PAPER Efficient Analyzing General Dominant Relationship based on Partial Order Models

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1 PAPER Eiint Anlyzing Gnrl Dominnt Rltionship s on Prtil Orr Mols Zhnglu YANG, Lin LI, Nonmmrs, n Msru KITSUREGAWA, Fllow SUMMARY Skylin qury is vry importnt us it is th sis o mny pplitions,.g., ision mking, usr-prrn quris. Givn n N-imnsionl tst D, point p is si to omint nothr point q i p is ttr thn q in t lst on imnsion n qul to or ttr thn q in th rmining imnsions. In this ppr, w stuy gnrliz prolm o skylin qury tht, usrs r mor intrst in th tils o th ominnt rltionship in tst, i.., point p omints how mny othr points n whom thy r. W show tht th xisting rmwork propos in [17] n not iintly solv this prolm. W in th intrrlt onntion twn th prtil orr n th ominnt rltionship. Bs on this isovry, w propos nw t strutur, PrCu, whih onisly rprsnts th ominnt rltionship. W propos som tiv strtgis to onstrut PrCu. Extnsiv xprimnts illustrt th iiny o our mthos. ky wors: skylin qury, lgorithm, ominnt rltionship nlysis, prormn vlution 1. Introution Th skylin qury [3] hs ttrt onsirl ttntion ths yrs us it is th sis o mny pplitions,.g., multi-ritri ision mking [3], usr-prrn quris [11] [9] n miroonomi nlysis [17]. Skylin mining ims to in thos points, whih r not omint y othrs, in -imnsionl sptil tst. This prolm n sn s spil lss o prto prrn quris [11], onvx hull [23] or mximum vtors [14]. Fig. 1 shows on lssi xmpl o skylin qury tht ustomrs r lwys intrst in thos st hotls tht r ttr thn othrs t lst t on o th two ritri, th istn n th pri, with smllr vlus. Th skylin o th xmpl tst in Fig. 1 onsists o n. Thr r mny issus rlt to skylin qury, inluing th gnrl ull-sp skylin points qurying [3] [7] [13] [21], susp skylin points mining [31] [26] [28], skylin points xtrting in strm [18] [27] [2], Top-k n high-imnsionl skylin points xtrting [5] [6], mining skylin in istriut nvironmnts [2] [1] [29], pproximt skylin qurying [12]. All ths issus, howvr, onrn only th pur ominnt rltionship mong tst, i.., point p is whthr omint y othrs or not, n got thos non-omint ons s rsults. Th uthor is rsrh ssoit in th Institut o Inustril Sin, th Univrsity o Tokyo. Th uthor is n ssistnt prossor in th Dprtmnt o Computr Sin n Thnology,& Wuhn Univrsity o Thnology, Chin. Th uthor is prossor in th Institut o Inustril Sin, th Univrsity o Tokyo. Fig. 1 istn pri Exmpl o th skylin qury Rntly, Li t l. [17] propos to nlyz th ominnt rltionship in usinss mol tht, usrs r mor intrst in th til o th ominnt rltionship in tst, i.., point p omints how mny othr points n is omint y how mny othrs. Hr w show n xmpl. Exmpl 1: Consir you r mngr o hotl ompny. You wnt to know th usinss position o lol hotl in th urrnt mrkt with rgr to your prrn, i.., pri n istn to th h, y hking how mny othr hotls r ttr/wors thn. For th smpl hotls shown in Fig. 1, you n ut th onlusion tht hotl is ttr thn 2 othr hotls ut wors thn nothr 2 hotls with rgr to your prrn. In rl worl, howvr, usrs r lwys intrst in not only how mny ojts r ominting/omint y spii ojt, ut lso whom thy r, whih ws not mntion in [17]. This prolm n sn s gnrl ominnt rltionship nlysis to th ons propos in [17]. It is nivly thought, n sily solv y ssoiting h ojt with its orrsponing uoi in DADA [17]. So whn usrs qury th ominnt rltionship, ths ojts will xtrt simultnously. Nvrthlss, u to hug numr o uplit xistn in DADA, th storg ovrh n th qury tim will unptl or usrs. In this ppr, w im t proposing iint n tiv mthos to nswr th whom prolm. Bus o th intrrlt onntion twn th prtil orr n th ominnt rltionship, w propos nw t strutur ll PrCu, whih onisly rprsnts th omplt inormtion o th gnrl ominnt rltionship s on th prtil orr nlysis. Spiilly, w ror th prtil orr s Dirt Ayli Grph (DAG) or h Not tht th nlysis hr n urthr us to trmin th pri o hotl, whih shoul omptitiv in th urrnt mrkt whil rsrving th most proit.

