1. Buffering in Query Evaluation over XML Streams Ziv Bar-Yossef Technion Marcus Fontoura Vanja Josifovski IBM Almaden Research Center

2. 2 XML Document 1: 2: 3: 4: Intro 5: 6: 7: bla bla bla 8: 9: 10: 11: 12: Results 13: 14: 15: yada yada yada 16: 17: 18: 19: 20: Conclusions 21: 22: 23: etc etc etc 24: 25: 26: On the Complexity of Database Queries 27: 28: 29: Papadimitriou 30: 31: 32: Yannakakis 33: 34:

3. 3 content XML Document Tree paper title section id title root section id title On the Complexity of Database Queries Intro 2 author content Papadimitriou Yannakakis Results yada yada yada section id title 1 etc etc etc 3 content Conclusions bla bla bla

4. 4 XPath Queries Results yada yada yada content paper title section id title root section id title On the Complexity of Database Queries Intro 2 author content Papadimitriou Yannakakis section id title 1 etc etc etc 3 content Conclusions bla bla bla /paper[author=“Papadimitriou”]/section[@id = “2” or title = “Intro”]/content

5. 5 XPath Queries Results yada yada yada content paper title section id title root section id title On the Complexity of Database Queries Intro 2 author content Papadimitriou Yannakakis section id title 1 etc etc etc 3 content Conclusions bla bla bla /paper[title != section/title]/author

6. 6 XPath Query = path pattern + predicates XPath 2.0 Forward axis only Eval(Q,D): nodes in D that match Q Two modes of XPath evaluation: Full fledged evaluation: given Q,D, output Eval(Q,D) Filtering: given Q,D, determine whether Eval(Q,D) is nonempty.

7. 7 XML Streams XML stream: sequence of SAX events startDocument(), endDocument(), startElement(name), endElement(name), text(str) Why XML streams? For transferring XML between systems For efficient access to large XML documents Critical resources Memory Processing time

8. 8 Streaming XML Algorithms XFilter and YFilter [Altinel and Franklin 00] [Diao et al 02] X-scan [Ives, Levy, and Weld 00] XMLTK [Avila-Campillo et al 02] XTrie [Chan et al 02] SPEX [Olteanu, Kiesling, and Bry 03] Lazy DFAs [Green et al 03] The XPush Machine [Gupta and Suciu 03] XSQ [Peng and Chawathe 03] FluX [Koch el al 04] TurboXPath [Josifovski, Fontoura, and Barta 05] … All of them use lots of memory on certain queries & documents All of them use lots of memory on certain queries & documents

9. 9 Memory Bottleneck I : Storage of Large Transition Tables Framework of most algorithms: Q  NFA Simulate NFA by DFA Caveat: exponential blowup However: exponential blowup is not necessary [Bar-Yossef, Fontoura, Josifovski 04] Algorithm for filtering XML streams whose space is linear in the query size

10. 10 Memory Bottleneck II : Buffering of Document Fragments Scenario 1: buffering nodes, which may or may not be part of the output. Results yada yada yada content paper title section id title root section id title On the Complexity of Database Queries Intro 2 author content Papadimitriou Yannakakis section id title 1 etc etc etc 3 content Conclusions bla bla bla /paper[author=“Papadimitriou”]/section[@id = “2” or title = “Intro”]/content

11. 11 Memory Bottleneck II : Buffering of Document Fragments Scenario 2: buffering nodes needed for evaluating pending predicates. Results yada yada yada content paper title section id title root section id title On the Complexity of Database Queries Intro 2 author content Papadimitriou Yannakakis section id title 1 etc etc etc 3 content Conclusions bla bla bla /paper[title != section/title]/author

12. 12 Memory Bottleneck II : Buffering of Document Fragments Scenario 3: buffering multiple candidate matches that are nested within each other. a root c a b a c b //a[b and c] Relevant only when document is “recursive” Space required:  (doc-recursion-depth) [Bar-Yossef, Fontoura, Josifovski 04]

