The M 2 -tree: Processing Complex Multi-Feature Queries with Just One Index

Transcription

1 The M -tee: Pocessng Complex Mult-Featue Quees wth Just ne Index Paolo Cacca, Maco Patella DEIS - CSITE-CNR, Unvesty of Bologna, Italy fpcacca,mpatellag@des.unbo.t Abstact Motvated by the needs fo effcent smlaty eteval n multmeda dgtal lbaes, we pesent basc pncples of a new paged and balanced ndex stuctue, the M -tee. The M -tee can be appled wheneve complex ange and/o best matches quees ove dffeent descptons (featues) of objects need to be solved. The poposed appoach combnes wthn a sngle ndex stuctue nfomaton fom multple metc spaces, thus beng able to effcently suppot quees on abtay combnatons of ndexed featues. Effcency of the stuctue s pesented though pelmnay expemental esults ove a eal-wold data-set. 1 Intoducton Smlaty quees ae a pmay concen fo suppotng content-based eteval n multmeda dgtal lbaes (MM-DLs). A common appoach vews smlaty seach as a seach fo close ponts n some (hgh-dmensonal) featue space, wth closeness measued though some doman- (and possbly use-) specfc dstance functon. Fo nstance, etevng mages wth smla colos can be appoached by epesentng the colo content of an mage though an hstogam (.e. a vecto) and then by measung the Eucldean dstance between hstogams. Fo the effcent pocessng of smple smlaty quees,.e., whee only one matchng cteon/pedcate s specfed, seveal ndex stuctues have been poposed, ncludng mult-dmensonal (spatal) tees, such as the X-tee [1] and the SR-tee [7], metc tees (M-tee [3], Slm-tee [9], etc.), and sgnatue-based appoaches, such as the VA-fle [10]. These solutons, howeve, ae not sutable fo the moe geneal case whee moe than one smlaty cteon s specfed, each efeng to a specfc object s featue, fo nstance: Fnd vdeo news whee Bll Clnton s talkng about the Kosovo Wa. Exploaton of MM-DLs usng such complex (multmodal) quees s the ule athe than the excepton. Fo nstance, a majo objectve of the Infomeda-II poject at CMU s to:... allow multdmensonal quees that may combne mage elements, vdeo clps, text and speech. 1 The poblem we addess n ths pape s how to effcently suppot complex smlaty quees ove lage MM- DLs. Snce we do not want to lmt ouselves to a patcula doman, the soluton we seek should also satsfy the two followng equements: 1. (Genealty) Due to the complex natue of MM objects, t should wok not only on vecto spaces, but t should also be able to deal wth the moe geneal case of metc spaces,.e. when featue values can only be compaed usng a dstance functon.. (Flexblty) We would also etan the possblty to ssue quees ove only a subset of objects featues, as well as to vay at quey tme the elevance of each featue n detemnng the fnal esult. Complex smlaty quees ae composed of smple pedcates, each nvolvng a sngle featue. Such pedcates ae then combned so as to yeld, fo each object, an oveall evaluaton scoe (see Secton.1). Cuent appoaches fo pocessng complex quees can be summazed as follows. Synchonzed evaluaton of pedcates. Ths appoach, well epesented by Fagn s A 0 [5] and Quck-Combne [6] algothms, deals wth best-matches (also called k-neaest neghbos o k-nn) quees, whee the k objects wth the best oveall scoes ae sought. It s assumed that each pedcate can be effcently evaluated by an ndex, and that ndces povde a soted access scan modalty, wth a GetNext() method that etuns the best match fo that pedcate among the not-seen-yet objects. At each step, the A 0 algothm checks whethe the best soluton computed so fa can be mpoved by usng objects not yet eteved 1 The CMU Infomeda-II Webste:

2 by the scans. If not, the soted access can be stopped; othewse, at least anothe step s equed. To compute the fnal esult, a second andom access phase s equed to evaluate, by takng nto account scoes fo smple pedcates, the oveall scoe fo each object eteved dung the pevous soted access phase. Unfotunately, the pefomance of ths appoach apdly deteoates wth the numbe of pedcates. Statstcs-based evaluaton. In [], the authos popose to tansfom a k-nn quey nto a mult-dmensonal ange quey by explotng statstcs on data dstbutons. Howeve, ths appoach woks only fo (lowdmensonal) vecto spaces, and can only suppot a lmted class of scong functons. Sgnatue-based appoach. The VA-fle [10], whch s a sequental stuctue that stoes bnay appoxmatons of hgh-d featue values, can be staghtfowadly used to pocess complex quees by buldng a VA-fle ove each featue n the quey. Howeve, ths soluton only deals wth vecto spaces and ts complexty scales lnealy wth the data-set sze. Collapsng appoach. Fnally, one could use a sngle ndex fo the combnaton of all featues. Snce we ae lookng fo a metc soluton (1st equement), ths would eque to pedetemne an oveall dstance functon accodng to whch objects could be oganzed, whch s n conflct wth ou nd equement. In concluson, no known pocessng technque can effcently suppot complex quees and, at the same tme, satsfes both ou equements. The M -tee Consde a collecton C of database objects than can be descbed by way of a set F = ff 1 ;::: ;F m g of featues (n ths wok, we do not consde the poblem of choosng the numbe and the type of featues whch ae moe sutable, fom the effcency and/o effectveness pont of vew, fo the doman at hand). Fo each featue F, whose values ae dawn fom a doman D = dom(f ), (ds-)smlaty between featue values s assessed by way of a dstance functon d : D!<+ 0, whch, fo any pa of featue values fom D, yelds a non-negatve eal value, beng undestood that low dstances coespond to smla values and hgh dstances to dssmla values. In the followng, we assume that d s a metc,.e. a non-negatve and symmetc functon whch also satsfes the tangle nequalty. Each couple of featue doman and dstance functon, thus, foms a metc space (D ;d )..1 Quey Model We consde genec (smple) pedcates p havng the fom F ο q, whee q D s a constant, also called quey value, and ο s a (ds-)smlaty opeato. Evaluatng p on an object Cequals to compute d (q; :F ), whee wth :F we denote the value of featue F extacted fom object. In the followng, fo ease of epesentaton, we wll wte ff(p; ) =d (q; :F ), meanng that assessng the dstance between a smple pedcate p : F ο q and an object equals to compute the d functon between the quey value q and the featue value :F. Metc tees, lke the M-tee [3], ae able to solve ange and k-nn quees based on smple pedcates. Snce ou objectve s to genealze to complex quees, we need a way to combne multple pedcates, possbly efeencng dffeent featues, nto a fomula f n ode to obtan a sngle dstance value, to compae wth the use-povded theshold, fo a complex ange quey, o on whch to ode objects, fo a complex k-nn quey. We only eque that, f f = f (p 1 ;::: ;p n ) s a fomula composed of pedcates p 1 ;:::p n, then the oveall dstance value of an object wth espect to f, denoted as ff(f;) s computed by way of a coespondng scong functon [5], d f, takng as nput the dstances of wth espect to each pedcate p j of f, that s: ff(f (p 1 ;::: ;p n );)=d f (ff(p 1 ;);::: ;ff(p n ;)) (1) Although, n lne of pncple, any knd of scong functon would do the job, n ths wok we only consde monotonc scong functons, n the sense of the followng defnton. Defnton 1 (Monotoncty) We say that a scong functon d f (ff(p 1 ;);::: ;ff(p n ;)) s monotonc nceasng (espectvely deceasng) n the vaable ff j f, gven any two n-tuples of dstance values (ff 1 ;::: ;ff j ;::: ;ff n ) and (ff 1 ;::: ;ffj 0 ;::: ;ff n) wth ff j» ffj 0,tsd f (ff 1 ;::: ;ff j ;::: ;ff n )» d f (ff 1 ;::: ;ffj 0 ;::: ;ff n) (espectvely d f (ff 1 ;::: ;ff j ;::: ;ff n ) d f (ff 1 ;::: ;ffj 0 ;::: ;ff n)). If a scong functon d f s monotonc nceasng (esp. deceasng) n all ts vaables, we smply say that d f s monotonc nceasng (esp. deceasng).