2 2 IEICE TRANS. INF. & SYST., VOL.Exx D, NO.xx XXXX 2x Fig. 2 DAG rprsnttion in 2- sp uoi in PrCu n propos iint t struturs n strtgis to nswr th gnrl ominnt rltionship quris. Morovr, w introu iint strtgis to onstrut PrCu. Th xprimntl rsults n prormn stuy onirms th iiny n tivnss o our strtgis. To illustrt th or i o this ppr, hr w show simpl xmpl. Fig. 2 rprsnts th prtil orr (no s DAG ormt) o th xmpl tst in Fig. 1 in 2-imnsionl sp. W n know th point omints th points n n is omint y th points n, y ounting th out-link n in-link o, rsptivly. Hr w solv not only th how mny prolm, ut lso th whom prolm. From this xmpl, w know tht th gnrl ominnt rltionships o tst n rprsnt into thir orrsponing prtil orr rprsnttion (i.., DAGs). In ontrst, th DADA t strutur [17] pplis th gri-s inx thniqu, whih os not iintly ror th ominnt rltionship, s will illustrt in th xprimntl vlution. Our ontriutions in this ppr r s ollows: W gnrliz th ominnt rltionship quris propos in [17], s Gnrl Dominnt Rltionship Qury (GDRQ). W in th intrrlt onntion twn GDRQ n th prtil orr nlysis. W propos t u, PrCu, whih onisly rprsnts th omplt inormtion o th gnrl ominnt rltionship s DAGs s on th prtil orr or h uoi. W introu tiv mthos to onstrut th PrCu. W onut omprhnsiv xprimnts to illustrt th tivnss n iiny o our mthos. Th rminr o this ppr is orgniz s ollows. In Stion 2, w isuss th rlt work. In Stion 3, w prsnt th prliminris o this ppr. A niv mtho s on xisting strtgy to nswr GDRQ is introu in Stion 4. Th omputtion o PrCu is prsnt in Stion 5.1 n th qury prossing strtgis or gnrliz ominnt rltionship nlysis using PrCu is sri in Stion 5.2. Th prormn nlysis r rport in Stion 6. W onlu th ppr n provi suggstions or utur work in Stion 7. oms rom som ol lssi topis, suh s onvx hull [23] n mximum vtors [14]. Skylin qury lgorithms n lssii into two tgoris. Th irst on is non-inx s mtho, i.., BNL [3], SFS [7], DC [3]. Th son tgory is inx s mtho, i.., NN [13], BBS [21], SUBSKY [26]. As xpt, th inx-s mthos hv n shown to suprior ovr th non-inx-s ons n urthrmor, th inx-s strtgis n progrssivly rturn nswrs without hving to sn th ntir t input. Spilly, SUB- SKY [26] ws propos to omput low-imnsionl Skylins n is th st lgorithm or susp skylin isovring. Bs on th t istriution, SUBSKY rts n nhor point or h lustr, n uils B+-tr on th L istn twn h ojt to its orrsponing nhor. Thn, SUBSKY sns th tr l nos oring to th sning orr o th points s smllst vlu o - imnsion to gt Skylins. From th viw point o imnsion onrn, th xisting lgorithms n lso lssii into two tgoris, i.., ull sp s mtho [13] [21], n susp s mtho [26] [31] [28]. Othr rlt work on skylin mining inlus mining skylin in istriut nvironmnts [2] [1] [29], skylin qury in t strm [18] [27] [2], pproximt skylin qury [12], intrsting skylin points in high-imnsionl sp [5] [6]. All th ov works onrn only th pur ominnt rltionship n, outputt thos points whih r not omint y othrs. Not tht in ition to th originl mning in [3], omint hr n vrint, i.., k-ominnt [6]. In ontrst, Li t l. propos to nlyz mor gnrl ominnt rltionship rom miroonomi spt [17]. Th usrs r lwys intrst in not only th inry ominnt rltion twn th points in tst, ut lso th sttistil inormtion, i.., how mny othr points r ominting/omint y spii point. In [17], th uthors propos thr si Dominnt Rltionship Quris (DRQs) n onstrut t u, DADA, to iintly orgniz th inormtion nssry to DRQs. Morovr, novl t strutur, D*-tr, ws propos to ulill iint omputtion or DRQs. Howvr, usrs r lwys intrst in not only how mny ojts r ominting/omint y spii ojt, ut lso whom thy r, whih ws not mntion in [17]. This prolm nnot sily solv y using th mthoologis propos in [17] us o th lrg uplit storg ost in DADA. In this ppr, w propos iint t strutur n strtgis to solv suh kin o gnrl ominnt rltionship qury s on our isovry tht GDRQ hs intrrlt onntion with prtil orr. 2. Rlt work 2.1 Skylin Qury Skylin qury ws irst introu in [3]. Th prolm W sri th til o SUBSKY us it is on o th slin lgorithms in th xprimntl vlution.

