Paper: "Performance Analysis of Algorithms to Reason about XML Keys"

1. Performance Analysis of Algorithms to
Reason about XML Keys
Emir Mu˜oz Jim´nez
n e
University of Santiago de Chile
Yahoo! Labs LatAm
Joint work with F. Ferrarotti, S. Hartmann, S. Link, M. Marin
DEXA 2012 @ Vienna, Austria, 3rd September 2012
Introduction RW & Motivation XML Keys Results Conclusion 1/24

2. Contribution
An efficient implementation of an algorithm that decides the
implication problem for a tractable and expressive class of
XML keys.
Performance Analysis:
Implication problem over large sets of XML Keys.
Non-redundant covers of XML keys and validation of XML
documents.
Our experiments show that reasoning about expressive notions
of XML keys can be done efficiently in practice and scales well.
Introduction RW & Motivation XML Keys Results Conclusion 2/24

3. Outline
1 Introduction
2 Related Work & Motivation
3 Reasoning about XML Keys
4 Results
5 Conclusion
Introduction RW & Motivation XML Keys Results Conclusion 3/24

4. Introduction
Keys
XML: de-facto standard for Web data exchange and
integration.
XML ↔ data storage ↔ data processing.
Both industry and academia have long since recognized the
importance of keys in XML data management.
Several notions of XML keys have been proposed. Most
influential is due to Buneman et al. who defined keys on the
basis of an XML tree model similar to the one suggested by
DOM and XPath.
Introduction RW & Motivation XML Keys Results Conclusion 4/24

5. Introduction
XML Tree Example (1/2)
Example (An XML tree representing an XML document)
db
E
project project
E pname E
pname team
team tname
E tname A E
A
A Riders MediaLow
employee A Coyotes
Phobia employee employee employee
rol rol rol employee rol rol rol employee
E E E E E E
A A A A A A
Team name Team name Team name Team name Team name Team name
Member Member Leader Member Member Leader
E E E E E E
fname lname fname lname fname lname fname lname fname lname fname lname
E E E E E E E E E E E E
S S S S S S S S S S S S
Dexter Cooper John Smith William Bell Dexter Cooper William Bell Brook Davis
Introduction RW & Motivation XML Keys Results Conclusion 5/24

6. Introduction
XML Tree Example (2/2)
Some Constraints: (gathered as business rules)
(a) A project node is identified by pname, no matter where the
project node appears in the document.
(b) A team node can be identified by tname relatively to a
project node.
(c) Within any given subtree rooted at team, an employee node
is identified by name.
Introduction RW & Motivation XML Keys Results Conclusion 6/24

7. Introduction
XML Tree Example (2/2)
Some Constraints: (gathered as business rules)
(a) A project node is identified by pname, no matter where the
project node appears in the document. Absolute key
(b) A team node can be identified by tname relatively to a
project node. Relative key
(c) Within any given subtree rooted at team, an employee node
is identified by name. Relative key
Introduction RW & Motivation XML Keys Results Conclusion 6/24

8. Related Work & Motivation
Keys are important in many perennial tasks in database
management.
The hope is that keys will turn out to be equally beneficial for
XML.
In practice, expressive yet tractable notions of XML keys have
been ignored so far.
We initiate an empirical study of the fragment of XML keys
with nonempty sets of simple key paths.
One of the most fundamental questions on keys is:
Can we decide if a new key holds given a set of known keys?
(Logical Implication Problem.)
Introduction RW & Motivation XML Keys Results Conclusion 7/24

9. Related Work & Motivation
Key implication example
Some Constraints: (gathered as business rules)
(a) A project node is identified by pname, no matter where the
project node appears in the document.
(b) A team node can be identified by tname relatively to a
project node.
(c) Within any given subtree rooted at team, an employee node
is identified by name.
Suppose, the consideration of a further key:
(d) A project node can be identified in the document by its child
nodes pname and team.
Introduction RW & Motivation XML Keys Results Conclusion 8/24

10. Related Work & Motivation
Key implication example
Some Constraints: (gathered as business rules)
(a) A project node is identified by pname, no matter where the
project node appears in the document.
(b) A team node can be identified by tname relatively to a
project node.
(c) Within any given subtree rooted at team, an employee node
is identified by name.
Suppose, the consideration of a further key:
(d) A project node can be identified in the document by its child
nodes pname and team. key (a) actually implies (d)!!
because (d) is a superkey of (a).
Introduction RW & Motivation XML Keys Results Conclusion 8/24

