queries? 20/ 33

26 C B D E A F G p 1 : /descendant :: [ancestor :: B[true]] p 2 : /descendant or self :: B[true]/descendant :: [true] 21/ 33

27 C B D E A F G p 1 : /descendant :: [ancestor :: B[true]] p 2 : /descendant or self :: B[true]/descendant :: [true] 21/ 33

28 Definition p 1 and p 2 are equal, if they select the same set of nodes, that means: p 1 NodeSet (x) p 2 NodeSet (x) for every context node x and every document. 22/ 33

29 Motivation - Reverse Axis Motivation Reverse Axis Anorther approach ancestor parent precedingsibling preceding Storing in memory Evaluation in more then one run Removing reverse axis 23/ 33

30 Motivation - Reverse Axis Motivation Reverse Axis Anorther approach ancestor parent precedingsibling preceding Storing in memory Evaluation in more then one run Removing reverse axis 23/ 33

31 Motivation - Reverse Axis Motivation Reverse Axis Anorther approach ancestor parent precedingsibling preceding Storing in memory Evaluation in more then one run Removing reverse axis 23/ 33

32 Deletion of Reverse Axis Motivation Reverse Axis Anorther approach p1 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] 24/ 33

33 Deletion of Reverse Axis Motivation Reverse Axis Anorther approach p1 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] 25/ 33

34 Deletion of Reverse Axis Motivation Reverse Axis Anorther approach p1 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] 26/ 33

35 Anorther approach Motivation Reverse Axis Anorther approach p1 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] 27/ 33

36 Anorther approach Motivation Reverse Axis Anorther approach p2 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] 28/ 33

37 Anorther approach Motivation Reverse Axis Anorther approach p2 arbitrarily many nodes between A B descendent descendent - or - self ancestor selected node p 1 : /descendant :: a[ancestor :: b[true]] p 2 : /descendant or self :: b[true]/descendant :: a[true] 29/ 33

38 Anorther approach Motivation Reverse Axis Anorther approach /descendant :: n[q 1 ancestor :: m[q 2 ]] /descendant or self :: m[q 2 ]/descendant :: n[q 1 ] for a proof, article page 9 30/ 33

39 Anorther approach Motivation Reverse Axis Anorther approach Definition /descendant :: n[q 1 ancestor :: m[q 2 ]] /descendant or self :: m[q 2 ]/descendant :: n[q 1 ] /descendant :: m[q 1 ]/preceding :: n[q 2 ] /desecendant :: n[q 2 following :: m[true]] N B M arbitrarily many nodes between descendent following preceding selected node 31/ 33

40 Anorther approach Motivation Reverse Axis Anorther approach Definition /descendant :: n[q 1 ancestor :: m[q 2 ]] /descendant or self :: m[q 2 ]/descendant :: n[q 1 ] /descendant :: m[q 1 ]/preceding :: n[q 2 ] /desecendant :: n[q 2 following :: m[true]] B B arbitrarily many nodes between descendent N M N M following preceding selected node 32/ 33

41 References Motivation Reverse Axis Anorther approach XPath: Looking Forward by Dan Olteanu, Holger Meuss, Tim Furche, Springer Verlag Berlin Heidelberg, 2002 An Introduction to XML and Web Technologies by Andreas Moller, Michael Schwartzbach Addison-Wesley, 2006 XPath Leashed by Michael Benedikt, Christoph Koch Oxford University Computing Labority, Cornell University Minimizing Simple XPath Expressions by Peter T. Wood University of London, School of Computer Science and Information Systems 33/ 33