Parallel Computation for Managing Transitive Relations. Yijie Han and Yugyung Lee. University of Missouri at Kansas City.

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1 Prlll Computtion or Mnging Trnsitiv Rltions Yiji Hn n Yugyung L Computr Scinc Tlcommunictions Progrm Univrsity o Missouri t Knss City Knss City, Missouri hn,yugig@cstp.umkc.u My 24, 2000 Abstrct This ppr ocuss on th mngmnt o inormtion or trnsitiv rltions. Trnsitiv rltions mol importnt constructs such s inhritnc rltions in objct orint tchnology. Inhritnc hirrchis to l with rl worl problms tn to b lrg, complx n icult to mintin n us. Mintining n cint storg o inhritnc hirrchis rquirs to support st subsumption. In this ppr, th ollowing issus will b rss: (1) W vlut xisting ncoings to unrstn th spc n tim lowr boun rquir or inhritnc hirrchis. (2) W prov tht in th gnrl cs t lst O(n 2 ) bits r n or ncoing trnsitiv rltions no mttr which ncoing schm is us. (3) W show tht in boun cyclic grphs lss thn O(n 2 ) storg is sucint or storing ncoing o th inhritnc hirrchis whil mintining st trnsitivity using prlllism. kywors: prlll lgorithms, trnsitiv rltions, inhritnc, subsumption, ncoing. 1 Introuction Trnsitiv rltions mol mny importnt constructs. In prticulr, it mols inhritnc rltions. Inhritnc ws originlly introuc to support concptul spciliztion. In rlity, it hs ply mny ltrntiv rols or inhritnc in tbs n othr rs, such s structuring or moling mchnism or rsoning bout progrms, n mchnism or co shring n rus in objct-orint prigm [20]. Th inhritnc rltionship mong objcts cn b prsnt in orm o hirrchis, cll inhritnc hirrchis. Th inhritnc hirrchis bcom ocus structur u to wi spr ploymnt o objct-orint tchnology. Th concpts n structur o inhritnc hirrchis hv bn wily ppli to mny rs o computr scinc, such s tbs mngmnt n qury procssing [11, 15], sotwr nginring n moling [19], objct-orint progrmming lngugs incluing C++ n Jv [23], knowlg rprsnttion n rsoning [6, 22]. Thy hv lso bn ppli to mny irnt ppliction rs such s igitl librry browsing [9], micl tbs n rsoning [16], CASE tool n gogrphicl inormtion systms [14]. Inhritnc hirrchis hv bn urthr us in t mining n rl-worl clssiction pplictions such s clssiying ocumnts into subjct ctgoris unr th librry o congrss schm, or clssiying worl-wi-wb ocumnts into topic hirrchis. Th pr r logicl intrrltions btwn th concpts in txonomy, th mor hv bn improv n xtnsiv clssiction by introucing cint hirrchy ncoings in igitl librry pplictions [9, 18]. At ny irction, sinc rl worl hirrchis r lwys lrg n complx, spcil tchniqus r n to chiv such gols n ny systms stning on spcil tchniqus cn gin bnt rom improvmnts in hirrchy rprsnttions. In ling with lrg 1

2 n ynmic hirrchis, mssivly prlll procssing ws us or st trnsitiv rsoning n ynmic upt [12]. Currntly, ynmic n complx spcts o objcts r itionlly consir s n importnt ctor in inhritnc hirrchis. Our rsrch is motivt by Bomml t l.'s [3] xprimntl rsults rom comprisons o svrl incrmntl upt mthos or multipl inhritnc hirrchis. Thir rsults show tht th storg n or hirrchis ws minimiz in multipl inhritnc hirrchis or crtin spcil css. Nvrthlss, th procss pproch will probbly not b usul or rl worl problms, which usully l with gigbyts o t, n rquir rpi growth o storg. As objct-orint tchnology bcoms prvlnt to l with th rl worl problms, th stuy o th storg n computtion o inhritnc hirrchis bcoms n urgnt n importnt rsrch topic. Inhritnc hirrchis cn b rprsnt by irct cyclic grphs (DAGs), i.., G = (V; E) whr V is st o vrtics with jv j = n n E is th st o gs. An xmpl o inhritnc hirrchis is shown in Fig. 1. By mpping irct cyclic grphs to inhritnc hirrchis, vrtics o th grph rprsnt clsss