3 YANG t l.: EFFICIENT ANALYZING GENERAL DOMINANT RELATIONSHIP BASED ON PARTIAL ORDER MODELS 3 D2 2.2 Prtil Orr Mining Prtil orr hs ppr in mny omputtionl mols n thr r lot o pplitions involvs with prtil orr issus, suh s onurrnt mols [15], optimisti rollk rovry [25], iology [16], surity [24] n prrn qury [11]. In this ppr, w minly onsir th prolm tht how to onvrt th sptil tst into prtil orr rprsnttion, whih r thn quri to gt th gnrl ominnt rltionship iintly. As r s w know, thr is no work on this prolm. An intrsting stuy invstigt th prolm o mining smll st o prtil orrs glolly itting t st [19]. Prtiulrly, [19] rss squn t. Vry irnt rom th prolm stui hr, [19] tri to in on or (smll) st o prtil orrs tht it th whol tst s wll s possil, whih is n optimiztion prolm. An impliit ssumption is tht th whol tst somhow ollows glol orr. Mor rntly, [4] wr intn or isovring svrl smll prtil orrs rom st o squns inst o only on tht sris ll or most o th st. Thy propos to us los prtil orrs to summriz squntil t in onis mnnr. Yt irnt rom this ppr, thy i not urthr xplor th prtil orrs or spii purpos (i.., ominnt rltionship xtrtion). In this ppr, howvr, w n to trmint th prtil orrs givn sptil tst. W propos simpl mtho o onvrting th sptil tst to th orrsponing squn tst n thn, pply xisting strtgis suh s tht us in [4] with moiition y onsiring skylin proprty to gnrt th prtil orrs. 3. Prliminris Givn -imnsion sp S={s 1,s 2,...,s }, st o points D={p 1,p 2,...,p n } is si to tst on S i vry p i D is -imnsionl t point on S. W us p i.s j to not th j th imnsion vlu o point p i. For h imnsion s i, w ssum tht thr xists totl orr rltionship. For simpliity n without loss o gnrlity, w ssum smllr vlus r prrr [3] (i.., MIN oprtion) in this ppr. Dinition 1 (omint). A point p is si to omint nothr point q on S i n only i s k S, p.s k q.s k n s t S, p.s t <q.s t. A prtil orr on D is inry rltion on D suh tht, or ll x,y,z D, (i) x x (rlxivity), (ii) x y n y x imply x=y (ntisymmtry), (iii) x y n y z imply x z (trnsitivity). W us (D, ) to not th prtil orr st (or post) o D. W not y th strit prtil orr on D, i.., x y i x y n x y. Givn x,y D, x n y r si to omprl i ithr x y or y x; othrwis, thy r si to inomprl. Th Dinition 1 n trnslt into th orring ontxt s ollows: D D D () Exmpl sptil tst Fig. 3 Exmpl tst D1 () Rprsnttion in 2- sp {D1,D2} Dinition 2 (omint in orring ontxt). A point p is si to omint nothr point q on S i n only i s k S, p.s k q.s k n s t S, p.s t q.s t. Th prtil orr (D, ) n rprsnt y DAG G =(D, E), whr (υ, ω) E i ω υ n thr os not xist nothr vlu x D suh tht ω x υ. For simpliity n without loss o gnrlity, w ssum tht G is singl onnt omponnt. Dinition 3 (ominting st, DGS(p, D, S )). Givn point p, w us DGS(p, D, S ) to not th st o points rom D whih r omint y p in th susp S o S. Dinition 4 (omint st, DDS(p, D, S )). Givn point p, w us DDS(p, D, S ) to not th st o points rom D whih omint p in th susp S o S. Th prolm tht w wnt to solv is s ollows: Prolm 1 (Gnrl Dominnt Rltionship Qury (GDRQ)). Givn tst D, imnsion sp S n point p, in DGS(p, D, S ) n DDS(p, D, S ). Not tht skylin point p hs th ollowing proprty: DDS(p, D, S )=. In othr wors, th skylin qury n thought s spil s o th gnrl ominnt rltionship qury. Exmpl 1. Consir th 3-imnsionl tst D = {,,,,, } in Fig. 3 (). Givn qury point, imnsion sp S ={D 1,D 2 }, th ominting st DGS(, D,S )= {, } n th omint st DDS(, D,S )={, }. W will us this tst s running xmpl in th rst o this ppr. 4. A Niv Mtho To solv th prolms in in Stion 3, nturl i is to xtn th rmwork propos in [17]. In this stion, w rily introu this niv strtgy n thn, illustrt its wk points. Th uthors in [17] prtition th t sp y using griing strtgy. For xmpl, Fig. 4 () shows tst in 2-imnsionl sp (i.., {D 1,D 2 }). In Fig. 4 (), h gri rors th numr o th points whih urrnt gri omints. For instn, th gry gris r ll thos whih omints thr points. Inst o roring h gri inormtion, [17] propos D -tr to ror th omprss inormtion (uppr/lowr oun o rgion tht omints

4 4 IEICE TRANS. INF. & SYST., VOL.Exx D, NO.xx XXXX 2x D 2 D 1 () D D 1 () () lowr oun uppr oun num o points omint () Fig. 4 Th strtgy o DADA or GDRQ lowr oun uppr oun num o points omint points omint <, 3> <1, 4> 3 {,, } <1, 2> <3, 4> 3 {,, } <3, > <4, 2> 3 {,, } Fig. 5 Th xtnsion o DADA or gnrl ominnt rltionship nlysis th sm numr o points). For xmpl, th gry gris n prtition into thr rgions, whih r rprsnt y thir uppr oun, i.., {1, 4}, {3, 4} n {4, 2}, rsptivly. Th whol D -tr is shown in Fig. 4 (), whih is onstrut s on th rul in in [17] (Dinition 4.7). Fig. 4 () shows th omprss inormtion out th thr gry rgions, i.., th lowr oun (1st olumn), th uppr oun (2n olumn) n th numr o points omint (3r olumn). Givn point P qury, to gt th numr o th points P qury omints, it ns to strt rom th root o th D -tr n mov own th no with th uppr oun tht n omint P qury. On it knows tht P qury is ontin in rgion tht th no omints, th sir numr o th points whih r omint y P qury n output. Rr [17] or mor til. Yt thr r two issus rising whn prossing th gnrl ominnt rltionship quris y using DADA s strtgy. Firstly, th whom prolm n not iintly solv. For xmpl, lthough th gry rgions in Fig. 4 () ll omint thr points, thy hv irnt ominting sts, i.., {,, } or lu n yllow rgions n {,, } or r rgion. Although y ing th ominting st into h no o th D -tr n nivly nswr th qustion (s shown in Fig. 5), this simpl solution will introu srious urn o t uplition prolm. Thror, th strtgy o DADA is not pproprit or th gnrl ominnt rltionship nlysis prolm. Anothr issu is tht th srh strtgy in DADA whil trvrsing th D -tr is iniint, spilly whn th tr hs mny lyrs. 5. A Prtil Orr Bs Mtho In this stion, w propos to iintly pply th proprtis o th prtil orr to nlyz th gnrl ominnt rltionship. Spiilly, w irst introu tiv strtgis to onstrut prtil orr t u (PrCu), whih onisly rprsnts th ominnt rltionship y using DAGs. Morovr, w propos iint lgorithms to nswr th Fig. 6 Th work low o PrCu onstruting gnrl ominnt rltionship quris s on PrCu. In th ollowing stion, w introu our mthos o onstruting PrCu. 5.1 Construting PrCu As sri in Stion 3, th ominnt rltionship n no in prtil orr rprsnttion (DAGs). In this stion, w xplin how to onstrut th prtil orr t u (PrCu) with sptil tst input. As r s w know, thr is no work on this prolm. In this ppr, w propos to pply strtgis rom nothr rsrh ontxt, squntil pttrn mining [1], to gt th prtil orr rprsnttion rom sptil tst. Th whol work low is shown in Fig. 6. W propos simpl mtho o onvrting th sptil tst to th orrsponing squn tst in th irst pross