11. Keys for XML
Definitions (1/2)
Definition (Value equality)
u is value equal to v (u =v v) if the trees rooted at u and v are
isomorphic by an isomorphism that is the identity on string values.
(e.g., element nodes employee for “Dexter Cooper”)
db
E
project project
E pname E
pname team
team tname
E tname A E
A
A Riders MediaLow
employee A Coyotes
Phobia employee employee employee
rol rol rol employee rol rol rol employee
E E E E E E
A A A A A A
Team name Team name Team name Team name Team name Team name
Member Member Leader Member Member Leader
E E E E E E
fname lname fname lname fname lname fname lname fname lname fname lname
E E E E E E E E E E E E
S S S S S S S S S S S S
Dexter Cooper John Smith William Bell Dexter Cooper William Bell Brook Davis
Introduction RW & Motivation XML Keys Results Conclusion 9/24

12. Keys for XML
Definitions (2/2)
Some Constraints: (in class K of XML keys)
(a) A project node is identified by pname, no matter where the
project node appears in the document.
(ε, (project , {pname}))
(b) A team node can be identified by tname relatively to a
project node.
(project , (team, {tname}))
(c) Within any give subtree rooted at team, an employee node is
identified by name.
( ∗ .team, (employee, {name}))
(d) A project node can be identified by its child nodes pname
and team.
(ε, (project , {pname, team}))
Introduction RW & Motivation XML Keys Results Conclusion 10/24

13. Deciding XML Key Implication
Notions
We say that Σ implies ϕ, denoted by Σ |= ϕ, iff every finite
XML tree T that satisfies all σ ∈ Σ also satisfies ϕ.
Implication Problem for a class C of XML keys: given any
Σ ∪ {ϕ} in C, decide whether Σ |= ϕ.
e.g., let be Σ = {(a), (b), (c)} and ϕ = (d), then Σ |= ϕ is
true.
e.g., let be Σ = {(a), (b)} and ϕ = (c), then Σ |= ϕ is false.
Hartmann & Link characterize the implication problem for the
class K of XML keys in terms of the of reachability problem
for fixed nodes in a suitable digraph.
Introduction RW & Motivation XML Keys Results Conclusion 11/24

14. Deciding XML Key Implication
Mini-trees and Witness Graphs (Hartmann & Link, 2009)
ϕ = (ε, (public. ∗ .project , {pname.S , year .S }))
rϕ = q ϕ
p 1111
0000
1111
0000 (1)
1111
0000 qϕ
1111
0000
1111
0000
rϕ1111
′ 0000 (2) E db
1111
0000
1111
0000
1111
0000
′
′
v1 E public ❦
qϕ E db
′
v1 E public
p
′
′ v2 E national E public
v2 E national
′
′ qϕ E project E national
qϕ E project (3)
′
w1 w1
pname ′❦
11111
00000 ϕ
E E year qϕ E project
r10000
ϕ1111
1111
0000 11111
00000r
11111
00000 2
1111
0000
1111
0000 11111
00000 xϕ xϕ
11111
00000 ′ 1 S S 2 pname E E year
p1 w1 w1
E pname E year
p2 × ×
S S
xϕ
1 S S xϕ
2 × ×
(a) Mini tree TΣ,ϕ (b) Witness graph GΣ,ϕ
Introduction RW & Motivation XML Keys Results Conclusion 12/24

15. An Efficient Implementation
Data structures and implementation
Suitable data structures for Mini-tree & Witness-graph.
L
Node 0 nodeEle
qϕ E db
vertexEle
(db)
Node 1
edgeEle
E public Node 1 nodeEle
vertexEle
(public)
Node 2
edgeEle
E national
Node 2 nodeEle
vertexEle
(national)
′ Node 3 Node 0
qϕ E project edgeEle edgeEle
Node 3 nodeEle
pname vertexEle
E E year (project)
Node 4 Node 6 Node 1
..... edgeEle edgeEle edgeEle
S S
Node 7
× × (S)
(c) Witness graph GΣ,ϕ (d) Adj. list for GΣ,ϕ
Introduction RW & Motivation XML Keys Results Conclusion 13/24

16. XML Key Reasoning for Document Validation
Applications
Fast algorithms for the validation of XML documents against
keys are crucial to ensure the consistency and semantic
correctness of data stored or exchanged between applications.
We use our implementation of the implication problem to
compute non-redundant covers for sets of XML keys.
Which in turn can be used to significantly speed up the
process of XML document validation against sets of XML
keys.
Introduction RW & Motivation XML Keys Results Conclusion 14/24