n rcs rprsnt inhritnc rltionships. I thr is n rc rom vrtx to vrtx b, thn b is cll prnt o n is chil o b. Bcus o th xistnc o multipl inhritnc, vrtx my hv multipl prnts. I thr is irct pth rom vrtx to vrtx b, thn b is cll n ncstor o n is scnnt o b. In this ppr, w us trm rch to spciy tht thr is pth rom to b, i.., is rchbl to b through th pth. For inhritnc hirrchis, th subsumption rltionship (i.. whthr b is n ncstor o ) is th mostly quri rltionship. A numbr o continuous orts r lso notworthy, trying to pply wll-vlop thoris o inhritnc hirrchis n thir mngmnt. Th orts ocus on srching or cint ncoing n storg mthos or inhritnc hirrchis [1, 2, 3, 4, 5, 8, 10, 13]. All ths stuis hv common intrst in iscovring n ctiv wy to rprsnt th rl worl problms n lso chiv st rspons. In mny spcil css, som o th propos ncoing schms o chiv cincy n rsult in ning much lss thn O(n 2 ) bits or storg. Howvr, ll xisting schms us t lst cn 2 bits in th worst cs, whr c is constnt, s will b monstrt by us in this ppr. Sinc th irct cyclic grph rprsnting th inhritnc hirrchy my hv bout n 2 rcs, in th wors cs cn 2 bits or storg sm to b th minimum mount o spc n or storing ncoing o th inhritnc hirrchy. Howvr, thr r importnt css whr inhritnc hirrchy is consirbly simplr. For xmpl, whn th numbr o inhritnc links (i.., rcs) is boun by quntity Q, irct cyclic grph o n vrtics hs only bout nq rcs (bcus ch vrtx hs t most Q rcs to its prnts). Thus, this xmpl supports tht thr is goo chnc whr th storg or this clss o inhritnc hirrchis cn b ruc to unr O(n 2 ) bits. Evn in this cs, currnt xisting schms us cn 2 bits. W hv prorm n xtn stuy to n out whthr cint storg schms with lss thn O(n 2 ) bits xist or storing ncoings o inhritnc hirrchis. Mny o th currntly vilbl ncoing schms rquir nonconstnt tim to comput inhritnc oprtions. Som schms r clim by thir invntors to us constnt tim computtion o subsumption oprtion [1, 8]. Howvr, th constnt tim clim is us to prorm oprtions on n bits. Whn n grows lrgr, it woul b icult to imgin tht n bits cn b stor in constnt numbr o wors. Thus, w cnnot count n oprtion on n bits s only on stp procss. It is mor likly tht w shoul ssum on wor cn stor log n bits n thror n bits woul b stor in n= log n wors. Thror, n oprtion such s bit-wis OR oprtion on n bits shoul b count s n= log n oprtions. In this ppr, w propos n ncoing schm tht supports st computtion o inhritnc oprtions. Ths schms shoul b sy to comput n cn b stor cintly. Th primry gol o this ppr is (1) to boun th ovrll spc complxity o ncoing o inhritnc hirrchis bs on n xtnsiv nlysis o xisting ncoing schms n (2) to prsnt n cint schm n its prlll lgorithms or slightly rstrict irct cyclic grphs o multipl inhritnc hirrchis. Th rst o this ppr is orgniz s ollows. Sction 2 nlyzs 2

3 known ncoing mthos n provs thir minimum storg rquirmnts or multipl inhritnc hirrchis. Sction 3 proposs our ncoing schm or inhritnc hirrchis n lgorithms to mng th ncoing. W conclu this ppr in Sction 4. 2 Known Encoing Mthos n Lowr Boun 2.1 Encoing Mthos in th Gnrl Cs Mny mthos or ncoing inhritnc hirrchis hv bn propos [1, 2, 3, 4, 5, 8, 10, 13]. Th most strightorwr wy o ncoing is to us n n n Booln mtrix A to stor th trnsitiv closur o th inhritnc hirrchis, rprsnting s irct cyclic grphs. Thus A[i; j] = 1, i i is n ncstor o j, othrwis A[i; j] = 0. This ncoing schm uss n 2 bits. Most rsrchrs bliv tht this mount o storg is xcssiv. Altrntiv schms hv bn propos to rplc it. Howvr, ll ths schms us t lst cn 2 bits or crtin irct cyclic grphs, whr c is constnt, s will b prov by us in this ppr. Ait-Kci t l. [1] propos mtho bs on comprss vrsion o trnsitiv closur. Thy us nw bit position