Yuqing Wu, Dirk Van Gucht Indiana UniversityMarc Gyssens Hasselt UniversityJan Paredaens University of Antwerp

A Study of a Positive Fragment of Path Queries: Expressiveness, Normal Form

and Minimization

Research in XMLXML data modelXML query languages

XPathXQuery……

XML data repositories Support from DB vendorsLORE, Niagara, TIMBER……

2

Research in XMLCharacteristics of XML query languages

XPath and fragmentsCharacteristics: expressiveness,

distinguishibility, complexity, …System design

Query processing and evaluationNew access methods: structural join

Integrity, security, ……

3

Theoretical Study System Design RDB did very well in this aspectOur work at Indiana University

Coupling the theoretical study of XML data and query language and the system design of XML search engines [ICDT-EROW 07]

Coupling the partition of XML documents induced by the structure of XML document with the partition induced by fragments of XPath algebras. [DBPL 07, IS 08]

Applying the coupling in the design of structural indices for XML [WebDB 08]

Designing workload sensitive structural indices for XML. [in submission]

4

OutlineWhat we studiedEquivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

5

OutlineWhat we studied

XML documentsPath+ algebraTree queries

Equivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

6

XML DocumentsA labeled tree (V, Ed, l),

whereV is the set of nodesEd is the set of edgesl : VL is a node-

labeling function.

7

Querying XML Documentfor $i in doc(…)//a/b

for $j in $i/c/*/d[e] for $k in $j/*/f

return ($i, $k)intersectfor $i in doc(…)//a/b

for $j in $i/c/a/d for $k in $j/c/f

return ($i, $k)

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Path+ Algebra – Path Semantics

¼ (1)

})(|),{()(ˆ)(

}|),{()(

lmVmmmDl

DVmmmD

l

1)(

)(

EdD

EdD

)()()(

))()(()(;)}(),(:|),{()()}(),(:|),{()(

2121

21324,121

2

1

DEDEDEE

DEDEDEEDEnmmnnDEDEnmnmmDE

9

Path+ Expression – An Example

¼ (1)

;ˆ;;ˆ;ˆ;;);))ˆ;;(

);ˆ;(((;;)ˆ;;ˆ(;)ˆ;;ˆ(;)(

1

12221

dccc

acacdE

E(D) = {(n8,n11), (n8,n12)}

10

;ˆ;;ˆ;ˆ

;

;);))ˆ;;();ˆ;(((

;

;)ˆ;;ˆ(;)ˆ;;ˆ(;)(

1

12

221

dcc

ca

cacdE

Interesting Sub-languages

212121 |||;|||||: EEEEEElE

Path+ :

Path+(1, 2) :

DPath+(1) : EEElE 121 |;||||:

11

EEEElE 2121 ||;|||||:

Tree Query for XML A tree query T is a 3-

tuple (T, s, d), with T: a labeled tree –

nodes of T are either labeled with a symbol of L or with a wildcard *.

s and d: nodes of T, called the source and destination nodes.

12

OutlineWhat we studiedEquivalences of query languages

Normal formResolution expressivenessEfficient query evaluationSummary and discussion

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Equivalences of Query LanguagesTheorem

The query languages Path+ , T and Path+

(1, 2) are all equivalent in expressive power, and there exist translation algorithms between any two of them.

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Path+ Expression Tree Query T

¼ (1)

})(|),{()(ˆ)(

}|),{()(

lmVmmmDl

DVmmmD

l

1)(

)(

EdD

EdD

)}(),(:|),{()(

)}(),(:|),{()(

2

1

DEnmmnnDEDEnmnmmDE

)()()(

))()(()(;

2121

21324,121

DEDEDEE

DEDEDEE

*

l*

*s

d

*

*d

s

15

Transformation of Composition

16 ))()(()(; 21324,121 DEDEDEE

Transformation of Intersection

E1 E2

17 )()()( 2121 DEDEDEE

Tree Query T Path+ Expression

Base cases: Empty tree(<{n},>, n,n)

