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 % ?On the Memory Requirements of XPath Evaluation over XML Streams@@(NZiv Bar-Yossef Marcus Fontoura Vanja Josifovski IBM Almaden Research Center *OP3Preliminaries: XML  Preliminaries: XPath 1.0  XML Streams  Streaming XML AlgorithmsXFilter 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] TurboXPath [Josifovski, Fontoura, and Barta 04] & S+3333"3 333 &33 ;       Our ResultsSpace lower bounds for evaluating XPath on XML streams A streaming XML algorithm Matches the lower bounds on a large fragment of the language Uses space sub-linear in the query size rather than exponential in the query sizexS$lP AH   Related Work  Data Complexity [Vardi 82]  3F(Q,D) Evaluation function of a query Q on document D. FQ(D) Evaluation function of a fixed query Q on document D. Data complexity on Q: Complexity of best algorithm for FQ on worst D. Worst-case data complexity: maxQ (complexity of FQ). We characterize the data complexity of FQ separately for each Q (not just the worst-case one). e3323333/3JB(%xA+XPath Fragment B,XPath Fragment (cont.)D.XPath Fragment (cont.) Query Frontier Size Document Recursion Depth Document Depth  New algorithm #Proof of Theorem 1 Critical Document   Main Lemmas  One-way Communication Complexity !  3! Reduction  3"Fooling Set Technique -&Proof of Lemma 1 .'Proof of Lemma 1 Proof of Lemma 1 (cont) E/Proof of Claim (example) 2 +$ AlgorithmUses the query as an NFA Based on three global data structures Pointer array Validation array Level array Matches the lower bounds for a fragment of XPath.6?+2?+2$Algorithm Example Run %&' (!)"*#1*Conclusion: our Contributions!Space lower bounds on the instance data complexity of XPath on XML streams: In terms of Query Frontier Size In terms of Document Recursion Depth In terms of Document Depth A streaming XML algorithm Matches the lower bounds on a fragment of the language Does not use finite-state automataL$ald$ZlP3 33 3 Z  XPath 1.0   XPath 1.0 ,% Reduction  3/P456789:;= > ? @  ` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PT     o (`0p>>  ''(  ZB  c $D"`   6D "M }  X Click to edit Master title style!!  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FS Y paper_cnt  3 F _ S M R F `@ S  g0 F a S  0.  b s *(_zC"?  :PODS 2f c s *XczC"?o  @ Josifovski 2 f d s *fzC"?o ;Fagin 2f e s * jzC"?&z 71 2f f s *lnzC"?C0* 73 2f g s *qzC"?dK Dx0& 2  h s *luzC"?5   Dx1& 2  i s *yzC"?5 ,  Dx2& 2  j s *~zC"?x W_  Dx3& 2  k s *dzC"? <  Dx6& 2  l s *4zC"? 4  Dx4& 2  m s *zC"?   Dx5& 2  n s *zC"? b  Dx7& 2  o s *zC"?Z 0A  Dx8& 2 H  0޽h ? %#& #$' #/5/16/37VQWQSXSTYSUZQR[Q\_\]`\^a 3fffff3̙3f̙___PPT10.IpU+Ӭ!DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(+8+0+-z +J  g  &(  r  S zM }  z   0z * y5XML stream: XML document arriving as a one-way stream(60 2 *$  0lzP lm",$ 0 ,Critical resources: Memory Processing timeH0 21 2g  0tz^!30 ,$ 0 aWhy XML streams? For transferring XML between systems For efficient access to large XML documentsH0 2Q1 2PjH  0޽h ? 3ff~\T___PPT104.IPFY+TD' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+z ++0+z +} g $(  r  S zM }  z r  S z;  z H  0޽h ? 3ff~___PPT10i.I`?+D=' = @B + g n(  r  S pM }     S z; `<$@ 0  *Ft`H  0޽h ? 3ff~___PPT10.I+EDZ' = @B D' = @BA?%,( < +O%,( < +DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9S%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ g ((  (r ( S 4zM }  z  ( 0&! ; ` 4Space 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] sZZZ* 3+363 33 33tIN\ H ( 0޽h ? 3ff~___PPT10i.vJ`+D=' = @B +  g ( (  r  S |M }  |   s * | "6 ?; W<$@ 0 | *XYYhH  0޽h ? 3ff~ZR___PPT102.I`|V+;Y"vD' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ce%(+ g  (  r  S T-|M }  | *  <9! ?"6@ NNN?Ny  1. Queries are subsumption-free "!0 2! 2  6d6!S"`? O  @ conference  32  0t  C<&0 22  6$|S"`?S d  @ conference  32  0,| 1 u&  Y paper_cnt  3 2  0l|' :root3F  S vS F @ S   2  08|   Z author_cnt  3    6?"0@NNN?N   ! 6?"0@NNN?N   " <| ?"6@ NNN?N   F< 30&0 2 # < | ?"6@ NNN?N *  D> 30$0 2H  0޽h ?       ! 3ff~___PPT10i.Op1j+D=' = @B +[ g rj (  x  c $ĄM }     < ?"6@ NNN?Nb[ f(3. Queries consist of conjunctions only ")0 2)*  <؊ ?"6@ NNN?Nf B4. Queries are  star-restricted ""0 2"H  0޽h ? 3ff~___PPT10i.Op1j+D=' = @B +!" g  <(  r  S M }   J ' 0( D<4___PPT9 BFrontier at u: u, its siblings, and the siblings of its ancestors.*C1 2 68X\ , s *C"?! ,$D 0 Theorem 1: For all queries Q in the fragment, stream-space(FQ) = W(FrontierSize(Q)).W0 2  333,7 - 0<6yV R Definitions:* 0 2 P . 0( H@8___PPT9 *FrontierSize(Q): size of largest frontier.*+1 28X2 / 6йS"`?Gv @ conference  32 0 0/ $  :name32 1 0 :root3F 2@ S RGF 3 S vRnd  4 0ŁrWa =Query0 22 5 0ʁ"`S .  =speaker32 6 0́"` ,  :name32 7 0 с R v  Y paper_cnt  3 L 8@ c $vRS L 9 c $  L :@ c $ 4  ; 6ց ?"6@ NNN?Nv ]  F= PODS$0 2 < <ځ ?"6@ NNN?NP ~7  C> 1$0 2H  0޽h ?_1/2/03/5856957: 3ff~___PPT10.IX]+RWDS' = @B D' = @BA?%,( < +O%,( < +D' =%(Dy' =%(D!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*7D' =1:B solid*a3>Bfill.type<*7D' =1:B true*]3>Bfill.on<*7D# ' =%(D ' =%(D!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*7D' =1:B solid*a3>Bfill.type<*7D' =1:B true*]3>Bfill.on<*7D!' =4@BBBB%(D' =