Institute for Advanced Computer Sciences. Abstract. are described for building these three data structures that make use of these

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1 Dt-Prlll Prmtvs or Sptl Oprtons Erk G. Holy Computr Sn Dprtmnt Unvrsty o Mryln Coll Prk, Mryln 074 ol@s.um.u Hnn Smt Computr Sn Dprtmnt Cntr or Automton Rsr Insttut or Avn Computr Sns Unvrsty o Mryln Coll Prk, Mryln 074 js@s.um.u Astrt Dt-prlll prmtvs or prormn oprtons on t PM qutr, ukt PMR qutr, n R-tr sptl t struturs r prsnt usn t sn mol. Alortms r sr or uln ts tr t struturs tt mk us o ts prmtvs. T t-prlll lortms r ssum to mn mmory rsnt. T lortms wr mplmnt on mnmlly onur Tnkn Mns CM-5 wt 3 prossors ontnn GB o mn mmory. Ts work ws support n prt y t Ntonl Sn Founton unr Grnt IRI

2 Introuton Sptl t onssts o ponts, lns, rons, rtnls, surs, volums, n vn t o r mnson w nlus tm. Sptl t rss n ppltons n mny rs nlun omputr rps, omputr vson, m prossn, pttrn ronton, roots, omputtonl omtry, sol moln, omputr- rtorpy, nry pyss, nt-lmnt nlyss, orp normton systms (GIS), n tss. T ny o solutons to prolms n ll o ts rs s nn y t o o n pproprt rprsntton (s,.., [Sm90, Sm90]). T ky ssu s tt t volum o t t s lr. Ts s l to n ntrst n prlll prossn o su t. Tr r two possl ppros trm m-sp n ojt-sp [Fol90]. In prtulr, t ojt-sp ppro ssns on prossor pr sptl ojt (.., [Bst9, Hol93, Hol94, Hol94]), wl t m-sp ppro ssns on prossor pr ron o sp (.., [Frn90]). In ts ppr our ous s on ojt-sp t-prlll rprsnttons o sptl t. T rprsnttons w w suss sort t t wt rspt to t sp tt t oups. Ts rsults n spn up oprtons nvolvn sr. T t o t sort s to ompos t sp rom w t t s rwn nto rons ll ukts. Our prsntton s or sptl t onsstn o ollton o lns su s tt oun n ro mps, utlty mps, rlwy mps, t. O ours, smlr rsults oul otn or otr typs o sptl t. On ppro known s n R-tr [Gutt84] ukts t t s on t onpt o mnmum ounn (or nlosn) rtnl. In ts s, lns r roup (opully y proxmty) nto rrs, n tn stor n notr strutur su s B-tr [Com79]. T rwk o t R- tr s tt t os not rsult n sjont omposton o sp tt s, t ounn rtnls orrsponn to rnt lns my ovrlp. Equvlntly, ln my sptlly ontn n svrl ounn rtnls, yt t s only ssot wt on ounn rtnl. Ts mns tt sptl qury my otn rqur svrl ounn rtnls to k or srtnn t prsn or sn o prtulr ln. T non-sjontnss o t R-tr s ovrom y omposton o sp nto sjont lls. In ts s, ln s ompos nto sjont sulns su tt o t sulns s ssot wt rnt ll. Tr r numrovrnts o ts ppro. Ty r n t r o rulrty mpos y tr unrlyn omposton ruls n y t wy n w t lls r rt. T pr p or t sjontnss s tt n orr to trmn t r ovr y prtulr ln, w v to rtrv ll t lls tt t oups. T rson s tt ln s ompos nto s mny ps (trm q-s) s tr r lls trou w t psss. Tr r two prnpl mtos: t R + -tr [Flo87] n vrnts o t PM qutr [Sm85, Nls86]. T prnpl rn twn tm s tt t lttr s s on rulr omposton o sp wl t ormr s not. Hr w stuy t lttr. Pror rsr n t prlll sptl omn s n lmt to qutrs, k-d-trs, n R-trs. T qutr rsr s prmrly ouss on r (or rstr) t n ron qutrs. Mu rsr s onntrt on lortms or uln (tr n top-own or ottom-up mnnr) ot pontr-s n lnr ron qutrs [Dn9, Ir93]. Otr orts v ouss on vlopn nor nn tnqus [Nn88] swll s xtrtn ron proprts n prormn st tort qurs [Bs88, Ks88, Dn9]. Som o t work s mploy proprtry prlll rtturs (.., two-mnsonl su xn ntwork [M86], or Drt [Mrt86]), or rnt prormmn lnus (.., Conurrnt Prolo [El85]) wl t mjorty s lt wt ypru rtturs. Bstul [Bst9] xtn t rsr unr t t prlll SAM (or Sn-An-Monoton-Mppn) mol o prlll omputton. In ton to ln wt lnr ron qutrs n t t prlll [Hll86] ontxt, lortms wr vlop y Bstul or uln n mnpultn (.., st tort sptl qurs) PR qutrs [Orn8, An83, Ros83] n PM qutrs. T k-d-tr [Bnt75] rsr ws lmt to smll ut mportnt srpton o t lortm or uln t t strutur or ollton o ponts usn t sn mol o omputton [Bll89]. T prlll R-tr rsr s n sprs n s onntrt on lortms or snl

3 pu{multpl prlll sk systms [Km9]. In ts ppr our ous s on t prmtvs tt r n to ntly onstrut ts rprsnttons. Our ol s on o sown t rr ow t nlos o rltvly smpl squntl oprtons n mplmnt n t-prlll nvronmnt. Our prsntton ssums tt t t-prlll lortms r mn mmory rsnt. Our lortms wr mplmnt n C on mnmlly onur Tnkn Mns CM-5 wt 3 prossors ontnn GB o mn mmory (t lortms v lso n run on 6K prossor CM-). T rst o ts ppr s ornz s ollows. Ston ry srs t sptl t struturs on w w ous. Ston 3 rvws numr o rnt prlll mols o omputton n ponts out ow ty l wt sptl t struturs. Ston 4 susss t t-prlll prmtvs tt r us to onstrut t t struturs, wl Ston 5 prsnts t lortms n trms o ts prmtvs. Ston 6 ontns som onlun rmrks. Sptl Dt Struturs In ts ston w rvw t tr t struturs tt r suss n t susqunt stons. W rst prsnt t PM qutr mly n xpln t vrnts tt w onsr, s wll s wy som o tm r not sutl or t-prlll nvronmnt. Ts s ollow y n xplnton o t R-tr. In nrl, w otn rtn t ornl nms o t t struturs ltou mor propr srpton woul us t qulr t prlll. W o not mk us o t unlss t stnton ns to mpsz n t s o potntl or msunrstnn lm.. PM Qutr T PM qutr [Sm85] svrtx{s mmr o t PM qutr mly. Wn nsrtn ln smnts nto ron, t ron s rptly suv untl rsultn ron ontns t most snl vrtx. Atonlly, ron ontns ln smnt vrtx (or npont), t my not ontn ny porton o notr ln smnt unlss tt otr ln smnt srs snl vrtx wt t ornl ln smnt n t sm ron. For xmpl, n Fur, ln smnts,, n sr ommon npont w lls n t ron ll A o t qutr. Do not tt t lr s ron ws suv s t ontns ln smnts n (w sr ommon npont tt lls outs t s rons). A Fur : PM qutr or n xmpl tst. T prmry prolm wt ts rprsntton s vnt wn two ln smnts v nponts tt r vry los totr, rsultn n lr numr o suvsons n orr to sprt t two nponts (or mor tls o ts ptolol vor, s [Nls86]). For xmpl, onsr Fur wr t nsrton o son ln smnt (ln n Fur ) rsults n v lvls o no suvson n t rton o tn nw nos n t PM qutr rprsntton (lvn o w r mpty). 3