only whn it is ncssry. During ncoing, th co o vrtx computs bs on binry OR oprtions o th cos o its chilrn. Whn th vrtx hs singl chil, nw bit is to istinguish th vrtx rom its chil. A nw bit is lso whn th ncoing gnrt hs conicts, inicting thr is rltionship btwn vrtics which r unrlt. Although this ncoing schm uss lss numbr o bits thn th Booln trnsitiv closur schm, in th worst cs it svs th mount o spc by only constnt rction o th n 2 bits. In Fig. 2, t lst k bits r us or ll vrtics, bcus ch vrtx b i rquirs nw bit to istinguish it rom othr vrtics. This rsult in t lst k 2 bits us or ncoing th inhritnc hirrchy. Not tht th totl numbr o vrtics in th grph is n = 2k + 2. Thus, bout n 2 =4 bits r us. Gnguly t l. [8] vlop n lgorithm or ncoing th inhritnc rltionship. Thy rst ivi th grph into svrl lyrs. Lyr i + 1 consists o thos prnts ll o whos chilrn hv lry bn clr to b in th lowr lyr 0 through i. Thir lgorithm us bitwis OR oprtions on chilrn's ncoing to riv prnt's ncoing. Thy cll vrtx impur, i it hs mor thn on prnt. I vrtx hs only on prnt, thn tht vrtx is rgr s pur vrtx. Thir lgorithm uss singl istinguish bit to rprsnt ch impur vrtx. Thus, whn thr r bout n impur vrtics, thir ncoing schm uss n 2 bits. As shown in Fig. 2, on istinguish bit is us or ch c i s ch c i is impur. Thus, gin k 2 bits (or bout n 2 =4 bits) r us or th ncoing. Schubrt [17] hs introuc goo t structur to rprsnt th trnsitiv closur o n inhritnc hirrchy. Wht is prticulrly ttrctiv bout this ncoing is tht it prmits trnsitiv closur rsoning in constnt tim, prcticlly inpnnt o th hight o th knowlg bs. Schubrt's originl pproch is limit to trs. Agrwl t l. [2] xtn th ncoing or irct cyclic grphs (DAGs) with n xtn tur o rng comprssion schm tht ssigns n intrvl to vrtx. First, thy obtin spnning tr o th inhritnc hirrchy, postorr trvrsl is us to obtin postorr numbr or ch vrtx. An inx numbr or ch vrtx v, which is th lowst postorr numbr o v's scnnts, is lso obtin. Th inx numbr n th postorr numbr orm n intrvl or ch vrtx, s shown in Fig. 3(). To xcut th subsumption oprtion or vrtics on tr, w just compr th intrvl o vrtx to th intrvl o vrtx b. I on intrvl is contin in th othr intrvl, thn th vrtx with th nrrow intrvl is scnnt o th vrtx with th wir intrvl. Whn thr r multipl inhritnc rltions, multipl intrvls r ssign to vrtx. This is on by propgting n intrvl o vrtx v to ll v's ncstors which r not ncstors o v in th tr. As shown in Fig. 3(b), th intrvl t ([1; 1]) is propgt to c n th intrvl t h ([5; 5]) is propgt to n b. Thir schm is vry wll suit to th css whr thr r w multipl inhritnc occurrncs. In Fig. 2, ch vrtx c i is inhrit rom k vrtics. Thus, ch vrtx b i will b 3

4 ssign k intrvls, on intrvl rom ch vrtx c i. This cn b rsult in t lst k 2 intrvls ssign to ll vrtics. Thus, t lst k 2 bits (or bout n 2 =4 bits) r us. Tlmo t l.[21] clim tht constnt tim cn b chiv or subsumption through t structur rquiring O(n p n) spc complxity, whr n is th numbr o vrtics in givn irct cyclic grph. Thir rsults, howvr, cn b pplicbl only to lttics which r subst o DAGs. On th othr hn our rsults cn b pplicbl to ny DAGs. 2.2 A Spc Lowr Boun or th Gnrl Cs W will now prov tht t lst cn 2 bits r n in th worst cs, no mttr which ncoing schm is us. Tht mns cn 2 bits is lowr boun in th gnrl cs. Although not ll th xisting schms r list in this ppr, ll th xisting schms will rquir t lst cn 2 bits or som constnt c bcus o th ollowing proo. Assum tht G is st o irct cyclic grphs, i.., G =G 1, : : :g. For ch grph G i, w construct n ncoing using th sm ncoing tchniqu. W rr to this ncoing s trnsitiv ncoing o G, not E t (G ). Lt G i, G j 2 G (ny cyclic irct grph), with G i 6= G j