n thn, pply xisting strtgis suh s tht us in [4] with littl moiition in th son n thir prosss to gnrt DAGs rom th trnsorm squn tst. Not tht w minly illustrt how to omput th u or ominting st sin omputtion o omint st n on in similr shion. Th irst pross in Fig. 6 is to onvrt th originl sptil tst to th squn tst. With k-imnsionl tst, w simply gt k-ustomr squn tst, y sorting th ojts in h ustomr (imnsion) oring to thir vlu in sning orr. For xmpl, Fig. 7 () shows th onvrt squn tst o th xmpl sptil tst in Fig. 7 (). Thorm 1. Th onvrt squn tst rors ll th ominnt rltionship o th points in th sptil tst. Proo. Trivil us th smll-lrg pir (ominnt) rltionship in th sptil tst is quivlnt to th rlylt pir (ominnt) rltionship in th onvrt squn tst. Th son n th thir prosss in Fig. 6 im to trmin prtil orr tht sris th point st in th susp S o t sp S in D. Th rlt prolm is

5 YANG t l.: EFFICIENT ANALYZING GENERAL DOMINANT RELATIONSHIP BASED ON PARTIAL ORDER MODELS 5 D D D Dim. Squn D1 D2 D3 root <> <> <> D 1 D 2 D 3 () Exmpl sptil tst Fig. 7 Pross 1 o onstruting PrCu () Trnsorm squn tst rss in [19] n mor rntly in [4]. In this ppr, w simply pply th pproh in [4] with minor moiition tht, inst o mining los squntil pttrns [3], w min gnrl squntil pttrns [1]. In pross 2 s shown in Fig. 6, w isovr th squntil pttrns rom th trnsorm squn tst y pplying PrixSpn lgorithm [22], whih is th stt-o-th-rt on. Spiilly, givn n-squn tst, w prtition it into svrl k-squn tsts, whr 2 k<n, n pply PrixSpn to thm, rsptivly, with minimum support qul to 1%. For xmpl, givn th squn tst s shown in Fig. 8 (), w prtition it into k-squn tsts whr k=2, i.., {D 1,D 2 }, {D 1,D 3 }, n {D 2,D 3 }. PrixSpn is thn ppli on thm. Not tht or k-squn tsts whr k=1, i.., {D 1 }, {D 2 }, n {D 3 },wo not n us PrixSpn us th mximl squntil pttrns r strightorwr (i.., th squn itsl). For k-squn tsts whr k=n, i.., k=3 or th tst shown in Fig. 8 (), w o not n to prtition it us th numr o th possil prtition tst is on, i.., {D 1,D 2,D 3 }. In t, th pross is th sm s uiling ommon t u, tht w trvrs vry possil susp ( k- squn tst, i.., {D 1,D 2 }), n pply PrixSpn on it with minimum support qul to 1%. To sv sp n onvnint th qury prossing, w mrg ths squntil pttrns s lol mximl squntil squns [1], whih r not th susqun o othr squntil pttrns in th sm susp. For xmpl, in susp {D 1,D 2 }, lthough thr r mny squntil pttrns, i..,,,,,,,,,,, n so orth. W only ror th mximl squntil pttrns, i..,, n, us ll th othr squntil pttrns r susquns o ths thr mximl squntil pttrns. Th mximl squntil pttrns o susp S ror th ominnt rltionship twn itms in S (s vrii y Thorm 1, Thorm 2). For xmpl, th pttrn inits tht omints, omints, n omints in susp {D 1,D 2 }. Th rsult t u (SqCu) got rom pross 2 or th xmpl tst is shown in Fig. 8 (). Thorm 2. SqCu rors ll th ominnt rltionship o th points in th squn tst D. Proo. (Proo y Contrition.) For simpliity, w only prov or spii susp o SqCu. Assum to th ontrry tht thr is ominnt rltionship twn two Du to limit sp, w skip th til o PrixSpn hr. Intrst usrs n rr [22]. Dim. D1 D2 D3 Squn () Trnsorm squn tst Fig. 8 <> <> <> D 1 D 2 <> <> <> D 1 D 3 <> <> D 1 D 2 D 3 <> <> <> D 2 D 3 () Th t u (ltti) whos uoi onsists o th lol mximl ommon squntil pttrns Pross 2 o onstruting PrCu points, omints in