(<{n1, n2},{(n1, n2)}>, n1, n2)

(<{n1, n2},{(n1, n2)}>, n2, n1)

s (d)

s d

d s

s (d)

l l̂

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Tree Query T Path+ Expression

dpTTtoP ,,; 2

Recursive case #1: s is not an ancestor of d.

s has no child and l(s)=*

d is parent of s, d has no ancestor, no other child and l(d)=*

d s

p

T1

T2 d

s

dpTTtoPssTTtoP ,,;;,, 21

;,,1 ssTTtoP

19

Recursive case #2: s is not the root.

s has no child and l(s)=*

Tree Query T Path+ Expression

dsTTtoPsrTTtoP ,,;,, 122

d

s

r

T1

T2

s

r

d srTTtoP ,,22

20

Recursive case #3: s is a strict ancestor of d.

d has no child and l(d)=*

s is parent of d, s has no child other than d and l(d)=*

Tree Query T Path+ Expression

;,,2 psTTtoP

ddTTtoPpsTTtoP ,,;;,, 12

ddTTtoP ,,; 1

d

s

d

s

T1

T2

p

21

Recursive case #4: s = d is the root.

l(s)=*

Tree Query T Path+ Expression

)(;,,;;,, 1111 scsTTtoPcsTTtoP nn l

s, d…

T1

s, d

Tn

c1

cn

nn csTTtoP

csTTtoP

,,

;;,,

1

111

22

Equivalences of Query LanguagesTheorem

The query languages Path+ , T and Path+

(1, 2) are all equivalent in expressive power, and there exist translation algorithms between any two of them.

Path+ exp T query Path+(1, 2) exp

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OutlineWhat we studiedEquivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

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Normal Form Observation about the tree query Path+(1,2)

transformation:The resultant Path+(1,2) expression is of the form

where m≥0 and n ≥0Ci (i = um ,…, u1, d1 ,…, dn) are of the formCtop is of the form

E is a DPath+(1) expression.

nm ddtopuu CCCCC 11

?*1 ]ˆ[)]([ lE?*

12 ]ˆ[)]()][([ lEE

EEElE 121 |;||ˆ||:

d

rt

s

25

Normal FormE(Tts) -1; E(Ttt) ; 2 (E(Trt)); E(Ttd)

E(Tts), E(Ttt),E(Trt),E(Ttd) areDPath+(1) expressions Tt

s Tt

d

Tr

t

Tt

t d

r

t

s

26

OutlineWhat we studiedEquivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

27

Resolution ExpressivenessResolution expressiveness: a language’s

ability to distinguish a pairs of nodes of a pair of paths in the document.

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Expression EquivalenceNodes m1 and m2 are expression-related

(m1 ≥exp m2), if for each expression E, E(D)(m1) implies E(D)(m2) , where E(D)(m) = {n | (m,n) E(D)}.

m1 =exp m2 if m1 ≥exp m2 and m2 ≥exp m1

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1-equivalence Nodes m1 and m2 are downward 1-related (m1 ≥

1 m2) iffl (m1) = l (m2);For each child n1 of m1, there exist a child n2 of m2 such that

n1 ≥1 n2.

Nodes m1 and m2 are 1-related (m1 ≥1 m2) iffm1 ≥

1 m2

if m1 is not the root and p1 is the parent of m1 , then m2 is not the root with parent p2 such that p1 ≥1 p2 .

m1 =1 m2 if m1 ≥1 m2 and m2 ≥1 m1.

(m1, n1) ≥1 (m2, n2) if m1 ≥1 m2 and n1 ≥1 n2 and sig(m1, n1) = sig(m2, n2) .