n E t (G i ) n E t (G j ) b ncoings o G i n G j, rspctivly. Lmm 1: I G 1 6= G 2 n G 1 n G 2 2 G, lt E 1 = E t (G 1 ) n E 2 = E t (G 2 ), thn E 1 6= E 2. Proo: Sinc G 1 6= G 2 n both hv no isolt vrtics, thr is n rc in ithr G 1 or G 2 but not in both. Without loss o gnrlity, ssum tht (u; v) is such n rc in G 1 but not in G 2. I E 1 = E 2 thn w cn obtin only on nswr rom E 1 n E 2 whn th qury \is v n ncstor o u" is ris. I this nswr is \Ys" thn th nswr is corrct or G 1 but incorrct or G 2, thror, th ncoing E 2 is not th corrct ncoing or G 2. I th nswr is \No" thn th nswr is corrct or G 2 but wrong or G 1, thror, th ncoing E 1 is not th corrct ncoing or G 1. Thror, th ncoing E 1 = E 2 cn b th ncoing or only on o G 1 n G 2 but not both. 2 Now w hv: Thorm 1: In th gnrl cs, t lst n 2 =4 bits r n to nco ll th possibl irct cyclic grphs. Proo: Lt's consir Fig. 2. Thr r k 2 rcs rom vrtics c j 's to vrtics b i 's whr 1 i; j k n k = (n 2)=2. I w numrt ll th possibl situtions o tking st o gs out rom th k 2 gs in th grph, w cn gnrt 2 k2 irnt grphs. Accoring to Lmm 1, w must hv 2 ncoings to rprsnt 2 irnt grphs. So w n t lst k2 k2 k 2 bits to rprsnt 2 k2 ncoings. Not tht th totl numbr o vrtics in th grph is n = 2k + 2. Thus, t lst n 2 =4 bits r us. 2 Thus w obtin lowr boun o n 2 =4 bits or spc rquirmnt. This shows tht, lthough ll known ncoing schms r goo or som spcil css, thr xist irct cyclic grph or which ths ncoing schms us t lst n 2 =4 bits. 2.3 Boun Dgr Cs In this subsction, w invstigt th cs whr ch vrtx hs boun numbr o prnts. Th xtrm cs is tht ch vrtx cn hv t most two prnts. Not tht whn ch vrtx hs only on prnt, th grph is tr n this cs is wll solv with th ncoing mtho o Schubrt t l.[17]. In th schm o Ait-Kci t l. [1] nw bit is whn vrtx hs singl chil to istinguish th vrtx rom its chil. A nw bit is lso whn th ncoing gnrt hs conicts, i.. inicting n xistnc o rltionship btwn vrtics which r unrlt. Fig. 4 shows irct cyclic grph whr ch clss hs only singl prnt. Yt ch vrtx rquirs 4

5 nw bit, bcus i bitwis OR oprtions o th ncoing or chilrn r us or n ncstor thn th ncoing or th ncstor woul b inistinguishbl rom on o its scnnt. Thror, or n vrtics, n 2 bits r us. For th ncoing schm o Gnguly t l. [8], w n to kp th numbr o impur vrtics to only cn in orr to orc thir ncoing schm to us t lst (cn) 2 bits. This is bcus thir ncoing schm uss istinguish bit or ch impur vrtx. As shown in Fig. 5, thr r k vrtics, which r impur. Bcus thir schm uss bitwis OR oprtions to riv ncoing, thir schm woul n n 2 bits or th grph in Fig. 4 s wll. Th schm o Agrwl t l. [2] uss multipl intrvls or multipl inhritnc. In prctic, thir schm usully gnrts vry compct ncoing. Howvr, w cn com up with n inhritnc hirrchy with th multiplicity o th inhritnc boun by 2, in which thir schm ils to gnrt ncoing using lss thn cn 2 bits or som constnt c. S Fig. 6. Bcus intrvl [ i ; b i ] ns to b propgt to t lst k i vrtics, w n up with k(k + 1)=2 intrvls us in th grph. Bcus o th conition o n < 3k + 3, t lst (n 3) 2 =18 intrvls r us. Not hr tht ch intrvl ns log n bits or ncoing. Thror, cn 2 log n bits r n 3 Propos Encoing n Its Mngmnt W hv shown tht ll known ncoing mthos rquir n 2 =4 bits s thir lowr boun (lst mount) o spc. In rlity, it rrly hppns tht clss is inhrit rom hug numbr o prnt clsss. Thus, th cs, whr th multiplicity o th multipl inhritnc is boun, is o mjor intrst. Th simplst cs woul b tht th multiplicity is boun by constnt. In this cs, th grph hs t most n rcs, bcus ch vrtx hs t most rcs connct to its prnts. W hv shown tht xisting schms still n cn 2 bits or storg, vn whn = 2. In othr wors, vn whn ch clss hv t most two prnts, prvious