4 () () Fur : () PM qutr onsstn o snl ln, n () t sm PM qutr tr t nsrton o son ln smnt (ln ) tt s vrtx tt s los to on o ln 's vrts.. PMR Qutr T PMR qutr (or polyonl mp rnom [Nls86, Nls87]) s n {s mmr ot PM qutr mly (s lso -Exll [Tmm8]). It mks us o prolst splttn rul wr lok s prmtt to ontn vrl numr o ln smnts. T PMR qutr s onstrut y nsrtn t ln smnts on-y-on nto n ntlly mpty strutur onsstn o on lok. E ln smnt s nsrt nto ll o t loks tt t ntrsts. Durn ts pross, t oupny o t lok sk to s t nsrton uss t to x prtrmn splttn trsol. I t splttn trsol s x, tn t lok s splt on, n only on, nto our loks o qul sz. T rtonl s to vo splttn no mny tms wn tr r w vry los lns n lok. In ts mnnr, w vo ptololly ss sr n t srpton o t PM qutr n lt n Fur. A ln smnt s lt rom PMR qutr y rmovn t rom ll t loks tt t ntrsts. Durn ts pross, t oupny o t lok n ts slns (t ons tt wr rt wn ts prssor ws splt) s k to s t lton uss t totl numr o ln smnts n tm to lss tn t prtrmn splttn trsol. I t splttn trsol xs t oupny o t lok n ts slns, ty r tn mr n t mrn pross s rursvly rppl to t rsultn lok n ts slns. Not t symmtry twn t splttn n mrn ruls. Fur 3: PMR qutr wt splttn trsol o two or t ollton o ln smnts o Fur. Fur 3 s n xmpl o PMR qutr orrsponn to st o 9 s ll trou nsrt n nrsn orr. Osrv tt t sp o t PMR qutr or vn tst s not unqu; nst, t pns on t orr n w t lns r nsrt nto t. Ts strutur ssums tt t splttn trsol vlu s two. T vnt o t PMR qutr ovr t PM qutr (n ts vrtx-s ppro) 4

5 s tt tr s no n to suv n orr to sprt ln smnts tt r vry \los" or wos vrts r vry \los" (rll Fur ). Ts s mportnt sn our loks r rt t suvson stp, n wn mny suvson stps our, mny mpty loks r rt, try ln to n nrs n t stor rqurmnts. Gnrlly, s t splttn trsol s nrs, t onstruton tms n stor rqurmnts o t PMR qutr rs wl t tm nssry to prorm oprtons on t wll nrs. It s ntrstn to pont out tt ltou ukt n ontn mor ln smnts tn t splttn trsol, ts s not prolm. In t, t n sown tt t mxmum numr o ln smnts n ukt s oun y t sum o t splttn trsol n t pt o t lok (.., t numr o tms t ornl sp s n ompos to yl ts lok), prov tt t ukt s not t t mxml pt llow y t prtulr mplmntton o t PMR qutr [Sm90]... Bukt PMR Qutr T sp o t PMR qutr s pnnt upon t nsrton orr o t nput t. In t tprlll nvronmnt, lns r nsrt smultnously urn t strutur onstruton. Tus, t orrn o t lns s unknown. Bus t sp o t PMR qutr rls upon t orrn o t t (or mor tls, s Ston 5.), t nton o t PMR qutr s sltly mo to yl t ukt PMR qutr. Fur 4: A ukt PMR qutr wt ukt pty o two n mxml tr t o tr w orrspons to t tst sown n Fur. T ukt PMR qutr [Hol94] svrnt o t PMR qutr wr nst o splttn n ovrown lok on, t lok (or ukt) s splt rptly untl su-ukt ontns not mor tn lns (wr s t mxml ukt pty). T motvton n t t strutur s t sr or t strutur wos sp s npnnt o t ln smnt nsrton orr. Not tt unlss t ukt pty s rtr tn or qul to t mxmlnumr o ntrston lns, t rursv omposton wll ontnu to t mxml pt llow y t ukt PMR qutr. For xmpl, onsr Fur 4 wr t rons orrsponn to t nponts o ln suv untl t mxml pt o t qutr (tr n ts s) s r..3 R-tr T R-tr (ornlly us to rprsnt olltons o rtnls [Gutt84]) n ts vrnts r sn to ornz ollton o rtrry omtr ojts n mnsons (most notly twomnsonl rtnls) y rprsntn tm s -mnsonl rtnls. E no n t tr orrspons to t smllst -mnsonl rtnl tt nloss ts son nos. T l nos ontn pontrs to t tul omtr ojts n t ts, nst o sons. T ojts (.., ln smnts n our s) r rprsnt y t smllst ln rtnl n w ty r ontn. 5

6 Otn t nos orrspon to sk ps n, tus, t prmtrs nn t tr r osn so tt smll numr o nos s vst urn sptl qury. Not tt t ounn rtnls orrsponn to rnt nos my ovrlp. Also, ln smnt my sptlly ontn n svrl nos, yt t s only ssot wt on no. Ts mns tt sptl qury my otn rqur svrl nos to vst or srtnn t prsn or sn o prtulr ln smnt. T s ruls or t ormton o n R-tr r vry smlr to tos or B-tr. All l nos ppr t t sm lvl. E ntry n l no s -tupl o t orm (R,O) su tt R s t smllst rtnl tt sptlly ontns ln smnt O. E ntry n non-l no s -tupl o t orm (R,P ) su tt R s t smllst rtnl tt sptlly ontns t rtnls n t l no pont t y P. An R-tr o orr (m,m) mns tt no n t tr, wt t xpton o t root, ontns twn m M= n M ntrs. T root no s t lst two ntrs unlss t s l no. R R4 R R5 R3 R6 R0 R3: R0: R R R: R3 R4 R: R5 R6 R4: R5: R6: () () Fur 5: () T sptl xtnts o t ounn rtnls n () t R-tr or t xmpl ollton o ln smnts. For xmpl, onsr t ollton o ln smnts vn n Fur. Lt M = 3 n m =. On possl R-tr or ts ollton s vn n Fur 5. Fur 5 sows t sptl xtnt o t ounn rtnls o t nos n Fur 5, wt rokn lns notn t rtnls orrsponn to t sutrs root t t non-l nos. Not tt t R-tr s not unqu. Its strutur pns vly on t orr n w t nvul ln smnts wr nsrt nto (n possly lt rom) t tr. T lortm or nsrtn ln smnt (.., ror orrsponn to ts nlosn rtnl) n n R-tr s nloous to tt us or B-trs. Nw ln smnts r to l nos. T pproprt l no s trmn y trvrsn t R-tr strtn t ts root n t stp oosn t sutr wos orrsponn ounn rtnl woul v to nlr t lst. On t l no s n trmn, k s m to s nsrton o t ln smnt wll us t no to ovrow. I ys, tn t no must splt n t M + rors must strut n t two nos. Splts r propt up t tr. Tr r mny possl wys to splt no. On possl ol s to strut t rors mon t nos so tt t lkloo tt t nos wll vst n susqunt srs wll ru. Ts s ompls y mnmzn t totl r o t ovrn rtnls or t nos (.., ovr). An ltrntv ol s to ru t lkloo tt ot nos r xmn n susqunt srs. Ts s ompls y mnmzn t r ommon to ot nos (.., ovrlp). O ours, t tms ts ols my ontrtory. For xmpl, onsr t our rtnls n Fur 6. T rst ol s sts y t splt n Fur 6, wl t son ol s sts y t splt n Fur 6. Guttmn [Gutt84] us n lortm s on t mnmzton o t totl r o t ovrn rtnls (.., t rst o t ols sr ov). Bkmnn [Bk90], owvr, mploy no splttn tnqu rsultn n wt s trm n R -tr. Ts tnqu ttmpts to mnmz t mount o ntrston r twn ovrn rtnls, w 6