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Resolution ExpressivenessTheorem :

m1 =exp m2 iff m1 =1 m2

Theorem : (m1,n1) E(D) implies (m2,n2) E(D) iff (m1,n1) ≥1 (m2,n2)

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OutlineWhat we studiedEquivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

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Tree Query Minimization1st Reduction: merging 1-equivalent nodes

in a tree query;*

a a

* c

*

c

*dd

sd

c

*d

*

a a

* c

*

c

*dd

sd

33

Tree Query Minimization 1-*-related (≥*

1 ): relax 1-related with l (m1) + l (m2) = l (m2);

2nd Reduction: deleting from a tree query in a top-down fashion every node m1 for which there exists another node m2 such that m1 ≥*

1

m2 . *

a a

* c

*

c

*dd

sd

*

a

c

* d

sd

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Efficient Query Evaluation Path+

Expression

E(Tts) -1; E(Ttt) ; 2 (E(Trt)); E(Ttd)E(Tts) , E(Ttt), E(Trt), E(Ttd)

are DPath+(1) expressionsTt

s Tt

d

Trt

Ttt d

r

t

s

Minimum Tree

Query

Tree Query

1st & 2nd Reduction

Normal Form

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Efficient Query EvaluationExp = E(Tts) -1; E(Ttt) ; 2 (E(Trt)); E(Ttd)

E(Tts) , E(Ttt), E(Trt), E(Ttd) are DPath+

(1) expressionsDPath+(1) queries can be evaluated via

index-only plan using P(k)-Trie index. [Bre08]

[Bre08]: Sofia Brenes, Yuqing Wu, Dirk Van Gucht, Pablo Santa Cruz. Trie Indices for Efficient XML Query Evaluation. WebDB 2008.36

OutlineWhat we studiedEquivalences of query languagesNormal formResolution expressivenessEfficient query evaluationSummary and discussion

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Summary

212121 |||;|||ˆ||: EEEEEElE

Objects of study:XML document: a treePath+ language : Tree queries

Areas of study:ExpressivenessEquivalenceNormal formQuery evaluation

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Extending the Path+ language

**

Adding operators:

Will the results hold? ExpressivenessEquivalenceNormal formQuery evaluation

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Yuqing Wu, Dirk Van Gucht Indiana UniversityMarc Gyssens Hasselt UniversityJan Paredaens University of Antwerp

A Study of a Positive Fragment of Path Queries: Expressiveness, Normal Form

and Minimization

Thank you.

Questions? A Study of a Positive Fragment of Path Queries: Expressiveness, Normal Form

and Minimization

41

SIGMOD/PODS 2010 Indianapolis, Indiana, USA

Conference date: Jun, 2010Deadlines: SIGMOD early Nov, 2009

PODS early Dec, 2009

P[k]-Trie IndexA1

D1C2

B3B2C1

B4A2B1

C3

B5

C4

Keep track of the P[k]-partitions

Use the reverse label path as key P

[2](A)(B)

(C)

(D)

{(A1, A1), (A2, A2)}{(B1, B1), (B2, B2), (B3, B3), (B4, B4), (B5, B5)}{(C1, C1), (C2, C2), (C3, C3), (C4, C4)}{(D1, D1)}

(A,A)(A,B)(B,B)(B,C)(B,D)

{(A1, A2)}{(A1, B1), (A2, B2), (A2, B3), (A1, B4)}{(B4, B5)}{(B1, C1), (B2, C2), (B3, C3), (B5, C4)}{(B2, D1)}

(A,A,B)(A,B,B)(A,B,C)(A,B,D)(B,B,C)

{(A1, B2), (A1, B3)}{(A1, B5)}{(A1, C1), (A2, C2), (A2, C3)}{(A2, D1)} {(B4, C4)}

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Query Evaluation with P[k]-Trie IndexQuery 1: //A/B/C

A1

D1C2

B3B2C1

B4A2B1

C3

B5

C4

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Query Evaluation with P[k]-Trie IndexQuery 2: //B/C

A1

D1C2

B3B2C1

B4A2B1

C3

B5

C4

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Query Evaluation with P[k]-Trie IndexQuery 3: //A/B[./D]/C A1

D1C2

B3B2C1

B4A2B1

C3

B5

C4

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Query Evaluation with P[k]-Trie IndexQuery 3: //A/B[./D]/C A1

D1C2

B3B2C1

B4A2B1

C3

B5

C4

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