susp S, is not rprsnt in th uoi S o SqCu. This mns tht th squntil pttrn is not list in S o SqCu, whih ontrits our ssumption tht th squntil pttrn mining pross n in ll th squntil pttrns. In pross 3, th omintions o th lol mximl squntil squns r numrt to gnrt prtil orrs with DAGs rprsnttion, y pplying th mtho propos in [4]. Th rsult t u (PrCu) got rom pross 3 or th xmpl tst is shown in Fig. 9 (). <> <> <> D 1 D 2 D 3 <> <> <> D 1 D 2 root <> <> <> D 1 D 3 <> <> D 1 D 2 D 3 () Th t u (ltti) whos uoi onsists o th lol mximl ommon squntil pttrns Fig. 9 <> <> <> D 2 D 3 D 1 D 2 D 3 D 1D 2 root D 1D 3 D 1D 2D 3 () DAG rprsnttion o prtil orr in t u PrCu Pross 3 o onstruting PrCu Thorm 3. PrCu rors ll th ominnt rltionship o th points in th sptil tst D. Proo. Proo n u s on Thorm 1, Thorm 2 in this ppr n [4]. 5.2 Qurying PrCu Dt Cu Th smnti mning kpt in th PrCu t u is th ky us to xtrt th gnrl ominnt rltionship iintly Gnrl Dominnt Rltionship Qury (GDRQ) Givn tst D, qury point P qury n susp S, D 2D 3

6 6 IEICE TRANS. INF. & SYST., VOL.Exx D, NO.xx XXXX 2x Fig. 1 DAG rprsnttion o th xmpl tst in 2-imnsionl sp {D 1,D 2 } th GDRQ is to omput th points omint or omint y P qury, whr P qury D. An importnt osrvtion in this s is tht, i P qury is in D, ll th gnrl ominnt rltionship rlt to P qury n sily isovr y trvrsing th DAG in spii susp. As n xmpl, Fig. 1 shows th DAG rprsnttion in susp {D 1,D 2 }. To ilitt th ounting pross, th numrs o points ominting/omint y urrnt no (point) r insrt into h no. This pross is xut in th promput-mo. Suppos th qury point is, w n gt th points omint y immitly, whih is 2. Upon usrs r intrst in whom ths two points r, it gos ownwr ollowing th out-link o, n gts th ominting st o s {, }. In DADA [17] rmwork, howvr, it ns to trvrs th D*-tr to gt th orrsponing lss. For xmpl, ssum th qury point is, th orr o th trvrs nos in D*-tr, s shown in Fig. 4 (), is { 1, 1, 2, 1, 2, 2, 3, 4 }. Thn it ins th ominting st o y hking th lss o { 3, 4 }. Oviously, DADA onsums mor tim ompr with our strtgy. 6. Exprimntl Evlution n Prormn Stuy To vlut th iiny n tivnss o our strtgis, w onut xtnsiv xprimnts. W prorm th xprimnts using Intl(R) Cor(TM) 2 Dul CPU PC (3GHz) with 3G mmory, running Mirosot Winows XP. All th lgorithms wr writtn in C++, n ompil in n MS Visul C++ nvironmnt. W onut xprimnts on oth synthti n rl li tsts. Dtil implmnttion o th lgorithms us to ompr is sri s ollows: 1. SUBSKY. SUBSKY ws tst with th lgorithm vlop in [26], whih is th stt-o-th-rt lgorithm or susp skylin qury. 2. Niv. Niv ws tst with th xtnsion o DADA [17], y storing th omint/ominting points in th orrsponing lss, s xplin in Stion PrCu. PrCu ws implmnt s sri in this ppr. 6.1 Dtsts W mploy th synthti t gnrtor [3] to rt our synthti tsts. Thy hv inpnnt istriution, with Fig. 11 Exution tim omprison twn SUBSKY n PrCu on skylin qury imnsionlity in th rng [3, 6] n t siz in th rng [1k, 5k]. Th ult vlus o imnsionlity wr 5. Th ult vlu o rinlity or h imnsion ws 5k. 6.2 Skylin Qury Prormn Bus th skylin qury is importnt n n sn s spil s o th gnrl ominnt rltionship qury, in this stion, w irst vlut th skylin qury nswring prormn o PrCu ompr with th stt-o-thrt lgorithm, SUBSKY [26]. Fig. 11() n 11 () show th skylin qury tim ginst numr o points in