schms us cn 2 bits or storg in th worst cs. On th othr hn, whn th multiplicity o multipl inhritnc is boun or lss thn O(n), w cn prov lowr boun bttr thn cn 2 bits or spc. Dnition 1: Th ingr o vrtx v in irct cyclic grph is th numbr o gs irct towr v, n th outgr o v is th numbr o gs irct wy rom v. Thorm 2: Givn irct cyclic grph G =(V, E), such tht V is st o vrtics v 1, : : :, v n n th outgr o ny vrtx v i is, th lowr boun o th ncoing G is (n=2) log(n=2) bits. Proo: Lt's consir Fig. 7 with th possibl wys o plcing! rcs rom c j 's to b i 's. Sinc ch k vrtx c j cn hv t most rcs to b i 's, thr r > > (k=) wys w cn plc ths rcs. Thror, or ll k j 's w cn construct > (k=) k irnt grphs. Thror, by Lmm 1, w n t lst k log(k=) > (n=2) log(n=2) bits or ncoing th grph. 2 Whn th multiplicity o multipl inhritnc is boun by quntity, ch vrtx hs t most rcs connct to its prnts. Thus th inhritnc grph cn b stor in O(n) wors ssuming ch wor hs log n bits (thror O(n log n) bits). Although th trivil storg schm (just stor th rc list or ch vrtx) stors th inhritnc grph with lss thn O(n 2 ) spc, th schm os not support st procssing o subsumption quris. W now prsnt nw schm which rucs th spc n or storing ncoing o th inhritnc hirrchis n support st qury procssing. Th nw ncoing schm uss much lss thn cn 2 bits whn th multiplicity o multipl inhritnc is smll, but it still supports st qury procssing. Our propos solution to ruc th storg n tim rquirmnts o inhritnc grph is to us n pproch o oubling th rchbl vrtics or grphs o boun outgr. Th propos ncoing consists o thr phss: (1) Construction o trnsitiv closur ncoing, 5

6 (2) Qury procssing, n (3) Gnrliztion. W hv chiv vry cint solution or th spc n tim rquirmnts by prtitioning th construction o trnsitiv closur ncoing into construction n qury procssing phss n lying th complt construction o th ncoing to th qury tim. In othr wors, w prtilly construct th ncoing in th construction phs n incrmntlly builing th ncoing in n uring qury procssing phs. W rst l with irct cyclic grph with outgr 2 (i.. w llow ch vrtx to hv t most 2 prnts). Thn, w monstrt how to gnrliz our mtho to th grphs o outgr. 3.1 Construction o Trnsitiv Closur Encoing Th propos ncoing rquirs to buil st o ncstors which r rchbl rom vrtx v in irct cyclic grph G. Hr w ssum tht ch vrtx hs t most two prnts n thror, it hs n initil st o ncstors consisting o itsl n t most two prnts. Initil storg rquirmnts or th ncoing is 2n log n, whr ch vrtx rquirs two wors to rprsnt thir prnts n ch wor uss log n bits. Our lgorithm to construct th ncoing is strting rom ch vrtx in prlll, incrmntlly xpning sts o ncstors stp by stp, which r rchbl rom th givn vrtx. Blow is th lgorithm. Algorithm 1: Construction o Trnsitiv Closur Encoing Lt v b ny vrtx in th irct cyclic grph G with outgr 2. Lt rch(v) b st o vrtics. or ll v o in prlll bgin rch(v) = vg [ prnts o vg; or i = 1 to t o /* t is prmtr. */ rch(v) = rch(v) [ rch(rch(v)); n In th bov lgorithm rch(v) is th st o vrtics rch rom v t ny crtin stp o xpnsion, i.., stp t whr t is prmtr or th stp. Initilly mximum o thr vrtics r rchbl rom givn vrtx v. In th subsqunt xpnsion stp, mximum 7 vrtics with 4 nw vrtics through two prviously rchbl vrtics r now rchbl rom vrtx v, in t xpnsion stp, mximum o 2 2t +1 1 vrtics r rchbl rom vrtx v. Fig. 8 shows n xmpl o inhritnc hirrchy n its boun grph ( = 2). In th originl grph (Fig. 8()), q hs thr prnts (k, m, o). In th boun grph (Fig. 8(b)), by introucing nw vrtx l, q's thr prnts (k, m, o) r splitt into two prnts (l, o) n l hs two prnts (k, m). For th boun grph (Fig. 8(b)), w show in Tbl 1 how Algorithm 1 works or construction o rch(v) or th