17. XML Key Reasoning for Document Validation
Cover Sets for XML Keys
Fact
Σ is non-redundant if there is no key ψ ∈ Σ such that
Σ − {ψ} |= ψ.
Algorithm 1: Non-redundant Cover for XML keys
Input : Finite set Σ of XML keys
Output: A non-redundant cover for Σ
1 Θ = Σ;
2 foreach key ψ ∈ Σ do
3 if Θ − {ψ} |= ψ then
4 Θ = Θ − {ψ};
5 return Θ;
6
This set can be computed in O(|Σ| × (max{|ψ| : ψ ∈ Σ})2 ) time.
Introduction RW & Motivation XML Keys Results Conclusion 15/24

18. Experimental Results
Performance Analysis
We analyze:
(i) The scalability of the implication problem.
(ii) The viability of computing non-redundant cover sets to speed
up the validation of XML documents against XML keys.
For (i) we generated large sets of XML keys in the following
two systematic ways:
1. using a manually defined set of 5 to 10 XML keys as seeds, we
computed new implied keys by successively applying the
inference rules from the axiomatization presented by
Hartmann & Link (2009).
2. we defined some non-implied XML keys, adding witness edges
′
keeping qϕ not reachable from qϕ .
For (ii) we generated sets of keys following the same strategy.
Introduction RW & Motivation XML Keys Results Conclusion 16/24

19. Experimental Results – Datasets (1/2)
Table: XML Documents.
Doc ID Document No. of No. of Size Max. Average
Elements Attributes Depth Depth
Doc1 321gone.xml 311 0 23 KB 5 3.76527
Doc2 yahoo.xml 342 0 24 KB 5 3.76608
Doc3 dblp.xml 29,494 3,247 1.6 MB 6 2.90228
Doc4 nasa.xml 476,646 56,317 23 MB 8 5.58314
Doc5 SigmodRecord.xml 11,526 3,737 476 KB 6 5.14107
Doc6 mondial-3.0.xml 22,423 47,423 1 MB 5 3.59274
Introduction RW & Motivation XML Keys Results Conclusion 17/24

20. Experimental Results – Datasets (2/2)
SigmodRecord (seed keys)
(ε, (issue, {volume.S, number.S}))
(ε, ( ∗ .issue, {volume.S, number.S})) Interaction rule
(issue, (articles, {article.title.S})) (Q, (Q′ , {P.P1 , . . . , P.Pk })),
(issue.articles, (article, {title.S})) (Q.Q′ , (P, {P1 , . . . , Pk }))
(issue, (articles.article, {initP age.S, endP age.S})) (Q, (Q′ .P, {P1 , . . . , Pk }))
(ε, (issue.articles.article.authors.author, {position})) (issue, (articles.article, {title.S}))
Sigmod.xml
<SigmodRecord>
<issue>
<volume>13</volume>
<number>2</number>
<articles>
<article>
<title>Deadlock Detection is Cheap.</title>
<initPage>19</initPage>
<endPage>34</endPage>
<authors>
<author position="00">Rakesh Agrawal</author>
<author position="01">Michael J. Carey</author>
<author position="02">David J. DeWitt</author>
</authors>
</article> ...
</articles>
</issue> ...
</SigmodRecord>
Introduction RW & Motivation XML Keys Results Conclusion 18/24

21. Experimental Results
Strategies to generate implied and non-implied XML keys
Non-implied keys
If we apply the interaction rule to
E db
- (issue, (articles, {article.title.S}))
qϕ E conference
- (issue.articles, (article, {title.S}))
E issue
We derived the implied key
- (issue, (articles.article, {title.S})) E ℓ0 = content
We applied the interaction, E articles
context-target, subnodes, context-path
E article
containment, target-path containment,
′
subnodes-epsilon and prefix-epsilon rules qϕ E author
whenever possible. first E E last
S S
Introduction RW & Motivation XML Keys Results Conclusion 19/24

22. Results for Implication Problem
Deciding implication of XML keys
2.5
abs-abs mix-rel
1.8 abs-rel wildcard
rel-abs
1.6 rel-rel
mix-abs 2
mix-rel
Time [ms]
Time [ms]
1.4
1.2
1.5
1
0.8
1
0.6
0.4 0.5
0.2
0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Size of Σ Size of Σ
(e) XML key implication, all cases. (f) Effect of wildcards presence.
Figure: Performance of the Algorithm for the Implication of XML Keys
(Σ |= ϕ).
Introduction RW & Motivation XML Keys Results Conclusion 20/24