7 () () () Fur 6: () Four rtnls n t splts tt woul nu, () y mnmzn t totl r o t ovrn rtnls n () y mnmzn t r o ntrston twn t ovrn rtnls o ot nos. orrspons to t son o t prvously sr ols. 3 Mols o Prlll Computton In ts ston w sr tr mols o prlll omputton: PRAM, Sn, n SAM. In t pross w lort on tr sutlty or oprtons on sptl t struturs. As w wll s, t sn mol s t most pproprt. For t sn mol w lso sr t typs o prmtv oprtons s ty wll us n t srpton o t sptl prmtv oprtons n Ston PRAMs An N prossor Prlll Rnom Ass Mn (or PRAM) onssts o prossors P ;P ;;P N n lol sr mmory [Kuk77, L9]. Fur 7 ontns smpl rprsntton o n N prossor PRAM. E o t N prossors n r or wrt rom ny loton wtn t sr mmory t stp o t omputton. lol sr mmory P P P n Fur 7: Smpl ur rprsntn N prossors onnt to ommon sr mmory n t PRAM mol. PRAMs r ommonly lss orn to onurrnt ss plts o t lol sr mmory. T most rstrtv o t mols s t xlusv-r, xlusv-wrt (EREW) PRAM. At ny st o t omputton, only on prossor s llow to tr r rom or wrt to sp mmory loton n t lol sr mmory. Iw rlx t xlusv r onstrnt n llow multpl prossors to smultnously r rom sp mmory loton, w otn t onurrnt-r, xlusv wrt (CREW) PRAM. Fnlly, t xlusv wrt onstrnts s smlrly rlx, w otn t onurrnt-r, onurrnt-wrt (CRCW) PRAM. T PRAM mol o prlll omputton rs t usr rom t tous tls o tully mplmntn prlll lortm on prlll mn. T prormmr os not n to worry out t prossor ntronnton topoloy n ommunton onts. Unortuntly, lr sr mmory prlll omputrs r ult to mplmnt, n sr-notn mns ppr to mor sll n wll sut to vlopn prlll mns wt lr numrs o prossors [DW9]. It s possl owvr to mult PRAMs on lr sr-notn mns (.., yprus [L9]) su s t CM-5, ut wt prormn pnlts [Alt87]. 7

8 3. Sn Mol T sn mol o prlll omputton [Bll88, Bll89] s n n trms o ollton o prmtv oprtons tt n oprt on rtrrly lon vtors (snl mnsonl rrys) o t. Tr typs o prmtvs (lmntws, prmutton, n sn) r us to prou rsult vtors o qul lnt. A sn oprton [Sw80] tks n ssotv oprtor L,vtor [ 0 ; ;; n, ], n rturns t vtor [ 0 ; ( 0 L ); ;( 0 L L L n, )]. Blllo [Bll90] ponts out tt t EREW PRAM mol wt t sn oprtons nlu s prmtvs s trm t sn mol. T sn mol onsrs ll prmtv oprtons (nlun sns) s tkn unt tm on ypru rttur. Ts llows sortn oprtons to prorm n O(lo n) tm. 3.. Snws Oprtons In ton to n lss s tr upwr or ownwr, sn oprtons my smnt. A smnt sn my tout o s multpl prlll sns, wr oprts npnntly on smnt o ontuous prossors. Smnt roups r ommonly lmt y smnt t, wr vlu o nots t rst prossor n t smnt. For xmpl, n Fur 8, tr r our smnt roups, orrsponn to smnts o sz 3, 4,, n 3. t s:smnt l up-sn(t,s,+,n) up-sn(t,s,+,x) own-sn(t,s,+,n) own-sn(t,s,+,x) Fur 8: Exmpl smnt sns or ot t upwr n ownwr rtons (s wll s nlusv n xlusv). Fnlly, sn oprtons my urtr lss s n tr nlusv or xlusv. For xmpl, n upwr nlusv sn oprton rturns t vtor [ 0 ; ( 0 L ); ;( 0 L L L n, )], wl n upwr xlusv sn rturns t vtor [0; 0 ;;( 0 L L L n, )]. Vrous omntons o smnt sns (wr L s oun to t ton oprtor) r sown n Fur Elmntws Oprtons An lmntws prmtv s n oprton tt tks two vtors o qul lnt n prous n nswr vtor, lso o qul lnt. T t lmnt n t nswr vtor s t rsult o t pplton o n rtmt or lol prmtv to t t lmnt o t nput vtors. In Fur 9, n xmpl lmntws ton oprton s sown. A n B orrspon to t two nput vtors, n w(+,a,b) nots t nswr vtor. A B w(+,a,b) Fur 9: Exmpl ltn n lmntws ton oprton. 8

9 3..3 Prmuttons A prmutton prmtv tks two vtors, t t vtor n n nx vtor, n rrrns (prmuts) lmnt o t t vtor to t poston sp y t nx vtor. Not tt t prmutton must on-to-on; two or mor t lmnts my not sr t sm nx vtor vlu. Fur 0 provs n xmpl prmutton oprton. A s t t vtor, nx s t nx vtor, n prmut(a,nx) nots t nswr vtor. poston A nx j poston prmut(a,nx) j Fur 0: Exmpl o prmutton. 3.3 SAM Mol A smlr ut mor rstrtv mol o prlll omputton, t SAM (Sn-An-Monotonmppn) mol o prlll omputton [Bst9] my n y on or mor lnrly orr sts o prossors w llow lmnt-ws n sn-ws oprtons to prorm. Bot wtn n twn lnrly orr st o prossors, monoton mppns my lso prorm. A monoton mppn s n s on n w t stnton prossor ns r monotonlly nrsn or monotonlly rsn unton o t sour prossor ns. For xmpl, onsr t stuton pt n Fur wr t sour prossors r ontn n prossor st A, n t stnton prossors r lot n prossor st B. Fur s vl monoton mppn, wl t mppn n Fur s not monoton mppn (s oms or n t lnr orrn). A 3 4 A 3 4 B j B j () () Fur : () An xmpl monoton mppn twn two sts o prossors, n () smlr mppn w s not monoton. Bn mor rstrtv tn t sn-mol y rqurn monoton mppns, t SAM mol lso onsrs sn oprtons s tkn unt tm, tus llown sortn oprtons to prorm n O(lo n). T SAM mol ws m npproprt or our rsr s t s unl to ntly ltt t mnpulton o R-trs. Ts s u to t ults nvolv n mntnn monoton mppns twn two rnt R-trs wn prormn sptl qurs su s mp ntrston (s [Hol94, Hol94] or mor tls). Not owvr tt our lortm or uln t-prlll R-trs s sr n Ston 5.3 os not volt t mor rstrtv SAM mol. Bukt PMR qutrs, wt tr rulr sjont ompostons, r strutur or w t SAM mol s wll-sut. 9

