Incremental DFT Based Search Algorithm for Similar Sequence

Transcription

1 Inrnl DFT Bsd Srh Algorih or Siilr Squn Qun hng, hiki Fng, nd Ming hu Dprn o Auoion, Univrsiy o Sin nd Thnology o Chin, Hi, 37, P.R. Chin qzhng@us.du.n nzhiki@il.us.du.n Absr. This ppr bgins wih nw lgorih or opuing i squn d pnsion disn on h i doin h, wih i opliy o On, solvs h probl o rind siilriy r h shiing nd sling o i squn on h is. Ar his, nohr lgorih is proposd or opuing i squn d pnsion disn on rquny doin nd srhing siilr subsqun in long i squn, wih i opliy o rly On, suibl or onlin iplnion or is high iiny, nd dpbl o h ndd diniion o i squn d pnsion disn. An inrnl DFT lgorih is lso providd or i squn d nd linr wighd i squn d, whih llows dinsion rduion on h window o long squn, sipliying h rdiionl On o On. Inroduion In i squn d ining, h nsivly pplibl hniqu, h undnl issu o i squn d siilriy oprison hs proising prosp. Currn hods o d siilriy oprison nd s siilr subsqun srhing inlud, pr ro Eulid hniqu, rquny doin hod [],[],[3],[4], sgnion hod [5],[6], wvor dsripiv lngug hod [7]. Prvious sudis produd h onp o i squn pnsion disn o prsrv h siilriy o i squn d r linr shiing. Mjor sudis on pndd siilr squn srhing hv bn ondud by Chu l [8] nd Agrwl l [9]. Howvr, h disn proposd in [8] is syril, whih y ld o rsuls gins oon sns, whil h lgorih is bsilly osly disn opuion on h i doin. Th siilr squn srhing lgorih in [9] surs ro:. sipl norlizion hniqu is usd o solv h shiing nd sling probls in subsqun siilriy oprison, whih no univrslly pplibl;. oplid nd osly. In his ppr, h bsi rquny doin hod is ndd o pply o h srh in pndd siilr squn. Min poins inlud: providing n nlyil rsul or opuing i squn d pnsion disn on i doin; An innoviv opuing hod or i squn d pnsion disn bsd on rquny doin nlyil soluion, nd h s srhing hniqu or rlvn siilr squn. Copuing on dinsion rdud rquny doin, h hniqu is L. Wng nd. Jin Eds.: FSKD 5, LNAI 363, pp. 85 9, 5. Springr-Vrlg Brlin Hidlbrg 5

2 86 Q. hng,. Fng, nd M. hu highly iin, wih i opliy o On, nd dpbl o pnsion disn o i squn d; In h siilr subsqun srhing, DFT dinsion rduion is nssry or h window o long squn. Th inrnl DFT or h window o long squn nd h inrnl DFT or linr wighd i squn d proposd blow rdu h i opliy ro h rdiionl On o On. Epndd Ti Squn D Disn nd Is Anlyil Soluion Diniion : Th pndd syri disn o on dinsion i sris d o h s lngh [,,, - ] T y[y,y,, y - ] T is dind s: b, i i /. i d, y in y b Th dvng o his diniion is h i inins h siilriy o i squn d r sling nd shiing, pplibl o dirn sling nd shiing oun o rnsdurs. This disn is syril nd gins oon sns, hror nohr diniion or i squn d disn is givn by [], s h iniu vlu o syri disn d,y nd dy,. Alhough Chu l [8] proposd n lgorih h ps h i squn d o shiing-liind pln whr h disn is opud, h hod is ovr-oplid. This ppr proposs dir nlyil hod or opuing i squn d pnsion disn hrough opuing h opiu prr o, b. Thor : Th opiu nlyil soluion o h syri disn o on dinsion i squn d y o h lngh is: Whr: /, d y i yi b. i y i i i yi i y y, b y Th syril is h iniu vlu o d,y nd dy,. Th i opliy o h srh lgorih or rlvn siilr squn is, pr hing, O, whil subsqun srhing, i is On, whr n is h lngh o h long i squn nd is h lngh o subsqun. This lgorih, i.. opuing i squn d disn on h i doin, voids srhing on h,b pln, nd y quikly obin h nlyi soluion o h pnsion disn ording o h vlus o i squn d.

