Performance Analysis of Algorithms to   Reason about XML Keys   Emir Mu˜oz Jim´nez          n      e   University of Santi...
Contribution          An efficient implementation of an algorithm that decides the          implication problem for a tracta...
Outline   1 Introduction   2 Related Work & Motivation   3 Reasoning about XML Keys   4 Results   5 ConclusionIntroduction...
Introduction   Keys          XML: de-facto standard for Web data exchange and          integration.          XML ↔ data st...
Introduction                    XML Tree Example (1/2)   Example (An XML tree representing an XML document)               ...
Introduction           XML Tree Example (2/2)   Some Constraints: (gathered as business rules)   (a) A project node is ide...
Introduction           XML Tree Example (2/2)   Some Constraints: (gathered as business rules)   (a) A project node is ide...
Related Work & Motivation          Keys are important in many perennial tasks in database          management.          Th...
Related Work & Motivation           Key implication example   Some Constraints: (gathered as business rules)   (a) A proje...
Related Work & Motivation           Key implication example   Some Constraints: (gathered as business rules)   (a) A proje...
Keys for XML                    Definitions (1/2)   Definition (Value equality)   u is value equal to v (u =v v) if the tree...
Keys for XML           Definitions (2/2)   Some Constraints: (in class K of XML keys)   (a) A project node is identified by ...
Deciding XML Key Implication           Notions          We say that Σ implies ϕ, denoted by Σ |= ϕ, iff every finite        ...
Deciding XML Key Implication           Mini-trees and Witness Graphs (Hartmann & Link, 2009)                              ...
An Efficient Implementation           Data structures and implementation          Suitable data structures for Mini-tree & W...
XML Key Reasoning for Document Validation           Applications          Fast algorithms for the validation of XML docume...
XML Key Reasoning for Document Validation           Cover Sets for XML Keys   Fact   Σ is non-redundant if there is no key...
Experimental Results           Performance Analysis          We analyze:            (i) The scalability of the implication...
Experimental Results – Datasets (1/2)                                   Table: XML Documents.          Doc ID   Document  ...
Experimental Results – Datasets (2/2)   SigmodRecord (seed keys)   (ε, (issue, {volume.S, number.S}))   (ε, ( ∗ .issue, {v...
Experimental Results            Strategies to generate implied and non-implied XML keys                                   ...
Results for Implication Problem                           Deciding implication of XML keys                                ...
Results for Validation Problem           Document Validation (1/2)        XML doc.    Key Set                  Time[ms]   ...
Results for Validation Problem           Document Validation (2/2)          There are some very expensive XPath queries.  ...
Main conclusion   This work was motivated by two objectives:      1 To demonstrate that there are expressive classes of XM...
Questions?                                            THANKS!                                              Emir Mu˜oz Jim´...
Axiomatization                                                               ∗Let be P L from the grammar Q → ℓ | ε | Q.Q ...
Deciding XML Key Implication         Mini-trees and Witness Graphs (2/2)  Theorem (Hartman & Link, 2009)  Let Σ ∪ {ϕ} be a...
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DEXA 2012 Talk

  1. 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 2012Introduction RW & Motivation XML Keys Results Conclusion 1/24
  2. 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. 3. Outline 1 Introduction 2 Related Work & Motivation 3 Reasoning about XML Keys 4 Results 5 ConclusionIntroduction RW & Motivation XML Keys Results Conclusion 3/24
  4. 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. 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 DavisIntroduction RW & Motivation XML Keys Results Conclusion 5/24
  6. 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. 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 keyIntroduction RW & Motivation XML Keys Results Conclusion 6/24
  8. 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. 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. 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. 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 DavisIntroduction RW & Motivation XML Keys Results Conclusion 9/24
  12. 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. 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. 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,ϕIntroduction RW & Motivation XML Keys Results Conclusion 12/24
  15. 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,ϕIntroduction RW & Motivation XML Keys Results Conclusion 13/24
  16. 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. 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 Σ1 Θ = Σ;2 foreach key ψ ∈ Σ do3 if Θ − {ψ} |= ψ then4 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. 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. 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 3.59274Introduction RW & Motivation XML Keys Results Conclusion 17/24
  20. 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. 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 SIntroduction RW & Motivation XML Keys Results Conclusion 19/24
  22. 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. 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 DocumentsIntroduction RW & Motivation XML Keys Results Conclusion 21/24
  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. 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. 26. Questions? THANKS! Emir Mu˜oz Jim´nez– emir@emunoz.org n eIntroduction RW & Motivation XML Keys Results Conclusion 24/24
  27. 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 concatoperator, and “ ∗ ” is the wildcard.For nodes v and v ′ of an XML tree T , the value intersectionof v[[Q]] and v ′ [ Q]] is given byv[[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} ℜA set of rules ℜ is sound (complete) if Σ+ ⊆ Σ∗ (Σ∗ ⊆ Σ+ ) ℜ ℜA sound and complete set of rules is called axiomatization 1/2
  28. 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, otherwise1 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

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