10 Bus o t ukt PMR qutr's rulr omposton, unqu lnr orrn my rly otn (vn prtulr lnr orrn mtooloy su s Pno urv [Pn90]). As wll sown ltr, t R-tr, wt ts rrulr omposton, os not v unqu lnr orrn. Wn prormn oprtons on two mps wt non-unqu lnr orrns, t mntnn o t monoton mppns oms xpnsv u to t nssry prossor rorrns n t t-prlll nvronmnt. For xmpl, onsr t stuton pt n Fur wr two sts o prossors (st (A,B) n st (C,D) orrspon to t ovrlppn rons n Fur. Suppos prossor n on roup must ommunt wt ntrstn prossor n t otr roup (.., A wt C n D, n B wt C n D). For t rst roun o ommunton sown n Fur, monoton mppn my mntn. T son roun (pt n Fur ) owvr volts t monoton mppn. I prossors A n B n t rst st r rorr (n xpnsv oprton or lr ollton o prossors), t monoton mppn my on n mntn s sown n Fur. C A A B A B B A B D C D C D C D () () () () Fur : () An xmpl ollton o ntrstn ounn oxs, () vl monoton mppn, () n nvl monoton mppn, n () vl monoton mppn ollown prossor rorrn. 4 Dt-Prlll Sptl Prmtv Oprtons In ts ston w sr t prmtv oprtons tt r n to onstrut PM qutr, ukt PMR qutr, n n R-tr. Svrl o t lowr-lvl prmtvs v n sr lswr (.., [Nss8, Hun89]). 4. Clonn Clonn (lso trm nrlz [Nss8]) s t pross o rpltn n rtrry ollton o lmnts wtn lnr prossor orrn. Fur 3 sows n xmpl lonn oprton. Clonn my ompls usn n xlusv upwr ton sn oprton, n lmntws ton, n prmutton oprtor. X lon l X' Fur 3: Exmpl o lonn oprton. Fur 4 tls t vrous oprtons nssry to omplt t lonn oprton. In t ur, lon l nts w lmnts o x must lon; n ts xmpl, lmnts,, n r to lon. T s tnqu s to lult t ost nssry tt xstn lmnt must mov towr t rt n t lnr orrn n orr to mk room or t nw 0

11 lon lmnts. Ts my ompls y mployn n upwr xlusv sn w sums t lon s, s not y up-sn(cf,+,x) n t ur. Atr t ost s n trmn, n lmntws ton on t ost vlu (F) n t poston nx (P) trmns t nw poston or lmnt n t orrn (w(+,p,f)). A smpl prmutton oprton s tn us to rposton t lmnts (prmut(x,f)). Fnlly, t lonn oprton s omplt wn wn o t lonn lmnts ops tsl nto t nxt lmnt n t lnr orrn (not y t smll urv rrows n t ur). P X lon l CF up-sn(cf,+,x) F w(+,p,f) F P prmut(x,f) Fur 4: Exmpl ltn t mns o t lonn oprton. 4. Unsun Unsun s t pross o pyslly sprtn two rtrry, mutully xlusv n olltvly xustv susts o n ornl roup. Ts oprton, wn ppl wtout monoton mppns, s lso n trm pkn [Krus85] or splttn [Bll89]. Unsun n ompls usn two nlusv sns (on upwr n on ownwr), two lmntws oprtons (n ton n sutrton), n prmutton oprtor. An xmpl unsun oprton s sown n Fur 5. Fur 5: Exmpl o n unsun oprton. P X up-sn(x=,+,n) F own-sn(x=,+,n) F {X=} w(-,p,f) F3 {X=} w(+,p,f) P prmut(x,f3) Fur 6: Exmpl ltn t mns o t unsu oprton. T tul mns o t unsu oprton or t t o Fur 5 r llustrt n Fur 6. T two rnt typs w must unsu v typ ntrs n. Assum

12 tt t 's r to rposton towr t lt, n t 's towr t rt n our lnr orrn. T s tnqu s, or lmnt o t two roups, to lult t numr o lmnts rom t otr roup tt r poston twn tsl n ts sr poston t tr t lt n or t rt n. An upwr nlusv sn (up-sn(x=,+,n)) s us to ount t numr o 's twn n t lt n o t orrn. Smlrly, ownwr nlusv sn (own-sn(x=,+,n)) s lso us to ount t numr o 's twn nvul n t rt n o t lnr orrn. On ts two vlus r lult, two lmntws oprtons r us to lult t nw poston nx or lmnt o t lnr orrn. For lmnt, n lmntws sutrton o t lult numr o ntrpos 's (F) rom t ornl poston nx P trmns t nw poston nx (w(-,p,f)). Smlrly, or lmnt, n lmntws ton o t lult numr o ntrpos 's (F) n t ornl poston nx P trmns tr nw poston ns (w(+,p,f)). Fnlly, vn t nw poston ns n F3, smpl prmutton oprton (prmut(x,f3)) wll rposton lmnt nto t propr poston n t lnr orrn. 4.3 Duplt Dlton Duplt lton (lso trm onntrt [Nss8]) s t pross o rmovn uplt ntrs rom sort lnr prossor orrn. An xmpl uplt lton (wt t uplt lmnts s) s sown n Fur 7. Duplt lton s ompls usn n upwr xlusv sn oprton, ollow y lmntws sutrton n nlly prmutton oprton. X X' Fur 7: Exmpl o uplt lton oprton. Assumn tt t lmnts n t lnr orrn v n sort y ntr, t s tnqu mploy wn ltn uplt ntrs s to ount t numr o uplts twn lmnt n t lt s o t orrn. E lmnt s tn mov towr t lt y ts numr o postons. Consr Fur 8 wr t lmnts r sort n t uplt tms r mrk (uplt l), n upwr xlusv sn oprton (up-sn(df,+,x)) s us to sum t numr o lmnts n t lnr orrn tt r to lt. An lmntws oprton (w(-,p,f)) s tn mploy to sutrt t numr ontrpos tms to lt (F) rom t lmnt's poston nx P. Ts vlu s tn us s t nw poston nx n smpl prmutton oprton (prmut(x,f)) n ompltn t uplt lton oprton. P X uplt l DF up-sn(df,+,x) F {DF=0} w(-,p,f) F P {DF=0} prmut(x,f) Fur 8: Exmpl ltn t mns o t uplt lton oprton.

13 4.4 No Cpty Ck For sptl ompostons su s t ukt PMR qutr n t R-tr wos no splttn rul ouss solly on t numr o tms n no, no pty k n us n trmnn wtr or not no n t tr s ovrown n ns to splt. Ts n ompls usn ownwr nlusv ton sn oprton, ollow y n lmntws wrt (or r) oprton. In Fur 9, t ownwr sn s sown or n xmpl tst. Follown t trmnton o t no ounts, nos wos ukt pty s x my mrk or suvson. 3 4 nos lns 3 4 ount Fur 9: Exmpl sown ow ownwr nlusv smnt sn oprton my mploy n no pty k. 4.5 Dtrmnn PM Qutr No Soul Splt For t PM qutr, t pross o trmnn wtr or not no soul splt rqurs mor normton tn smply t numr o lns tt ntrst t no. Gvn t mxmum n mnmum numr o nponts ssot wt ll lns wtn no, t s possl to trmn wtr or not som o t nos must suv. T no must suv tr t mxml numr o nponts s qul to two, or t mxml numr s on n t mnml numr s zro. I, owvr, t mxmum n mnmum numrs r qul to otr (.., 0 or ), tn tonl normton s nssry or t suvson trmnton n m. T tonl normton tt s nssry n t s o no wr t mxmum n mnmum r ot on, s wtr or not snl npont xsts wtn t no. I tr r two or mor nponts wtn t no, tn t no must suv. Ts npont ount my trmn y ormn t mnml ounn ox o t nponts tt l wtn t no [Bst9]. I t npont ounn ox s trvlly pont, tn ts nts tt ll lns wtn t no sr ommon vrtx, tus tr s no n to urtr suv t no. Otrws, t no must suv s tr s mor tn on npont n t no. In t s wr ot t mnm n mxm r qul to zro, t s nssry to trmn t numr o lns wtn t no. I t numr o lns wtn t no s rtr tn on, tn t no must suv. In prlll, ln rst trmns t numr o ts nponts tt xst wtn t no; tr 0,, or. In Fur 0, ts numr s rprsnt y t EPs (or nponts) l. Usn squn o ownwr nlusv smnt sn oprtons, t mxmum n mnml numr o nponts ssot wt ll lns wtn t no s trmn. Fur 0 rprsnts ts vlus n t mn EPs n mx EPs ls. Ts two numrs r tn ommunt y t rst ln n smnt roup to t orrsponn no n t tr. Bs upon t lult mxmum n mnmum npont vlus, t n trmn tt no n Fur 0 must suv. For t rmnn nos n t xmpl, tonl normton s nssry n orr to trmn wtr or not t no must suv. For nos wr t mnmum n mxmum numr o nponts s qul to on, t rqur normton s wtr t no ontns snl npont tt s sr mon ll lns n t no. Ts n trmn y ormn t mnmum 3