xpnsion stp t = 1. Th bol lttr is us to spciy nwly rchbl vrtx t th spcic stp. Lt us nlyz th spc n tim us or constructing trnsitiv closur ncoing. In th construction, w r tking vntg o prlll computtion. This rsults in st ncoing construction tim. Suppos tht w tk t = log log n itrtions in th bov lgorithm, thn th totl numbr o vrtics which cn b rch rom th givn vrtx v is 2 2log log n +1 1 = 2 log n+1 1 = 2n 1. In rlity, this quntity is lwys boun by n, bcus thr r only n vrtics in th grph. Thror, totl spc to stor th ncoing or ll vrtics will b O(n 2 log n) (Not tht ch vrtx rquirs O(log n) storg to nco). Howvr, i w tk lss thn log log n itrtions in th bov lgorithm, i.., lt t = log log n whr is quntity to b x ltr, w cn stor th ncoing with much lss thn O(n 2 ) bits. Th totl vrtics w cn rch tr th t stps r 2 2log log n +1 1 = 2 log n= = 2n 1=2 1 < 2n 1=2. Thus, or ll vrtics w n only 2n 1+1=2 log n bits which is much lss thn n 2 bits. Th prlll tim n to ccomplish computtion or th bov lgorithm is log log n stps with th ploymnt o O(n 1=2 ) procssors or ch vrtx or O(n 1+1=2 ) procssors or ll vrtics on Concurrnt R Concurrnt Writ 6

7 Vrtx b c g h i j k l m n o p q r g j k l m o q b c Initiliztion b c h i g k h n k p l i m i i n j o First Pss b c b c g c b h c i c c j k g i l k m g i h m hi n i o k n g i p j q l o k m n r p q r p q j l o Tbl 1: Construction o Trnsitiv Closur Encoing or Fig. 8 (CRCW) Prlll Rnom Accss Mchin (PRAM) [7]. I w xcut th lgorithm on squntil mchin th tim complxity woul bcom O(n 1+1=2 ). 3.2 Qury procssing Now w show how to prorm th qury procssing or subsumption. As th scon phs o our computtion, w procss qury \Is v n ncstor o u?" Th procss is to vriy whthr v is n lmnt o rchbl st which is builing incrmntlly strting rom vrtx u using th ncoing A rom Algorithm 1. I vrtx w cn b rch rom u in A by pth o lngth 1 (i.. singl rc) thn w cn b rch rom u in th grph G by pth o lngth 2 t = 2 log log n. Sinc G hs n vrtics, th longst shortst pth (i.. mx p min p p) in G hs lngth no mor thn n 1. Bcus ch pth o lngth 2 t is rprsnt by singl rc in A, th longst shortst pth in A is < n=2 log log n. Algorithm 2: Subsumption /* Qury whthr v is n ncstor o u.*/ bgin rch A (u) = ug; rch ront A (u) = prnt A (rch A (u)); or i = 1 to n=2 log log n o bgin rch A (u) = rch A (u) [ rch ront A (u); i v 2 rch A (u) thn print \v is n ncstor o u"; xitg; rch ront A (u) = prnt A (rch ront A (u)); n print \v is not n ncstor o u"; n In th bov lgorithm, prnt A (S) is th st o prnts (with rspct to ncoing A, in our cs th lst row o Tbl 1) o ll vrtics in st S. To sp up th procssing w us prlll procssing in st oprtions such s union n mmbr oprtions using O(n 1+1=2 ) procssors. rch A (u) is th st o ll th vrtics which hv bn rch rom u (with rspct to ncoing A) t th currnt stp. rch ront A (u), which is n xpning ront o vrtics rch rom u 7

8 t spcic stp, is inclu in th rch A (u) in th subsqunt stp. W illustrt th procss o Subumption in Algorithm 2 with n xmpl (s Tbl 1). Lt u n v b r n c. Initilly, rch A (r) = rg n rch ront A (r) = p; q; j; l; og (lst column o lst row in Tbl 1). Atr th rst pss, rch A (r) = r; p; q; j; l; og n rch ront A (r) = ; c; ; ; g; h; i; j; k; l; m; n; og. In th scon pss, rch A (r) = r; p; q; j; l; o; ; c; ; ; g; h; i; k; m; ng. Thus in th scon pss Algorithm 2 conclus with \c is n ncstor o r" sinc c is n lmnt in rch A (r). Now i lgorithm Subsumption runs on CRCW PRAM, th tim n is th numbr o itrtions in th bov lgorithm, i.. O(n=2 log log n ). Th spc us in th lgorithm is O(n 1+1=2 ) bcus rch ront A (u) cn hv t most n vrtics n thror prnt A (rch ront A (v)) my involv O(n 1+1=2 ) rcs. Thus w hv chiv