Roadmap. XML Indexing. DataGuide example. DataGuides. Strong DataGuides. Multiple DataGuides for same data. CPS Topics in Database Systems

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1 Roadmap XML Indxing CPS Topics in Databas Systms Indx fabric Coopr t al. A Fast Indx for Smistructurd Data. VLDB, 2001 DataGuid Goldman and Widom. DataGuids: Enabling Qury Formulation and Optimization in Smistructurd Databass. VLDB, 1997 T-indxs Milo and Suciu. Indx Structurs for Path Exprssions. ICDT, 1997 Som rcnt paprs Grust; Chung t al.; Kaushik t al., SIGMOD, 2002 Kaushik t al., ICDE, DataGuids DataGuid xampl Can handl graph data and arbitrary rgular path xprssions Givn a smistructurd/xml databas instanc DB, a DataGuid for DB is a graph G such that: Evry labl path in DB also occurs in G Complt covrag Evry labl path in G also occurs in DB Accurat covrag (no bogus path) Evry labl path in G (starting from a particular objct) is uniqu (i.., G is a DFA) Efficint sarch: to procss a labl path of lngth n, just xamin n nods in G 3 Databas DataGuid Each nod in th DataGuid can point to a st of databas nods 4 Multipl DataGuids for sam data Strong DataGuids Lt p, p b two labl path xprssions and G a graph; dfin p G p if p(g) = p (G) That is, p and p ar indistinguishabl on G G is a strong DataGuid for a databas DB if th quivalnc rlations G and DB ar th sam Databas DataGuid G 1 DataGuid G 2 Which is bttr? 5 Exampl G 1 is strong; G 2 is not A.C(DB) = { 5 }, B.C(DB) = { 6, 7 } Not qual A.C(G 2 ) = { 20 }, B.C(G 2 ) = { 20 } Equal 6

2 Siz of DataGuids If DB is a tr, thn G DB Linar construction tim In th worst cas, howvr, th siz of a strong DataGuid may b xponntial in DB o1 o2 A B B B A o3 B T-indxs Can handl graph data and, in gnral, multipl path xprssions chaind in squnc 1-indx indxs all objcts rachabl through an arbitrary path xprssion P from a root 2-indx indxs all pairs of objcts connctd by an arbitrary path xprssion P T-indx indxs all squncs of objcts connctd by a squnc of path xprssions o4 o5 B o6 7 8 A first attmpt at 1-indx (slid 1) Lt L v b th st of words on paths from som root nod to v l 1 l 2 l n L v = { l 1 l 2 l n root v 1 v } That is, all path quris that lad to v Dfin quivalnc rlation on th nods in DB u v if L u = L v That is, u and v ar indistinguishabl by path quris starting from th root Notation: [u] is th quivalnt class containing u 9 A first attmpt at 1-indx (slid 2) Indx is also a graph (no biggr than DB) Each indx nod corrsponds to an quivalnt class; it points to th st of DB nods in that quivalnt class Thr is an indx dg labld from s to s if thr is a DB dg labld from a nod in s to a nod in s!any accurat indx should hav at last this many nods!expnsiv to construct (PSPACE-complt) 10 1-indx Ida: us simulation/bi-simulation instad of Strongr conditions " finr quivalnc classs " mor indx nods Simulation and bi-simulation ar much asir to comput (PTIME) Dtails in othr paprs To b practical, still nd Extrnal-mmory construction algorithm Incrmntal indx updat algorithm 11 Simulation/bi-simulation (slid 1) A binary rlation ~ on DB nods is a (backward) bisimulation if If v ~ v and v is a root, thn so is v (and vic vrsa) Root nods can b bi-similar only to root nods If v ~ v, thn for any dg u v thr xists u v such that u ~ u (and vic vrsa) Edgs ar mappd consistntly u v ~ v Simulation: no vic vrsa (not symmtric in gnral) " u ~ u v ~ v 12

3 Simulation/bi-simulation (slid 2) 1-indx xampl Two nods u and v ar bi-similar (u b v)if thy ar rlatd in som bi-simulation Two nods u and v ar similar (u s v) if thr ar two simulations ~ and ~ s.t. u ~ v and v ~ u Fact: u b v u s v u v Why? x y z x s y s z x b y b z (using bi-simulation) Analyzing 1-indx For a tr-structurd DB, 1-indxs using b, s, ar all idntical to DataGuid Always: siz(1-indx) siz(db) Unlik DataGuid But w ar back to NFS; is lookup tim boundd? Always: can construct indx in O( DB log DB ) Still nd: xtrnal-mmory construction algorithm and incrmntal updat algorithm Dsignd to answr arbitrarily complx path xprssions, but such xprssions may not show up oftn in quris 15 2-indx 1-indx is for quris of th form: root x Givn P, find all x s that satisfy th qury * P 2-indx is for quris of th form: root x 1 x 2 Givn P, find all (x 1, x 2 ) pairs that satisfy th qury Again, indx is a graph What ar th nods? What ar th dgs? P 16 Nods of 2-indx Lt L (u, v) b th st of words on th paths from u to v l L (u, v) = { l 1 l 2 l n u 1 l n v } That is, all path quris that rturn (u, v) as on of its answrs Dfin quivalnc rlation on pairs of nods in DB (u, v) (u, v ) if L (u, v) = L (u, v ) That is, thy ar indistinguishabl by path quris of th form: root * x 1 P x 2 Nods in a 2-indx corrspond to quivalnt classs dfind by ; ach 2-indx nod points to [(u, v)], a st of pairs in th sam quivalnt class as (u, v) Again, w can us a rfinmnt of that is asir to comput 17 Edgs of 2-indx Dfin 2-indx dgs in a way such that: A path qury P on th 2-indx rturns a st of 2- indx nods that point to th answr to th qury * P root x 1 x 2 in DB If u u in DB, thn for ach nod v in DB, [(v, u)] [(v, u )] in th 2-indx Intuitivly, if v and u ar connctd via P, thn v and u ar connctd via P. A root of a 2-indx has th form [(u, u)] bcaus L (u, u) contains th mpty word 18