14 X Y nos 3 4 W 3 4 Z lns EPs 0 0 mn EPs mx EPs 0 0 Fur 0: Intl onurton o nos n lns. Usn squn o ownwr smnt sns, t mxmum n mnmum numr o nponts ssot wt ll lns n no s trmn. T ry no s trmn to rqur splt. X Y nos 3 4 W 3 4 Z lns MBBs W X X - Z Z Z Z Z Z Fur : Clulton o t npont mnmum ounn oxs (MBBs) or nos wr t mxmum n mnmum numr o nponts r qul. T rk ry no ws prvously trmn n Fur 0 to rqur splt, t lt ry no s urrntly trmn to rqur splt, wl t ross no 4 os not rqur suvson. ounn ox o t nponts tt l wtn t no. I t vrtx ounn ox s trvlly pont, tn ts nts tt ll lns wtn t no sr ommon vrtx. Tus tr s no n to urtr suv t no. T mnmum ounn oxs n trmn usn smll squn o ownwr nlusv smnt sn oprtons. In Fur, t mnmum ounn oxs r rprsnt y t ollton o npont lls (.., W, X, Y, n Z) nt ln. For xmpl, t npont mnmum ounn ox or no ontns nponts X n W, wl t mnmum ounn ox or no 4 ontns only npont Z. Bs upon t lult ounn oxs, no must suv, wl no 4 os not n to suv. In t s wr ot t mnm n mxm r qul to zro, t s nssry to trmn t numr o lns wtn t no. I t numr o lns wtn t no s rtr tn on, tn t s nssry to suv t no. In Fur, t ln ount s lult wt smpl ownwr nlusv smnt sn usn t ton oprtor. For t rmnn no n quston (no 3), ln ount o mpls tt t no os not n to suv. Ts nl oprton omplts t trmnton o wtr or not PM qutr no must suv. 4.6 Splttn Qutr No T tnqu mploy to splt qutr no s two st pross. Atr trmnn tt no soul splt, t no s rst splt vrtlly, n tn orzontlly. Ts rsults n t suvson o t no nto qul sz qurnts. A no pty k rst s mploy to ount t numr o lns ssot wt t no n trmn wtr or not t no soul splt. Fur 3 pts ts pross or snl 4

15 X Y W nos Z lns ount Fur : Clulton o t ln ount or t rmnn no (3). Bs upon t ount o, t no s not rqur to suv. Not tt prvously, nos n wr trmn to rqur suvson, wl no 4 not rqur suvson. nos lns ount Fur 3: Exmpl ntl ln to no ssoton urn no splttn pross. T no pty k ps o t pross s lt. no n v ssot ln smnts. I t numr o lns ssot wt t no prossor xs t prn no pty (4 n ts xmpl), tn t no must splt nto our sunos n o t lns must rroup, orn to t nos t ntrsts. nos lns lon Fur 4: Dtrmnn w lns ntrst t orzontl splt xs n must lon. No splttn ours n two sts, wt t rst st orrsponn to vrtl splt o t no nto two ps. In prlll, ln n t splttn no trmns wtr or not t ntrsts t splt xs. I t ln ntrsts t splt xs, t must lon. For t xmpl tst, ntrstn ln (lns n ) ssown wt t lon vlu o. A lonn oprton, s sr n Ston 4., s tn prorm on t lns n t no tt ntrst t splt xs. Ts s sown n Fur 4. On t ntrstn lns v n lon, t s nssry to rroup t lns orn to wtr ty l n t top or t ottom l o t splttn no. In prlll, ln my mk 5

16 nos lns s T B T B B T B Fur 5: Follown ln lonn, ln n prlll trmns wtr t ls n t top (T) or ottom (B) l o t two rsultn nos. An unsu oprton s tn ppl s upon w l t ln rss n. ts trmnton us ln stors t sz n poston o t no tt t rss n. In Fur 6, t s vlu rprsnts wtr t ssot ln s n t top (T) or ottom (B) l o t splttn no. T rroupn o t lns s v wt n un-su oprton s tl n Ston 4.. T un-su s us to onntrt t lns totr nto two nw smnts, o w orrspons to ll o t ln prossors lyn tr n wol or n prt ov orlow t y oornt vlu o t ntr o t splttn no. T un-su oprton omplts t rst l o t qutr no splttn oprton. T rsult o ts un-su oprton s pt n Fur 6. nos lns lon Fur 6: Rsult o t vrtl no splt. T son ps o t no splt ns wt ln trmnn wtr ty ntrst t orzontl splt xs. Intrstn lns must lon. T son l o t no splttn oprton uss nloous tnqus n splttn t two rsultn nos n n l orzontlly. Ts orzontl splt wll rsult n t ornl no pt n Fur 3 n suv nto our qul sz rons. T son st ns wt ln trmnn wtr or not t ntrsts t orzontl splt n soul lon. In Fur 6, t ntrstn lns (ln n no ) s sown wt ts lon vlu st to. Follown t ln lonn, ln n prlll trmns wtr t ls on t lt (L) or rt (R) s o t splt xs. Bs upon t ln's poston rltv to t splt xs, n un-su oprton s us on o t two nos n prlll to rt two smnt roups or o t two splttn nos. E smnt roup wll orrspons to ll o t ln prossors w l tr n wol or n prt to t lt or t rt o t splt xs. T un-su oprton s sown or t xmpl tst n Fur 7. T rsult o t un-su oprton s pt n Fur 8. At ts pont, t qutr no splttn oprton s omplt. 6

17 nos lns s R L L R L R R R Fur 7: Follown ln lonn, ln n prlll trmns wtr t ls n t lt (L) or rt (R) l o t two rsultn nos. An unsu oprton s tn ppl s upon w l t ln rss n. nos lns Fur 8: Fnl rsult o t xmpl no splt oprton. 4.7 Sltn n R-tr No Splt T top o ow to splt n ovrown no s n t sujt o mu rsr on squntl R-trs. For xmpl, t R -tr [Bk90], s n R-tr vrnt tt uss mor sopstt no nsrton n splttn lortm tn tos prov wt t onvntonl nton o t R-tr [Gutt84]. For t t-prlll R-tr, w v vlop two no splttn lortms, o vryn omputtonl omplxty. In t rst n smplst lortm, t splttn xs (.., x or y-xs n t two-mnsonl s) n t oornt vlu r trmn y nn t mn vlus lon xs o t mponts o ll ounn oxs n t ln prossor st n prlll v squn o sn oprtons. For xs n smnt roup, t mponts o t ounn oxs r rst summ usn ownwr nlusv smnt sn oprton (wt t ton oprtor). T rst no n t smnt roup tn vs t sum y t numr o ounn oxs n t smnt roup. Ts vlu s tn rost [Hun89] to ll otr nos n t smnt roup wt n upwr smnt sn (usn t opy oprtor). E no tn trmns wtr t ls n t lt or rt rsultn ounn oxs. Fnlly, smll squn o upwr n ownwr nlusv sn oprtons (usn tr mn or mx oprtor, pnn upon t ntur o t sn) s us to trmn t pysl xtnts o t two ounn oxs. T splt xs n oornt vlu r osn rom t two possl splts (.., t mn lon t x-xs n t y-xs) so s to mnmz t mount o r ommon to t two rsultn nos. Ts oprton s o omplxty O() t st o t uln oprton s onstnt numr o sns omnts t omputton. T son no splttn lortm rst sorts ll lns n t smnt orn to t lt o tr ounn oxs. A squn o upwr sn oprtons r us to trmn t xtnts o t ounn ox orm y ll lns prn ln n t sort smnt. A smlr squn o ownwr sns wll trmn t ounn ox or ll ollown lns n t smnt. For ll ll splts (.., wr o t two rsultn nos rvs t lst m=m o t lns n 7