3 Inrnl DFT Bsd Srh Algorih or Siilr Squn 87 3 Copuing Ti Squn D Disn on h Frquny Doin Th srh lgorih or siilr squn bsd on i doin nlyi soluion, s dsribd in hor, is ondud on h i doin, hror osly nd unsuibl or onlin ppliion, whrs rquny doin hods r gnrlly no pplibl o h pnsion disn o i squn d. Our onrn is o sohow nd h rquny doin hod or opuion o i squn, nd dp i o h diniion o i squn d pnsion disn. L : l h orrsponding Fourir oiin o i squn d b, nd h i squn d r linr rnsorion b yh+b hn h Fourir prr or h nubr i in i squn d y is: b +. 3 Whr j π, Ar h nubr oponn o i squn d nd y, rspivly. Thor : Th pndd syri disn o i squn d nd y r pproily: /, b y d. 4 whr: b is h liiing rquny; is opl nubr squn inrodud or onvnin's sk; is h Fourir prr o h nubr i in h i squn d, nd is h Fourir prr o h nubr i in h i squn d y; union is pping ro opl nubr o rl nubr, i.. h produ o h rl prs o wo opl nubrs plus h produ o hir iginry prs.

4 88 Q. hng,. Fng, nd M. hu Th rlvn subsqun srhing lgorih llows, h s i, inrnl DFT nd pnsion disn opuion wih rquny doin nlyi soluion, hrby o pror siilriy oprison. Th i opliy o i is On. Coprd o h siilr subsqun srhing on i doin, h i opliy o whih bing On, his lgorih is or iin nd suibl or onlin ppliion, bus h rngs ro -5, -3 gniuds lowr hn. Furhror, his lgorih inins h siilriy o i squn d r linr shiing, nd is hror dpbl o h pndd diniion o disn. To siply h r, h siilr subsqun srhing lgorih on h rquny doin h uilizs inrnl DFT, nd solvs h issu o shiing nd sling is hnorh lld: Endd rquny doin hod. 4 Inrnl Fourir Shiing o Ti Squn D nd Linr Wighd Ti Squn D Rgrding h srhing o siilr subsqun, h lgorih dsribd in sion 3 rquirs disr Fourir shiing or h subsqun window. Aording o rdiionl DFT orul, i opliy or obining low ordr Fourir prrs is On, whih is osly. W now prsn n inrnl Fourir shiing lgorih h grly nhns h iiny, nd is suibl or onlin ppliion. Th long i squn is dividd ino n-+ inrlpping i windows h lngh. w i rprsns h nubr i window, pilizd W i, rprsns h nubr rquny oponn o h i window. Thor 3: Th rlion bwn W i,, h nubr Fourir prr o h d i window w i,, nd W i-,, h nubr Fourir prr o h prvious i window, is: Wi, W i, + i, w i, wi,. 5 i, i+ i Whr: On so osions, i h i squn d is losr o h urrn i -, i is rgrdd s or iporn hn h or disn poins. For onvnin sk, w inrodu orging union o onribu o h wigh o disn. s 6. z + k k + k + k. 6 Diniion : Th linr orging disn d w,y or on dinsionl i squn d y o h lngh o is: d w / y. 7, y

5 Inrnl DFT Bsd Srh Algorih or Siilr Squn 89 In h i squn, h nubr du in h nubr i window is rprsnd s w i,. h du r wighing is w i,. Thir rlions r: Fro diniion, w' i, wi, wi, k + k + k. 8 / / d w, y y ' y' d ', y' Thror opuing h wighd disn bwn subsquns is quivln o opuing h Eulidn disn bwn wo wighd i squns. Aording o Prsvl Ruls, w y k h irs w rquny oponns ro h rquny doin o h wighd i squn d o pror n pproi disn opuion, llowing s siilr squn srh. Th issu now is how o obin h Fourir prrs o h window r linr wighing, nd in n inrnl nnr. Ti window w is i window w r wighing. Whn h DFT prrs o h prvious window W i-, linr wighd Fourir prr W i-, nd uiliry prr WT i-, r givn, how o obin h DFT prrs W i-, o his linr wighd d window in n inrnl nnr. Whr, WT i, WT w, w, + w, WT, W i i, + Th ollowing ls n b obind: L : i, L 3: Th nubr Fourir prr W i, o h i squn window W i r linr orging is: W ' i, k + k Wi, + kwti, Thror, inrnl lgorih or Fourir prr or h i squn window W i, r linr orging n b obind,nly, hor 4. I s sy o prov by obining hor 3, l nd l 3. Thor 4: Rursion orul or inrnl opuion o h Fourir prr o linr orging i squn window r shown in 9-3: W,,. 9 WT,,. W,,, i w Wi + w,. + WT WT W w, w, i, i, i, +.