4 2-indx xampl T-indx In gnral, siz of th 2-indx may b quadratic in DB 19 T-indx handls tmplat: root x 1 x n Each T i can b A constant path xprssion, or An arbitrary path xprssion!exampl tmplat: Rstaurant x 1, x 1.Px 2!Th papr also handls an arbitrary formula (singl-stp path), but w will not considr it hr for simplicity Givn T 1,, T n, find (x 1,, x n ) tupls that satisfy th qury Quris matching th xampl tmplat: Rstaurant x 1, x 1.ownr x 2 Rstaurant x 1, x 1.managr.lastnam x 2 T 1 T 2 T n 20 Nods of T-indx Edgs of T-indx T 1 T 2 T n Qury tmplat: root x 1 x n Lt T (v1,, vi) b th languag gnratd by rgular xprssion R 1 $ R 2 $ $ R i, whr $ is a spcial symbol, and If T j rprsnts an arbitrary path xprssion, thn R j = L (vj 1, vj) If T j rprsnts a constant path xprssion, and if thr is such a path from v j 1 to v j, thn R j = S j (a spcial symbol); othrwis R j = (v 1,, v i ) (u 1,, u i ) if T (v1,, vi) = T (u1,, ui) Nods of th T-indx includ Equivalnc classs of th form [(v 1,, v i )], whr i n For ach [(v 1,, v i )] a nw nod [(v 1,, v i )] $ 21 For ach [(v 1,, v i 1, v i )] $, thr is an dg in T-indx [(v 1,, v i 1, v i )] $ $ [(v 1,, v i 1, v i, v i )] Intuition: aftr binding x i to v i, start matching T i + 1 from v i If T i rprsnts an arbitrary path xprssion If v i v i in DB, thn [(v 1,, v i 1, v i )] [(v 1,, v i 1, v i )] Intuition: can b part of T i ε [(v 1,, v i 1, v i )] [(v 1,, v i 1, v i )] $ Intuition: T i can b of any lngth and trminatd right hr If T i rprsnts a constant path xprssion T If v i i S v i in DB, thn [(v 1,, v i 1, v i )] i [(v 1,, v i 1, v i )] $ Intuition: spcial symbol S i rprsnts a complt match of T i 22 Roots, trminals, and an xampl Roots hav th form [(v)], whr v is a root of DB Trminals hav th form [(v 1,, v n 1, v n )] $ Rmov all nods not rachabl from root or not having any path to trminal [(1)] Exampl: t x 1, x 1.* x S 2 1 [(2)] $ $ [(2, 2)] ε a [(2, 2)] $ b a c d [(2, 7)] [(3, 8)] [(3, 9)] [(4, 11)] ε ε ε ε [(2, 7)] $ [(3, 8)] $ [(3, 9)] $ [(4, 11)] $ 23 Indxing XPath axs Most indxing work so far concntrats on spding up parnt-child travrsals What about othr typs of XPath axs such as following, prcding, tc.? Exampl: prcding axis contains all nods that ar bfor th contxt nod in documnt ordr, xcluding any ancstors //vnt[nam= nd ]/prcding::vnt[nam= bgin ]! Grust. Acclrating XPath Location Stps. SIGMOD,

5 Pr- and post-ordr travrsal Nod dscriptor indxing Pr-ordr travrsal (slf; lft subtr; right subtr) a, b, c, d,, f, g, h, i, j Pr-ordr ranks of nods: pr(a) = 0, pr(b) = 1, pr(c) = 2, Post-ordr travrsal (lft subtr; right subtr; slf) d,, c, b, g, i, j, h, f, a Post-ordr ranks of nods: post(d) = 0, post() = 1, Dscriptor of a nod v: dsc(v) = h pr(v), post(v), par(v), att(v), tag(v) i par(v): th pr-ordr rank of v s parnt att(v): tru if nod is attribut; fals othrwis tag(v): lmnt tag or attribut nam of v Us R-tr or B-tr on nod dscriptor tabl Ida: us ths ranks to dtrmin nod rlationship Adaptiv path indxing Most indxing work indxs all possibl paths in th data, but fw paths actually com up in quris Indx only th frquntly usd paths (mind from a qury workload)! Chung t al. APEX: An Adaptiv Path Indx for XML Data. SIGMOD, Mor XML indxing work! Kaushik t al. Exploiting Local Similarity to Efficintly Indx Paths in Graph-Structurd Data. ICDE, 2002 Instad of (bi-)similarity, considr (bi-)similarity w.r.t. paths of up to lngth k (may gt fals positivs) Considr indx updats! Kaushik t al. Covring Indxs for Branching Path Quris. SIGMOD, 2002 Considr branching path quris such as //part[bolt AND nut] Indx ach dg both forward and backward Rduc th siz of th indx by ignoring unimportant tags, limiting k, and limiting th tr dpth of branching quris 28

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