18 rstrut), t mount o ounn ox ovrlp s lult, wt t splt orrsponn to t mnml mount o ovrlp n slt s t x-xs nt. An nloous prour s mploy or t y-xs n otnn t y-xs nt splt oornt vlu. On t two nt splt oornt vlus r trmn, t on orrsponn to t mnml ounn ox ovrlp s slt. In t vnt o t, som otr mtr su soosn t splt wt t mnml ounn ox prmtr lnts my mploy. Ts splttn lortm s o omplxty O(lo n) t st o t uln oprton s w mploy twoo(lo n) sorts n onstnt numr o sn oprtons. L Box A B C D R Box ls: lt s rs: rt s L Box lt s L Box rt s R Box lt s R Box rt s A B C D sn typ up-sn(ls,-,mn,n) up-sn(rs,-,mx,n) own-sn(ls,-,mn,x) own-sn(rs,-,mx,x) Fur 9: Exmpl ltn t vrous sn typs n tr pplton to trmnn t x-oornt vlus or t lt n rt ounn oxs. Consr t xmpl sown n Fur 9 onsstn o our ounn oxs ll A{D wr t nos v n sort orn to tr lt x-oornt vlus. In ts xmpl, w r only onsrn t x-oornt vlus o t ounn oxs tou norporton o y-oornt vlus s strtorwr. In t ur, t lt n rt oornt vlus o t our nos r nt on t lns ll ls:lt s n rs:rt s rsptvly. For xmpl, no B s lt n rt x-oornt vlus 0 n 50 rsptvly, wl no C s lt n rt x-oornt vlus 40 n 70. Assumn tt no s roup wt ll nos on ts lt wn ormn t ounn oxs (.., no C s roup wt nos A n B wn ormn no C's lt n rt ounn oxs), t ollown squn o sn oprtons n us to trmn t ounn oxs on t lt n t rt or no. As sown n Fur 9, n upwr mnmum nlusv sn on t lt oornt vlu, s us to trmn t lt x-oornt vlu or t ounn ox on nos lt s (L Box lt s). Smlrly, n upwr mxmum nlusv sn on t rt x-oornt vlus wll stls t rt x-oornt vlu or t ounn ox on nos lt s (L Box rt s). Tus, or no B, w s tt t lt n rt oornt vlus or t ounn ox to ts lt (.., t ounn ox ontnn nos A n B, ll L Box n Fur 9), v x-oornt vlus 0 n 50. Ts vlus r oun n t rows o Fur 9 ll L Box lt s n L Box rt s. Anloous ownwr mn/mx xlusv sns r us to trmn t lt n rt x-oornt vlus o t ounn ox to t rt o no. W my osrv tt t lt n rt x-oornt vlus or t ounn ox to t rt o no B(.., ounn ox ontnn nos C n D, ll R Box 8

19 n Fur 9) v vlus 40 n 80, rsptvly. Atr t two ounn oxs v n trmn or ll splt (.., no splt wr o t two rsultn nos rvs t lst m=m o t lns n rstrut), t mount o ounn ox ovrlp s lult, wt t splt orrsponn to t mnml mountoovrlp n slt s t x-xs nt splt oornt vlu. An nloous prour s mploy or t y-xs n otnn t y-xs nt splt oornt vlu. On t two nt splt oornt vlus r trmn, t on orrsponn to t mnml ounn ox ovrlp s slt. In t vnt o t, som otr mtr su soosn t splt wt t mnml ounn ox prmtr lnts my mploy. Ts splttn lortm tks O(lo n) tm t st o t uln oprton s w mploy twoo(lo n) ln sorts n onstnt numr o sn oprtons. 5 Dt-Prlll Bul Alortms In ts ston w sow ow to ul PM qutr, ukt PMR qutr, n n R-tr. T lortms r r n mk us o t prmtvs sr n Ston PM Qutr Construton T t-prlll PM -qutr uln pross ns wt ln ssn to snl qutr no s pt n Fur 30. T s PM qutr onstruton s n trtv pross wr nos r suv untl tr splttn rtron (rr to Stons. n 4.5 or tl srpton) s no lonr sts. Usn t sm tnqu s sr n Ston 4.5, t root no s mrk or suvson s upon t mxmum numr o nponts n qul to two. T no s suv n t lns r splt n rstrut usn t qutr no splttn mto sr n Ston 4.6. nos lns Fur 30: Intl onurton. Follown t suvson o t root no o t PM qutr, w r lt wt t stuton sown n Fur 3. Not tt lns,, n wr lon urn ts no splt s ty ntrst on o t splt xs. Ts omplts t rst trton o no suvsons. E susqunt trton s smlr to t rst: no s rst k to s t must suv, n tn nssry, t no s suv usn t stnr qutr no splttn prmtv rom Ston 4.6. In Fur 3, t nw, n, n s nos must suv. T rsult o t son trton o no splttn s sown n Fur 3. At ts pont, on rmnn suvson must prorm on t nw l o t s qurnt (no 0). T nl trton rsults n t omposton sown n Fur 33. Bus no mor nos must splt, t PM qutr onstruton pross s omplt. For n ln smnts, t t-prlll PM qutr onstruton oprton tks O(lo n) tm, wr o t O(lo n) suvson sts rqurs O() omputtons ( onstnt numr o sns, lonns, n un-sus). 9

20 3 4 nos lns 3 4 Fur 3: Rsult o t rst roun o no splttn nos lns Fur 3: Atr son roun o no splttn. 5. Bukt PMR Qutr Construton In t t-prlll nvronmnt, ll lns r nsrt smultnously wn onstrutn sptl t strutur. Tus tr s no prtulr orrn o t t upon nsrton. T onvntonl PMR qutr's no splttn rul s on tt splts no on n only on wn ln s n nsrt. Ts s t s vn t numr o lns tt rsult xs t no's pty. Su splttn rul s nontrmnst n t sns tt t omposton pns on t orr n w t lns r nsrt. For xmpl, onsr t stuton pt n Fur 34 wr nn t nsrton orr o lns 3 n 4 rsults n rnt ompostons. Ts nontrmnsm s unptl wn mny lns r nsrt n no smultnously s w o not know ow mny tms t no soul splt. In orr to vo ts stuton, w os t ukt PMR qutr or t t-prlll nvronmnt s ts sp s npnnt o t orr n w t lns r nsrt n ts wll-v ukt splttn rul (.., tr s no muty wt rspt to ow mny suvsons tk pl wn svrl lns r nsrt smultnously) nos lns Fur 33: Fnl rsult o t PM qutr onstruton pross or t xmpl tst. 0

21 () () Fur 34: () An xmpl PMR qutr wt splttn trsol o two, wt t lns nsrt n numrl orr, n () t rsultn PMR qutr wn t nsrton orr s sltly mo so tt ln 4 s nsrt or ln 3. nos lns Fur 35: Intl ukt PMR qutr prossor ssnmnts. A ukt PMR qutr s ult n n trtv son, smlr to tt mploy wt t PM qutr onstruton lortm. Intlly, snl prossor s ssn to ln n t t st, n on prossor to t rsultnt ukt PMR qutr s pt or t smpl t st n Fur 35 (wt t xmpl tst, ssum w v n 8 8 qutr o mxml t 3). T rst trton ns wt t qutr no splttn prmtv s sr n tl n Ston 4.6. Bslly, no trmns t numr o lns ontn n ts ssot smnt roup, n ts numr xs t ukt pty, t no s splt usn squn o lonn n unsun oprtons. In Fur 35, t snl qutr no s suv s t t numr o lns (9) xs t ukt pty o n ts xmpl. T rsult o t rst suvson s sown n Fur 36. Contnun wt ts trtv pross, n Fur 36, t nw n s nos wll suv, rsultn n t stuton pt n Fur nos lns 3 4 Fur 36: Rsult o t rst no suvson, ln lonn, n un-sun.