3 Such popety s suely easonable fom a use s pont of vew. It s also the case that commonly used scong functons ae monotonc n all the aguments: Fo example, the mn and max functons ae monotonc nceasng. As anothe example, weghted sums, wth d f (ff 1 ;::: ;ff P n n ) = j=1 jff j, whee j s ae postve weghts, and P n j=1 j =1, ae monotonc nceasng [4]. Gven a fomula f whose scong functon d f s monotonc n all ts aguments, we consde ange quees ange f; (C), selectng all the objects whose oveall dstance (computed by way of the scong functon d f )to f s not hghe than a specfed theshold (ff(f;)» ), and k-nn quees NN f;k (C), selectng the k objects havng the lowest ff(f;) dstance to f. It should be noted that metc tees ae aleady able to effcently pocess complex smlaty quees when all the pedcates efe to a sngle featue (F 1 = F = ::: = F n ) [4]. In ths wok we deal wth the moe geneal case of mult-featue quees.. Pncples u appoach econcles the two equements of Secton 1 by povdng a soluton whch can be seen as a multdmensonal extenson of the M-tee [3], much lke as spatal access methods can be vewed, at some extent, as genealzatons of the B + -tee to D-dmensonal vecto spaces. The followng table summazes ths pont. no. of coodnates 1 many Space type Vecto B + -tee R-tee, X-tee, ::: Metc M-tee, ::: M -tee Fo metc ndces, lke the M-tee, we say that they use 1 coodnate snce they can oganze objects by usng only 1 dstance functon. Gven a quey value q j, a metc space can be vewed as an half-lne depatng fom q j, the so-called dstance space. All ponts of D j ae mapped to ths lne wth a dstance fom the ogn whch s equal to the dstance fom q j. A smple ange quey s equvalent to an nteval quey ove such space (see Fgue 1 ). If we consde a complex fomula f (p 1 ;::: ;p n ), wth p j : F j ο q j, each q j nduces an ndependent dstance space on D j, and we obtan what can be conventonally called a multple dstance space. Thus, a complex ange quey s equvalent to a egon quey ove such space (see Fgue 1 ). Theefoe, just as spatal access methods ae genealzatons of the B + -tee, snce they ae able to oganze mult-dmensonal featue (scala) spaces, the M -tee genealzes the M-tee n the sense that t can suppot multple dstance spaces q d(q,.f) q=(q,q ) Fgue 1: A smple ange quey n a dstance space and a complex ange quey n a multple dstance space. Fom the above table t can be seen that the M -tee: (1) can deal wth the moe geneal case of metc spaces, and () can ndex objects by usng many dstance functons at the tme. These ae ndeed the key ngedents to satsfy both equements expessed n Secton 1. Just as the M-tee, the M -tee s a balanced and dynamc tee whose fxed-sze nodes ae mapped to dsk pages and whee ndexed featue values ae stoed n the leaf nodes. To hghlght the man dffeence between M-tee and M -tee, t s useful to show how egons assocated wth each node of these stuctues ae defned: ffl Each node N of the M-tee coespond to a egon of the ndexed metc space (D;d). The egon of node N s Reg(N ) = fv Djd(v; v [N] )» [N] g, whee v [N] s the outng value of node N and [N] s ts coveng adus. All the objects n the sub-tee ooted at N ae then guaanteed to belong to Reg(N ), thus the dstance fom v [N] does not exceed [N] (the egon s equvalent to the nteval [0; [N] ] of the dstance space nduced by v [N] ). It should be noted that each coodnate n such space coesponds to a sngle (possbly mult-dmensonal) featue space.