o(n 2 ) (smll o) spc with sublinr tim qury procssing tim. Th numbr o procssors us is lso O(n 1+1=2 ). Th ollowing tbl shows th summry o tim n spc rquirmnts or construction o ncoing schm n qury procssing or givn inhritnc hirrchis or vrious pprochs. Encoing TC Mtrix Ait-Kci t l. Gnguly t l. Agrwl t l. Our Approch Spc O(n 2 ) O(n 2 ) O(n 2 ) O(n 2 ) O(n 1+1=2 log n) Cons. Op. O(n 3 log n) O(n 2 log n) O(n 2 log n) O(n 2 ) O(n 1+1=2 ) Prlll Cons. Tim O(log n) O(n) O(n) O(log 2 n) O(log n) Supsumption O(1) O(n= log n) O(n= log n) O(log n) O(n=2 log log n ) Tbl 2: Comprison o Inhritnc Hirrchis Encoings Assuming tht thir lgorithms procss log n bits in stp inst o procssing n bits in stp s i in thir ppr. Cons. Op. is th numbr o oprtions (tim procssor prouct) n or constructing th ncoing. 3.3 Gnrliztion Th im o th gnrliztion is to convrt n input grph to irct cyclic grph such tht ch vrtx in th grph my hv t most 2 rcs going to its prnts. Suppos w hv schm which uss (n) storg or irct cyclic grph o n vrtics n outgr = 2, thn w immitly hv schm using (n) storg or n rbitrry outgr. This is bcus w cn xpn irct cyclic grph o outgr to irct cyclic grph o outgr 2 by ing t most n vrtics, s shown in Fig. 9. This pproch is vry goo on, i unction is slow growing. For xmpl, i (n) = n log n, thn (n) = n(log n + log ) 2n log n. This mns i w multiply th numbr o outgr by, thn w n to multiply th mount o spc by only 2. Howvr, i is st growing, thn this pproch osn't sm to b pproprit. For xmpl, i (n) = n 3=2, thn (n) = 3=2 n 3=2 > n 3=2. This mns i w multiply th numbr o outgr by, thn w hv to multiply th mount o spc by 3=2. In our cs, it is obvious tht th squntil tim complxity to convrt irct cyclic grph o outgr to irct cyclic grph o outgr 2 is O(n). Th prlll tim complxity is O(log ) with n= log procssors. Sinc our schm uss cn 1+1= log n bits or constnt c, whn irct cyclic grph o outgr is convrt to irct cyclic grph o outgr 2, th spc us bcoms c(n) 1+1= log(n). Thus, i is not vry lrg, our schm is still vry much vorbl. Tk, or xmpl, = n 1=2 n = 10, w will rch spc rquirmnt o O(n 33=20 log n), which is vry goo. 8

9 4 Conclusions Th inhritnc hirrchis rprsnt th importnt rltionships in objct-orint tchnologis. Currntly vilbl ncoing n mngmnt or inhritnc hirrchis r not stisctory, bcus th computtionl rquirmnts hv not bn sucintly rss. In this ppr, th tim n spc lowr boun o inhritnc hirrchis ncoings is nlyz n compr. Bs on th xtnsiv stuy, w invstigt nw ncoing schm o inhritnc hirrchis n nw lgorithms to support its cint mngmnt. In ovrll, our pproch is signicntly improv in trms o spc n tim rquirmnts or trnsitiv closr mngmnt o inhritnc hirrchis. Rrncs [1] H. Ait-Kci, R. Boyr, P. Lincoln, R. Nsr. Ecint implmnttion o lttic oprtions. ACM Trnsctions on Progrmming Lngugs n Systms 11, 1 (Jnury 1989), [2] R. Agrwl, A. Borgi, J.V. Jgish. Ecint mngmnt o trnsitiv rltionships in lrg t n knowlg bss. Proc. ACM-SIGMOD Intrntionl Conrnc on th Mngmnt o Dt (Jun 1989), [3] M. F. vn Bomml, T. J. Bck. Incrmntl ncoing o multipl inhritnc hirrchis. Proc. Eighth Int. Con. on Inormtion Knowlg Mngmnt (ACM CIKM'99), Knss City, Missouri, pp (Nov. 1999). [4] Y. Csu. An objct-orint uctiv lngug. Annls o Mthmtics n Articil Intllignc 3, 2 (Mrch 1991). [5] Y. Csu. Ecint hnling o multipl inhritnc hirrchis. ACM SIGPLAN Notics, 8(28):271, Octobr [6] I. Dun I n G. Tlns, A sign mtho or multi-xprt objct cs-bs-rsoning systm, Procings o th 9th Intrntionl Conrnc Articil Intllignc EXPERSYS- 97. Articil Intllignc Applictions. Gourny sur Mrn, Frnc, pp.49-54, [7] S. Fortun, J. Wylli. Prlllism in rnom ccss mchins. Proc. 10th ACM Symposium on Thory o Computing, Sn