22 nos lns Fur 37: Rsult o t son roun o no suvsons. Ts trtv suvson pross ontnus untl ll nos n t ukt PMR qutr v ln ount lss tn or qul to t ukt pty, or t mxml rsoluton o t qutr s n r (.., no o sz ). Ts s not prolm s or prtl ukt pts (.., 8 n ov), ts stuton s xnly rr n wll not us ny lortm ults prov tt t ukt PMR qutr lortms o not ssum n uppr oun on t numr o lns ssot wt vn no nos lns Fur 38: Rsult o t ukt PMR qutr ul pross. Bus no 7's ukt pty s x (sown n Fur 37), n t mxml rsoluton s not yt n r, notr roun o suvson s nssry. T rsult o t tr n nl suvson or our xmpl t st s sown n Fur 38. Not tt on o t qutr nos (no 9) stll s ts ukt pty x. In t xmpl, t mxml rsoluton s n r (.., 8 8). Tror, no 9 wll not urtr suv. T t-prlll ukt PMR qutr uln oprton tks O(lo n) tm, wr o t O(lo n) suvson sts rqurs O() omputtons ( onstnt numr o sns n un-sus). N 0 lns ount Fur 39: Intl prossor ssnmnt or t R-tr onstruton lortm.

23 5.3 R-tr Construton T t-prlll R-tr onstruton lortm rs rom t squntl R-tr lortm s nst o nsrtn ln smnts squntlly nto t t strutur, ll ln smnts r nsrt smultnously. T t-prlll R-tr onstruton lortm pros s ollows. Intlly, on prossor s ssn to ln o t t st, n on prossor to t rsultnt t-prlll R-tr s pt or smpl tst n Fur 39. Our xmpl ssums n orr (; 3) R-tr. In t ur, t ll N 0 nots t R-tr no prossor st, wt t ssot squr ron ontnn t ntr o t R-tr no ssot wt t R-tr no prossor. W us t trm smnt to rr to t ollton o ln prossors ssot wt prtulr R-tr no prossor. Wtn t ln prossor st, t nn squr rons ontn t ln ntrs. A ownwr sn oprton s prorm on t ln prossor st to trmn t numr o lns ssot wt t snl R-tr no prossor. Ts s sown n Fur 39 s t ount l nt t ln prossor st. T numr o lns n t smnt s tn pss y t rst ln n t lnr orrn to t snl R-tr no prossor (pt n Fur 39 y t rrow rom ln to no ). I t numr o lns n t smnt xs t no pty M, tn t t-prlll R-tr root no must splt nto two l nos n root no (s s smlrly on wt t squntl R-tr). T two nw l nos r nsrt nto t R-tr no prossor st, wt t ormr root no/prossor upt to rt t two nw lrn. N 0 lns Fur 40: Un-su oprton. T son o t two R-tr no splttn lortms s tl n Ston 4.7 s us to slt t splttn xs n oornt vlu. On t splttn xs n t oornt vlu r osn, n un-su oprton s us to onntrt tos ln prossors totr nto two nw smnts, o w wll orrspon to on o t two R-tr l no prossors s pt n Fur 40. For xmpl, ll lns w v mpont tt s lss tn t splt oornt vlu r monotonlly st towr t lt, wl tos wos mpont s rtr tn t splt oornt vlu r monotonlly st towr t rt mon t ln prossors. T rsult o t un-su oprton on t lns n Fur 40 s sown n Fur 4. Not tt t root no o t t-prlll R-tr s ssot wt two smnts n t ln prossor st A (.., (,,,) n (,,,,)), n must tsl suv n n nloous mnnr. Tus, t ts st tr t rst root no splt n ln rstruton, w wll wn up wt two smnts n t ln prossor st, n two rnt R-tr prossor sts N 0 n N ( st orrsponn to no t rnt t n t t-prlll R-tr), s sown n Fur 4. T uln lortm wll now pro trtvly, wt smnt n t ollton o ln prossors trmnn t numr o lns t ontns, n trnsmttn t ount to t ssot R-tr no prossor. I t numr o lns n t smnt xs t no pty M, tn t smnt (n orrsponn R-tr no prossor) wll or to suv. Not tt ts suvson pross my rsult n prossors tt orrspon to ntrnl nos n t t-prlll R-tr splttn tmslvs (wt ts splts possly proptn upwr trou t t-prlll R-tr). T uln pross trmnts wn ll nos n t R-tr no prossor st v t most 3

24 lns N 0 Fur 4: Rsult o t un-su oprton. 3 N N 0 3 lns Fur 4: Complton o root no splt oprton. M l prossors (tr ntrnl R-tr nos or ln prossors) s sown n Fur 43 or our xmpl tst. T t-prlll R-tr root no orrspons to t snl prossor n st N, t l nos r ontn n prossor st N 0, n ll lns r roup n smnts o lnt lss tn or qul to 3 n t ln prossor st (rll tt w r ln wt n orr (; 3) R-tr n our xmpl). Gvn n lns, t t-prlll R-tr uln oprton tks O(lo n) tm, wr o t O(lo n) sts rqurs O(lo n) omputtons ( onstnt numr o sns, lonns, n two sorts). 6 Conluson Anumr o t-prlll prmtv oprtons us n uln sptl t struturs su s t PM qutr, ukt PMR qutr, n t R-tr wr sr s wll s t lortms. Ts prmtvs v n us n t mplmntton o otr t-prlll sptl oprtons su s polyonzton n sptl jon [Hol93, Hol94, Hol94]. It woul ntrstn to s 3 N N 0 lns 3 ount: Fur 43: Brostn t ln ounts to t ssot nos. 4

25 N N N lns Fur 44: Complton o t t-prlll R-tr uln oprton. wtr ts prmtvs r sunt or otr sptl oprtons n wtr mnml sust o oprtons n n. Ts s sujt or utur rsr. Rrns [Alt87] [An83] [Bk90] H. Alt, T. Hrup, K. Mlorn, n F. Prprt. Dtrmnst smulton o lz omputrs on mor rlst ons. SIAM Journl on Computn, 6:808{835, 987. D. P. Anrson. Tnqus or run pn plottn tm. ACM Trnstons on Grps, (3):97{, July 983. N. Bkmnn, H.-P. Krl, R. Snr, n B. Sr. T R*-tr: An nt n roust ss mto or ponts n rtnls. In H. Gr-Moln n H. V. Js, tors, Prons o t 990 ACM SIGMOD Intrntonl Conrn on Mnmnt o Dt, ps 3{33, Atlnt Cty, NJ, My 990. [Bnt75] J. L. Bntly. Multmnsonl nry sr trs us or ssotv srn. Communtons o t ACM, 8(9):509{57, Sptmr 975. [Bst9] [Bs88] [Bll88] [Bll89] [Bll89] [Bll90] T. Bstul. Prlll Prms n Prts or Sptl Dt. PD tss, Unvrsty o Mryln, Coll Prk, MD, Aprl 99. (lso Unvrsty o Mryln Computr Sn Tnl Rport CS-TR-897). S. K. Bskr, A. Rosnl, n A. Y. Wu. Prlll prossn o rons rprsnt y lnr qutrs. Computr Vson, Grps n Im Prossn, 4(3):37{380, Jun 988. G. E. Blllo n J. J. Lttl. Prlll solutons to omtr prolms on t sn mol o omputton. In D. H. Bly, tor, Prons o t 988 Intrntonl Conrn on Prlll Prossn (ICPP), volum 3, ps 8{, St. Crls, IL, Auust 988. G. E. Blllo. Sns s prmtv prlll oprtons. IEEE Trnstons on Computrs, 38():56{538, Novmr 989. (lso Prons o t 987 Intrntonl Conrn on Prlll Prossn, St. Crls, IL, Auust 987). G. E. Blllo. Sn Prmtvs n Prlll Vtor Mols. PD tss, Mssustts Insttut o Tnoloy, Cmr, MA, Otor 989. (lso Lortory or Computr Sn Tnl Rport MIT/LCS/TR-463). G. E. Blllo. Vtor Mols or Dt-Prlll Computn. MIT Prss, Cmr, MA,