4 ffl Fo an M -tee bult ove a set of m (m 1) featues F = ff 1 ;::: ;F m g, wth coespondng metc spaces (D ;d ) ( =1;::: ;m), the egon assocated to node N s now defned as: whee v [N] Reg(N )=f(v 1 ;::: ;v m ) D 1 :::D m jd (v ;v [N] s the outng value fo the -th featue and [N] Reg(N ) conssts of the hype-ectangle [0; [N] 1 ] ::: [0; [N] nduced by the outng values..3 Fomat of M -tee Entes )» [N] ; =1;::: ;mg s the coespondng coveng adus. Thus, m ] n the m-dmensonal dstance space Each enty e n an M -tee node conssts of a set of m values v D fo each consdeed featue F. Fo entes n a leaf node, such values coespond to the featues extacted fom each ndexed object, wheeas, fo ntenal nodes, such values ae obtaned by way of a specfc pomoton algothm (fo space eason, we do not gve detals hee of M -tee mantenance algothms). Entes of ntenal nodes also nclude a set of m coveng ad [N] and a ponte pt(t (e)), efeencng the oot N of a sub-tee T (e), wheeas entes n leaf nodes nclude an dentfe od(e), whch s used to povde access to the whole object, whch may esde n a sepaate data fle. The semantcs of the coveng ad s smla to that of M-tee: All the objects stoed n the sub-tee ponted by pt(t (e)) ae wthn dstance [N], consdeng the metc d, fom v [N],.e. 8 T (e); 8 = 1;::: ;m;d (v [N] ;:F )» [N)]. As fo M-tee, the dstances between enty featue values and paent outng values d (v ;v [N] ) ae stoed wthn each enty n ode to pune sub-tees dung the seach phase. Example 1 We have to ndex a collecton of mages, whch have been manually annotated by dffeent human opeatos, who assgned a set of keywods to each of them. We also would lke to seach the collecton fo mages havng smla colo dstbutons. To ths end, a colo descpto s extacted fom each mage by usng the technque descbed n [8], esultng n a 9-dmensonal vecto (see Fgue ). Dstance between mages fo the descpton featue s assessed as the numbe of common keywods dvded by the total numbe of keywods fo the mages,.e. d 1 (v 1 ;v j1 ) = 1 kv vj k 1 1 kv 1 [v j1, wheeas dstance between colo descptos s computed as the k q P9 Eucldean dstance between assocated vectos,.e. d (v ;v j )= h=1 jv [h] v j [h]j mage name keywods colo dstbuton vecto tge.bmp natue, anmals, mammals, felne, tge (1:73, 0:381, 0:483, 1:97, 0:407, 0:518, 0:957, 0:133, 0:0903) lon.jpg natue, anmals, mammals, felne, lon (1:3, 0:380, 0:79, 1:60, 0:480, 0:776, :7, 0:686, 1:34) tge cat.jpg anmals, domestc, felne, cat (0:754, 0:35, 0:730, 0:9, 0:409, 0:757, 1:58, 0:645, 1:41) tge shmp.jpg natue, anmals, custacean, tge, shmp (3:3, 0:104, 0:853, 3:4, 0:189, 0:884, 1:4, 0:, 0:8) Fgue : Fou mages wth the descpton and the coespondng colo dstbuton vectos. Suppose that the mages of Fgue ae nseted n an M -tee node: The two chosen outng values could be, fo example, v [N] 1 =(natue, anmals, felne, tge) and v [N] =(1:76, 0:97, 0:699, 1:98, 0:371, 0:734, 0:43, 0:0786, 0:093). The coespondng stuctue of leaf node N s depcted n Fgue 3. The epesentatve of node N n the paent node P (N ) s shown n Fgue 4, along wth a gaphcal epesentaton of Reg(N ) n the multple dstance space nduced by v [N]..4 Solvng complex quees wth M -tee In ths Secton we show how complex quees can be esolved by the M -tee access stuctue. Suppose that we want to solve the quey ange f; (C): In ode to see whethe a node N of the M -tee has to be accessed,