th tsts n imnsionlity, rsptivly. W n s tht th PrCu lgorithm outprorms th SUBSKY in oth ss y up to n orr o mgnitu. This is us th SUBSKY lgorithm ns to trvrs th tr t strutur (i.., B-tr) to xtrt th skylin on th ly. On ontrry, PrCu pr-omputs n stors th skylin points into prtil orr t strutur, whih n sily xtrt out us thy xist in th irst lyr o DAG grph (no othr points omint thm). Morovr, rom th igurs w n know tht imnsionlity hs mor t on qury prormn ompr with th numr o points in th tsts. 6.3 Dominnt Rltionship Qury Prormn To tst th t o th Gnrl Dominnt Rltionship qury (GDRQ), w rnomly slt 1 irnt points s on th synthti tst. Fig. 12 () n () show th qury tim ginst numr o points in th tsts n i-

7 YANG t l.: EFFICIENT ANALYZING GENERAL DOMINANT RELATIONSHIP BASED ON PARTIAL ORDER MODELS Fig. 12 Exution tim omprison twn Niv n PrCu on gnrl ominnt rltionship qury mnsionlity, rsptivly. W n s tht th PrCu pproh is ttr thn th Niv strtgy. Th prormn o Niv oms wors s numr o points or imnsionlity is lrgr, whil PrCu rmins lmost th sm. Th rson is similr to tht xplin in Stion 6.2. Niv ns to trvrs th inx t strutur (i.., D - tr) to ompr n xtrt ll th rquir points. In ontrst, PrCu only trvrs th DAG grph to irt xtrt vry no it pss n no omprison is nssry. 6.4 Inx Dt Strutur Constrution Prormn Th iiny o PrCuis root in th omprss t strutur it isovris, prtil orr t u (PrCu). In this stion, w show th onstrution tim or PrCu ompr with ost o uiling othr inx t strutur (i.., D -tr) in th Niv lgorithm. Fig. 13 () n () show th xution tim or inx uiling ginst numr o points in th tsts n imnsionlity, rsptivly. W n s tht th PrCu is snsitiv to th numr o points in th tsts, tht whn th numr gts lrgr, th prormn o PrCuonstrution is wors thn tht o D -tr uiling. Howvr, s illustrt in Fig. 13 (), D - tr onstrution oms wors s imnsionlity grows, whih mns tht D -tr inx uiling is mor snsitiv to th imnsionlity ompr with PrCuinx uiling. Th rson why th prormn o PrCu onstrution is goo, us in high imnsionl sp, th proility o on point omints nothr on, is vry low. Hn, th squntil pttrn is vry w in high imnsionl sp n th mining pross n trmint quikly Fig. 13 Exution tim omprison on inx uiling twn Niv n PrCu 6.5 Etivnss o Comprssion In this xprimnt, w xplor th omprssion nits o PrCuompr with Niv mtho. Fig. 14 () n () show th omprssion t on uiling th t u y prtil orr rprsnttion (PrCu), ompr with D -tr. Thy illustrt tht using th omprss t ormt, DAG, is vry iint on sp usg. Similr to qury prormn, imnsionlity hs mor t on th omprssion tor ompr with th numr o points in th tsts. 7. Conlusions In this ppr, w hv introu Gnrl Dominnt Rltionship Anlysis, whih oul not sily solv y xisting strtgis. Du to th intrrlt onntion twn th prtil orr n th ominnt rltionship, w hv propos nw t strutur ll PrCu, whih onisly rprsnts th omplt inormtion o th gnrl ominnt rltionship s on th prtil orr nlysis. W hv introu iint strtgis to onstrut PrCu. Th xprimntl rsults n prormn stuy onirm th iiny n tivnss o our strtgis. In th utur, w will invstigt how to urthr improv th iiny whil qurying th gnrl ominnt rltionship. Rrns [1] R. Agrwl n R. Sriknt. Mining squntil pttrns. In ICDE 95: Proings o th 11th Intrntionl Conrn on Dt Enginring, pp. 3-14, 1995.

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