26 [Com79] D. Comr. T uqutous B-tr. ACM Computn Survys, ():{37, Jun 979. [Dn9] F. Dn, A. G. Frrr, n A. Ru-Cpln. Ent prlll onstruton n mnpulton o qutrs. In K. So, tor, Prons o t 99 Intrntonl Conrn on Prlll Prossn (ICPP), volum 3, ps 55{6, St. Crls, IL, Auust 99. [DW9] [El85] [Flo87] [Fol90] D. J. DWtt n J. Gry. T utur o prormn ts systms. Communtons o t ACM, 35(6):85{98, Jun 99. S. Elmn n E. Spro. Qutrs n onurrnt Prolo. In D. Droot, tor, Prons o t 985 Intrntonl Conrn on Prlll Prossn (ICPP), ps 544{55, St. Crls, IL, Auust 985. C. Floutsos, T. Slls, n N. Roussopoulos. Anlyss o ojt ornt sptl ss mtos. In U. Dyl n I. Trr, tors, Prons o t 987 ACM SIGMOD Intrntonl Conrn on Mnmnt o Dt, ps 46{439, Sn Frnso, My 987. J. D. Foly, A. vn Dm, S. K. Fnr, n J. F. Hus. Computr Grps Prnpls n Prt. Ason{Wsly, Rn, MA, son ton, 990. [Frn90] W. R. Frnkln n M. Knknll. Prlll ojt-sp n sur rmovl. Computr Grps, 4(4):87{94, Auust 990. (lso Prons o t SIGGRAPH'90 Conrn, Atlnt, Auust 990). [Gutt84] [Hll86] A. Guttmn. R-trs: A ynm nx strutur or sptl srn. In Prons o t 984 ACM SIGMOD Intrntonl Conrn on Mnmnt o Dt, ps 47{57, Boston, Jun 984. W. D. Hlls n G. L. Stl Jr. Dt prlll lortms. Communtons o t ACM, 9():70{83, Dmr 986. [Hol93] E. G. Hol n H. Smt. Dt-prlll R-tr lortms. In Prons o t 993 Intrntonl Conrn on Prlll Prossn (ICPP), ps III{49{53, St. Crls, IL, Auust 993. [Hol94] [Hol94] [Hun89] [Ir93] [Km9] [Ks88] E. G. Hol n H. Smt. Dt-prlll sptl jon lortms. In Prons o t 994 Intrntonl Conrn on Prlll Prossn (ICPP), ps III{7{34, St. Crls, IL, Auust 994. E. G. Hol n H. Smt. Prormn o t-prlll sptl oprtons. In Prons o t Twntt Intrntonl Conrn on Vry Lr Dt Bss (VLDB), ps 56{67, Snto, Cl, Sptmr 994. Y. Hun n A. Rosnl. Prlll prossn o lnr qutrs on ms-onnt omputr. Journl o Prlll n Dstrut Computn, 7:{7, 989. O. H. Irr n M. H. Km. Qutr uln lortms on n SIMD ypru. Journl o Prlll n Dstrut Computn, 8():7{76, My 993. I. Kml n C. Floutsos. Prlll R-trs. In Prons o t 99 ACM SIGMOD Intrntonl Conrn on Mnmnt o Dt, ps 95{04, Sn Do, Jun 99. S. Ks. Optml prlll lortms or qutr prolms. Computr Vson, Grps n Im Prossn, 59(3):8{85, My 994. (lso Prons o t Ft Isrl Symposum on Artl Intlln, Vson, n Pttrn Ronton). 6

27 [Krus85] [Kuk77] [L9] [Mrt86] [M86] [Nn88] [Nss8] C. P. Kruskl, L. Rnolp, n M. Snr. T powr o prlll prx. IEEE Trnstons on Computrs, 34(0):965{968, Novmr 985. D. Kuk. A survy o prlll mn ornzton n prormmn. ACM Computn Survys, 9():9{59, Mr 977. F. T. Lton. Introuton to Prlll Alortms n Artturs. Morn Kumnn, Sn Mto, CA, 99. M. Mrtn, D. M. Crull, n S. S. Iynr. Prlll prossn o qutrs on orzontlly ronurl rttur omputn systm. In Prons o t 986 Intrntonl Conrn on Prlll Prossn (ICPP), ps 895{90, St. Crls, IL, Auust 986. G.-G. M n W. Lu. Prlll prossn or qutr prolms. In Prons o t 986 Intrntonl Conrn on Prlll Prossn (ICPP), ps 45{454, St. Crls, IL, Auust 986. S. K. Nny, R. Moon, n S. Rjopln. Lnr qutr lortms on t ypru. In Prons o t 988 Intrntonl Conrn on Prlll Prossn (ICPP), ps 7{9, St. Crls, IL, Auust 988. D. Nssm n S. Sn. Dt rostn n SIMD omputrs. IEEE Trnstons on Computrs, C-30():0{07, 98. [Nls86] R. C. Nlson n H. Smt. A onsstnt rrl rprsntton or vtor t. Computr Grps, 0(4):97{06, Auust 986. (lso Prons o t SIG- GRAPH'86 Conrn, Dlls, Auust 986). [Nls87] R. C. Nlson n H. Smt. A populton nlyss or rrl t struturs. In U. Dyl n I. Trr, tors, Prons o t 987 ACM SIGMOD Intrntonl Conrn on Mnmnt o Dt, ps 70{77, Sn Frnso, My 987. [Orn8] J. A. Ornstn. Multmnsonl trs us or ssotv srn. Inormton Prossn Lttrs, 4(4):50{57, Jun 98. [Pn90] [Ros83] [Sm85] [Sm90] [Sm90] [Sw80] G. Pno. Sur un our qu rmplt tout un r pln. Mtmts Annln, 36:57{60, 890. A. Rosnl, H. Smt, C. Sr, n R. E. Wr. Applton o rrl t struturs to orpl normton systms: Ps II. Computr Sn TR{37, Unvrsty o Mryln, Coll Prk, MD, Sptmr 983. H. Smt n R. E. Wr. Storn ollton o polyons usn qutrs. ACM Trnstons on Grps, 4(3):8{, July 985. (lso Prons o Computr Vson n Pttrn Ronton 83, Wsnton DC, Jun 983, 7{3; n Unvrsty o Mryln Computr Sn Tnl Rport CS-TR-37). H. Smt. T Dsn n Anlyss o Sptl Dt Struturs. Ason{Wsly, Rn, MA, 990. H. Smt. Appltons o Sptl Dt Struturs: Computr Grps, Im Prossn, n GIS. Ason{Wsly, Rn, MA, 990. J. T. Swrtz. Ultromputrs. ACM Trnstons on Prormmn Lnus n Systms, (4):484{5, Otor 980. [Tmm8] M. Tmmnn. T EXCELL mto or nt omtr ss to t. At Polytn Snnv, 98. (Mtmts n Computr Sn Srs No. 34). 7