23. Results for Validation Problem
Document Validation (1/2)
XML doc. Key Set Time[ms]
321gone & Processed Keys: 23 3.458 35000 >30min >25min 1.88min 5min
Full-set
yahoo Discarded keys: 15 Cover-set
(Doc1 & 2) Cover set: 8 keys 30000
DBLP Processed Keys: 36 12.757
25000
Running Time [ms]
(Doc3) Discarded keys: 24
Cover set: 12 keys 20000
nasa Processed Keys: 35 9.23
(Doc4) Discarded keys: 28 15000
Cover set size: 7 keys
10000
Sigmod Processed Keys: 24 5.294
Record Discarded Keys: 19 5000
(Doc5) Cover set: 5 keys 952ms 1329ms
mondial Processed Keys: 26 4.342 0
Doc1 Doc2 Doc3 Doc4 Doc5 Doc6
(Doc6) Discarded Keys: 16
XML documents
Cover set: 10
(a) Non-redundant Cover Sets. (b) Validation Against Cover Sets.
Figure: Non-redundant Cover Sets of XML keys and Validation of XML
Documents
Introduction RW & Motivation XML Keys Results Conclusion 21/24

24. Results for Validation Problem
Document Validation (2/2)
There are some very expensive XPath queries.
SigmodRecord
(issue, (articles.article, {title.S})) - there are 1503 nodes
in the XPath query “//issue/articles/article”.
(ε, (issue.articles.article.authors.author, {S})) - there are
3737 nodes in the XPath query
“//issue/articles/article/authors/author”.
Introduction RW & Motivation XML Keys Results Conclusion 22/24

25. Main conclusion
This work was motivated by two objectives:
1 To demonstrate that there are expressive classes of XML keys that can be
reasoned about efficiently.
2 To show that our observations on the problem of deciding implication has
immediate consequences for other perennial task in XML database
management.
[1.1] We studied a fragment of XML keys with nonempty sets of simple key
paths.
[1.1.1] Presenting an efficient implementation for the implication problem.
[1.1.2] Showing through experiments that the proposed algorithm runs fast in
practice and scales well.
[2.1] We studied the problem of validating an XML document against a set of
XML keys.
[2.2] Presenting an optimization method for this validation via the
computation of non-redundant cover sets of XML keys.
[2.3] Our experiments show that enormous time savings can be achieved in
practice.
Introduction RW & Motivation XML Keys Results Conclusion 23/24

26. Questions?
THANKS!
Emir Mu˜oz Jim´nez– emir@emunoz.org
n e
Introduction RW & Motivation XML Keys Results Conclusion 24/24

27. Axiomatization
∗
Let be P L from the grammar Q → ℓ | ε | Q.Q |
where ℓ ∈ L is a label, ε is the empty word, “.” is the concat
operator, and “ ∗ ” is the wildcard.
For nodes v and v ′ of an XML tree T , the value intersection
of v[[Q]] and v ′ [ Q]] is given by
v[[Q]] ∩v v ′ [ Q]] = {(w, w′ ) | w ∈ v[[Q]], w′ ∈ v ′ [ Q]], w =v w′ }
We define semantic closure by: Σ∗ = {ϕ ∈ C | Σ |= ϕ}
and also syntactic closure by: Σ+ = {ϕ | Σ ⊢ℜ ϕ}
ℜ
A set of rules ℜ is sound (complete) if Σ+ ⊆ Σ∗ (Σ∗ ⊆ Σ+ )
ℜ ℜ
A sound and complete set of rules is called axiomatization
1/2

28. Deciding XML Key Implication
Mini-trees and Witness Graphs (2/2)
Theorem (Hartman & Link, 2009)
Let Σ ∪ {ϕ} be a finite set of keys in the class K. We have Σ |= ϕ
′
if and only if qϕ is reachable from qϕ in GΣ,ϕ .
Algorithm 2: XML key implication in K
Input : finite set of XML keys Σ ∪ {ϕ} in K
Output: yes, if Σ |= ϕ; no, otherwise
1 Construct GΣ,ϕ for Σ and ϕ;
′
2 if qϕ is reachable from qϕ in G then return yes; else return no; end if
2/2