5 od(e) :F 1 :F d 1 (:F 1 ;v [N ] ) d (:F ;v [N ] e 1 tge.bmp natue, anmals, mammals, felne, tge (1:73; 0:381; 0:483; 1:97; 0:407; 0:518; 0:957; 0:133; 0:0903) 0: 1:5 e lon.jpg natue, anmals, mammals, felne, lon (1:3; 0:380; 0:79; 1:60; 0:480; 0:776; :7; 0:686; 1:34) 0: :60 e 3 tge cat.jpg anmals, domestc, felne, cat (0:754; 0:35; 0:73; 0:9; 0:409; 0:757; 1:58; 0:645; 1:41) 0:667 :80 e 4 tge shmp.jpg anmals, custacean, tge, shmp (3:3; 0:104; 0:853; 3:4; 0:189; 0:884; 1:4; 0:1; 0:8) 0:5 :33 Fgue 3: The stuctue of an M -tee node N. 1 ) ptt (e) :F 1 :F [N ] ::: ::: ::: ::: ::: ::: (1:76; 0:97; 0:699; 1:98; 0:371; e pt(t (e )) natue, anmals, felne, tge 0:667 :80 1 [N ] [N] [N] d (v,.f ) 0:734; 0:43; 0:0786; 0:093) ::: ::: ::: ::: ::: ::: [N] v [N] 1 d 1(v 1 [N],.F 1) Fgue 4: The outng object fo node N and a gaphcal epesentaton of Reg(N ). we compute a lowe bound on the dstance between any object eachable fom N and the complex quey fomula f, that s, between f and the egon assocated to N, ff mn (f;reg(n )), just as we would do fo a smple quey wth an M-tee. To compute such bound we compute a lowe bound fo each pedcate p j : F j ο q j n f, and combne such bounds by way of the scong functon d f. Theefoe, ff mn (f;reg(n )) s computed as d f (ff mn (p 1 ;Reg(N ));::: ;ff mn (p n ;Reg(N ))). Snce the scong functon d f s monotonc nceasng n all ts aguments, no object eachable fom N could lead to a value of ff lowe than ff mn (f;reg(n )). Bounds on ndvdual pedcates can be easly computed by takng nto account nfomaton about N stoed n ts paent,.e. the outng values v [N] and the coveng ad [N] : 3 ff mn (p j ;Reg(N )) = mnfff(p j ;v [N] j ) [N] ; 0g = mnfd j (q j ;v [N] ) [N] ; 0g nly nodes N fo whch ff mn (f;reg(n ))» ae accessed dung the seach. When the leaf level s eached, we can easly compute the oveall dstance ff mn (f;) fo each object n the leaf node usng Equaton 1. It should be noted that above consdeaton can be also used fo complex k-nn quees, by substtutng the theshold value wth the k-th lowest oveall dstance encounteed so fa. 3 Expemental Results Exstng appoaches, lke those poposed n [5,, 6], solve complex ange quees by sepaately ndexng each featue, e.g. wth an M-tee, and by ndependently accessng the ndces to solve a coespondng smple ange quey; fnally, esults of all quees ae combned. In the case of Fgue 5, the M 1 ndex would eteve objects, 3, 5, and 6, wheeas ndex M would eteve objects 1,, and 5 ; then, objects and 5 ae coectly etuned as esults. The M -tee appoach, on the othe hand, combnes nfomaton fom all the metc spaces, such that, n ode to solve a complex ange quey, only those nodes whose egon ovelaps the quey egon ae accessed dung the seach (see Fgue 5 ) and no wok s wasted to access objects that do not satsfy the quey (lke objects 1, 3, and 6 n Fgue 5 ). To show the effcency of the M -tee, we compaed ts pefomance fo complex k-nn quees ove a eal data-set composed of ove 15,000 mages wth espect to those of the A 0 algothm [5] and of a smple sequental scan. Featues used to assess object smlaty wee (1) a stng epesentng the mage name (compaed usng the L edt dstance,.e. the mnmum numbe of chaactes to be nseted, deleted o substtuted to tansfom a stng nto anothe) and a 3-bns hstogam epesentng