Digo, Cliorni, (1978). [8] D. Gnguly, C. Mohn, S. Rnk. A spc-n-tim cint coing lgorithm or lttic computtions. IEEE Trnsctions on Knowlg n Dt Enginring 6, 5 (Octobr 1994), [9] S. Gnr, D. Agrwl, A. El Abbi, T. Smith. Browsing lrg igitl librry collctions using clssiction hirrchis. Proc. Eighth Int. Con. on Inormtion Knowlg Mngmnt (ACM CIKM'99), Knss City, Missouri, pp (Nov. 1999). [10] M. Hbib, L. Nourin. Embing prtilly orr sts into prouct o chins. Proc. First Intrntionl Symposium on Knowlg Rprsnttion, Us n Storg or Ecincy (KRUSE'95), Snt Cruz, USA,

10 [11] J. Hn n R. T. Ng n Y. Fu n S. K. Do, Dling with Smntic Htrognity by Gnrliztion-Bs Dt Mining, Tchniqus, (itor) M. P. Ppzoglou n G. Schlgtr, Cooprtiv Inormtion Systms: Currnt Trns & Dirctions, Acmic Prss Nthrlns, pp. 207{231, [12] E. Y. L n J. Gllr, Constnt Tim Inhritnc with Prlll Tr Covrs, Procings o th Flori AI Rsrch Symposium (FLAIRS) Ky Wst, Flori, 243{250, [13] C. Mllish. Trm-ncobl scription spcs. In Logic Progrmming 1990 Pr-Conrnc Procings, 1-15, Assoc. Logic Progrmming, UK Brnch, [14] J. Montilv, Formlizing hyprmp objct mol or multimi-gogrphicl inormtion systms, Procings o th Intrntionl Conrnc on Inormtion Systms Anlysis n Synthsis. ISAS'96, Orlno, FL, pp [15] M. G. Tylor n K. Stol n J. A. Hnlr, Ontology-bs Inuction o High Lvl Clssiction Ruls, SIGMOD Dt Mining n Knowlg Discovry workshop procings, Tuscon, Arizon, [16] A. Rctor n P. Znstr n W. Solomon n J. Rogrs n R. Bu n W. Custrs n W. Clssn n J. Kirby n J. Rorigus n A. Mori n E. Vn r Hring n J. Wgnr, Rconciling Usrs' Ns n Forml Rquirmnts: Issus in Dvloping Rusbl Ontology or Micin, IEEE Trnsctions on Inormtion Tchnology in Biomicin, 2(4), pp. 229{242, [17] L. K. Schubrt n M. A. Pplskris n J. Tughr, Acclrting Ductiv Inrnc: Spcil Mthos or Txonomis Colors n Tims, (itor) N. Crcon n G. McCll, Th Knowlg Frontir, Springr Vrlg Nw York, NY, pp , [18] D. Schwrtz Intllignt Inxs with Inhritbl Fuzzy Rltions 18th Intrntionl Conrnc o th North Amricn Fuzzy Inormtion Procssing Socity Pisctwy, NJ, 371{375, [19] T. Shih, Yul-Chyun Lin, W. Pi, Chun-Chi Wng, An objct-orint sign complxity mtric bs on inhritnc rltionships, Intrntionl Journl o Sotwr Enginring & Knowlg Enginring 8(4), pp , [20] A. Tivlsri, On th notion o inhritnc, ACM Computing Survys, 28(3), pp , [21] M. Tlmo n P. Vocc, An cint tt structur or lttic oprtions. SIAM J. on Computing, vol. 28, no. 5, 1999, pp [22] C. Yu, W. Mng, K-L. Liu, W. Wu, N. Rish. Ecint n ctiv mtsrch or lrg numbr o txt tbss. Proc. Eighth Int. Con. on Inormtion Knowlg Mngmnt (ACM CIKM'99), Knss City, Missouri, pp (Nov. 1999). [23] K. Wih, Using tmplts to improv C++ sign, C++ Rport, 10(2), pp.14-21,

11 n n-1 b c Figur 1: An inhritnc hirrchy. Figur 4: A worst cs o Ait-Kci's ncoing. b b 1 2 b3 bk c 1 c 2 c 3 c k k-3 k-2 k-1 k Figur 2: Ait-Kci t l. ncoing. n Gnguly t l. Figur 5: A worst cs o Gnguly's ncoing. [1,9] [1,9] [, b ] k k [, b ], k k [, b ] k-1 k-1 [, b ] 1 1 [, b ] k-1 k-1 [, b ] 1 1 b [1,4] c [5,8] b [1,4][5,5] c [5,8][1,1] [, b ] k-1 k-1 [, b ] k-2 k-2 [, b ] [, b ] k-2 k g [1,1] [3,3] [2,2] [5,7] [1,1] [2,2] [3,3][5,5] g [5,7] [, b ],[, b ], [, b ], 3 3 [ 4, b 4 ] () h i [5,5] [6,6] (b) h i [5,5] [6,6] [, b ] 4 4 [, b ] 3 3 [, b ] 2 2 [, b ],[, b ], [ 3, b 3 ] [, b ],[, b ] [, b ] 1 1 Figur 3: Agrwl t l. ncoing. Arcs with ott lins r nontr rcs. [, b ] 1 1 Figur 6: A worst cs o Agrwl's ncoing. 11

12 b b 1 2 b3 bk c 1 c 2 c 3 c k Figur 7: Lowr boun or outgr. b. c.... g. h. i.. j k. m. n.. o. p q.. () r. b. c.... g. h. i.. j k. m. n. l.. o. p q.. (b) r. Figur 8: An xmpl o Encoing Construction. b c b c Figur 9: Convrting gr to 2. 12