6 9 Q. hng,. Fng, nd M. hu W ' i, k + k Wi, + kwti,. 3 Whn h wighd Fourir prrs o h window r obind, h pproi wighd disn o i squn d n b opud on dinsion rdud rquny doin, hiving high iiny in siilr squn srhing. Obviously, h i opliy o inrnl DFT lgorihi is On uh lowr hn h i opliy o rdiionl DFT lgorih On. 5 Eprin A oprison o h running i bwn ndd rquny doin hod nd i doin hod is shown in Tbl nd Tbl. In Tbl, h lngh o i squn n liiing rquny 3; in Tbl, lngh o subsqun ; liiing rquny in boh bls 3. Th rsuls indi h wih h ndd rquny doin hod, h i is bou / /5 o h i doin hod, grly iproving h iiny o srh lgorih. Th orr is lso dpbl o h diniion o pnsion disn o i squn. Tbl. Th running i o i doin hod nd ndd rquny hod long wih subsqun lngh subsqun Ti doin hod Endd rquny doin Lngh Sond hod Sond Tbl. h running i o i doin hod nd ndd rquny hod long wih squn lngh Squn Ti doin hod Endd rquny doin lngh n Sond hod Sond Conlusion In his ppr, n nlyil lgorih is proposd or opuing i squn d pnsion disn on h rquny doin, oring nw hniqus or siilr

7 Inrnl DFT Bsd Srh Algorih or Siilr Squn 9 subsqun srhing. I is provn, hrough prin, his lgorih is or iin hn h i doin bsd lgorih, nd suibl or onlin ppliion, dpbl o h diniion o i squn d pnsion disn. An inrnl DFT lgorih is lso providd or i squn d nd linr wighd i squn d, whih grly iprovs h iiny o DFT dinsion rduion on h window o long squn, sipliying h rdiionl On i opliy o On. Rrn. R. Agrwl, C.Flousos nd A.swi: Eiin siilriy srh in squn dbs. In FODO, Evnson, Illinois, Oobr Flousos Chrisos. Rngnhn M. nd Mnolopulos nnis: Fs subsqun hing in i sris dbss. Pro ACM SIGMOD, Minnrpolis MN, My D.Rii nd A.O.Mndlzon: Eiin rrivl o siilr i squns using DFT. In FODO, Kob,Jpn K.P.Chn nd A.W.C.Fu: Eiin i sris hing by wvls. In ICDE, Sydny, Ausrli Kogh Eonn, Pdhri syh: A probbilisi pproh o s prn hing in i sris dbss. Prodings o h Third Conrn on Knowldg Disovry in Dbss nd D Mining, AAAI Prss, Mnlo Prk, CA Kogh Eonn, Mihl J.Pzzni: An Enhnd rprsnion o i sris whih llow s nd ur lssiiion, lusring nd rlvn dbk. Proding o h 4 h Inrnionl Conrn o Knowldg disovry nd D Mining, AAAI Prss, Mnlo Prk, CA Rksh Agrwl, Giuspp Psil, Edwrd L.Wirs, Mohd i: Qurying shps o hisoris. Prodings o h s VLDB Conrn, urih, Swizrlnd K.K.W. Chu, M.H Wong: Fs i-sris srhing wih sling nd shiing. In prodings o h l g ACM Syposiu on Prinipls o Dbs Syss, Phildlphi, PA Agrwl R, Lin K I, Swhny H S, Shi K: Fs siilriy srh in h prsn o nois, sling nd rnslion in i-sris dbss. In Pro. 995 In. Con. Vry Lrg D BssVLDB 95, urih, Swizrlnd