13. 13 Our Results Quantitative space lower bounds for: Full-fledged evaluation of queries with predicates (Scenario 1) Filtering/full-fledged evaluation of queries with “multi-variate” predicates (Scenario 2) Matching upper bound Eager evaluation of predicates In all other scenarios: no buffering required Filtering of queries with “univariate” predicates over non-recursive documents is possible without buffering [Bar-Yossef, Fontoura, Josifovski 04]

14. 14 Related Work Space complexity of XPath evaluation over non- streaming XML documents [Gottlob, Koch, Pichler 03], [Segoufin 03] Space complexity of XPath evaluation over streams of indexed XML data [Choi, Mahoui, Wood 03] Space complexity of select-project-join queries over relational data streams [Arasu et al 02]

15. 15 Document Concurrency Q: query D =  1,…,  n : document Each  i is an SAX event  t = (  1,…,  t ) Definition: x  D is alive at step t if x   t and    s.t. x  Eval(Q,  t  )  x  Eval(Q,  t  ) t-concurrency(D,Q): number of nodes that are alive at step t concurrency(D,Q): max t t-concurrency(D,Q)

16. 16 Concurrency: Example 1: 2: 3: 4: Intro 5: 6: 7: bla bla bla 8: 9: 10: 11: 12: Results 13: 14: 15: yada yada yada 16: 17: 18: 19: 20: Conclusions 21: 22: 23: etc etc etc 24: 25: 26: 27: On the Complexity of Database Queries 28: 29: 30: Papadimitriou 31: 32: 33: Yannakakis 34: 35: alive dead /paper[author=“Papadimitriou”]/section[@id = “2” or title = “Intro”]/content

17. 17 Lower Bound Notions A “normal” lower bound: For every algorithm A, there exist Q and D s.t. A uses on Q and D  (concurrency(D,Q)) bits of space. Q and D may be “pathological” Doesn’t say much about real-world queries/documents An “ideal” lower bound: For every A, every Q, and every D, A uses on Q and D  (concurrency(D,Q)) bits of space. Too good to be true A can have D and Q “hard-coded”, and then know the result a priori Space of A on D and Q = minimum description length of Q and D

18. 18 Our Lower Bound Theorem: For every A, every Q, and every D, there exists an almost isomorphic document D’, s.t. A uses on Q and D’,  (concurrency(D,Q)) bits of space. D’ is the same as D, except for a few extra empty nodes with auxiliary names. Theorem holds only if: Q is “star-free” D is non-recursive

19. 19 Why isn’t this Obvious? Reason 1: we want the theorem to work for every Q and D, not only ones with high MDL. Reason 2: Obvious: If x is alive at step t  A has to remember x Because: A may or may not need to output x Not obvious: If x and y are alive at step t  A has to remember both If x and y are not “independent”, maybe it’s enough to remember just x (or just y)

20. 20 Proof of Lower Bound C = t-concurrency(D,Q) x 1,…,x C = nodes that are alive at step t Recall: for every x i there exist  i and  i s.t. x i  Eval(Q,  t  i ) x   Eval(Q,  t  i ) Lemma: there exist a single  and a single  s.t. for all i, x i  Eval(Q,  t  ) x i  Eval(Q,  t  )

21. 21 Proof of Lower Bound (cont.) For every S  { 1,…,C } define document D S : D S is the same as D, except For every i  S, we “mark” x i Marking: an extra empty child with an auxiliary name Note: D S is almost-isomorphic to D A = any algorithm Note: From output of A on D S, one can “reconstruct” the set S.

22. 22 Proof of Lower Bound (cont.) Consider state of A at step t when running on D S If suffix = , none of the x i ’s should be output  A could not have output any x i by step t If suffix = , no information in suffix about S but S can be reconstructed from output  state of A at step t must have all information about S Conclusion: space  ≥  (C) Actual proof: by one-way communication complexity

23. 23 Conclusions Our contributions: Quantitative space lower bounds Full-fledged evaluation of queries with predicates Filtering/full-fledged evaluation of queries with “multi- variate” predicates Matching upper bound Open problems: Quantitative lower bounds for XQuery evaluation over streams Address larger fragments of XPath