colo nfomaton fo the mage (compaed usng the Eucldean dstance). We bult an M -tee ove the featues and M-tees ove each sngle featue, and executed seveal k-nn conjunctve quees (d f =maxfff 1 ;ff g) by vayng the value of k. Fgues 6 and show the aveage computed dstances and the aveage page eads fo each quey as a functon of k. Fom the gaphs, we see that the M -tee s ndeed moe effcent than the A 0 algothm, fo both CPU and I/ costs. Moeove, we also see that, fo hgh values of k, pefomance of the A 0 algothm apdly deceases beyond that of the sequental scan, wheeas M -tee costs ae always the lowest. 3 If the d f functon s monotonc deceasng n ts j-th agument, we compute a uppe bound ff max(pj ;Reg(N )) = d j (q j ;v [N] j )+ [N] j and use t n place of ff mn (p j ; Reg(N )). j j j

6 4 3 6 q M 5 1 q=(q,q ) f(q,q ) M 1 M 5 1 Reg(N) q 1 q=(q,q ) Fgue 5: Solvng a complex ange quey (d f = ff 1 + ff ) wth M-tees and wth an M -tee computed dsts M tee A 0 Sequental scan I/s M tee A 0 Sequental scan k k 4 Conclusons Fgue 6: Aveage CPU and I/ costs fo solvng a complex k-nn conjunctve quey. In ths pape we have pesented basc pncples of M -tee, a balanced seach tee desgned fo ndexng multple metc spaces. We have shown how the M -tee can be used to pefom complex smlaty seach ove multmeda objects epesented by multple featues. Stuctue of M -tee nodes and the sketch of seachng algothms have been llustated. Fnally, we have demonstated the effcency of the poposed stuctue ove othe state-of-the-at appoaches though some pelmnay expements. Cuent and futue wok ncludes the complete specfcaton of ndex mantenance algothms (choosng the node n whch a new object should be nseted, splttng a node, choosng the outng values, etc.), as well as a wde quey model, to nclude othe quees that can be effcently solved by the M -tee (e.g. ntesecton and composton of complex quees). Thoough expementaton wth dffeent eal data-sets s also planned. Refeences [1] S. Bechtold, D. A. Kem, and H.-P. Kegel. The X-tee: An ndex stuctue fo hgh-dmensonal data. In VLDB 96, pp. 8 39, Mumba (Bombay), Inda, Sept [] S. Chaudhu and L. Gavano. ptmzng quees ove multmeda epostoes. In SIGMD 1996, pp , Monteal, Canada, June [3] P. Cacca, M. Patella, and P. Zezula. M-tee: An effcent access method fo smlaty seach n metc spaces. In VLDB 97, pp , Athens, Geece, Aug [4] P. Cacca, M. Patella, and P. Zezula. Pocessng complex smlaty quees wth dstance-based access methods. In EDBT 98, pp. 9 3, Valenca, Span, Ma [5] R. Fagn. Combnng fuzzy nfomaton fom multple systems. In PDS 96, pp. 16 6, Monteal, Canada, June [6] U. Güntze, W.-T. Balke, and W. Keßlng. ptmzng mult-featue quees fo mage databases. In VLDB 000, pp , Cao, Egypt, Sept [7] N. Katayama and S. Satoh. The SR-tee: An ndex stuctue fo hgh-dmensonal neaest neghbo quees. In SIGMD 1997, pp , New Yok, NY, May [8] M. Stcke and M. engo. Smlaty of colo mages. In Stoage and Reteval fo Image and Vdeo Databases SPIE, volume 40, pp , San Jose, CA, Feb [9] C. Tana J., A. J. M. Tana, B. Seege, and C. Faloutsos. Slm-tees: Hgh pefomance metc tees mnmzng ovelap between nodes. In EDBT 000, pp , Konstanz, Gemany, Ma [10] R. Webe, H.-J. Schek, and S. Blott. A quanttatve analyss and pefomance study fo smlaty-seach methods n hgh-dmensonal spaces. In VLDB 98, pp , New Yok, NY, Aug