Data-Parallel Primitives for Spatial Operations Using PM. Quadtrees* primitives that are used to construct the data. concluding remarks.

Transcription

1 Dt-rlll rmtvs or Sptl Oprtons Usn M Qutrs* Erk G. Hol Hnn Smt Computr Sn Dprtmnt Computr Sn Dprtmnt Cntr or Automton Rsr Cntr or Automton Rsr Insttut or Avn Computr Sns Insttut or Avn Computr Sns Unvrsty o Mryln Unvrsty o Mryln Coll rk, Mryln 07 Coll rk, Mryln 07 Astrt Dt-prlll prmtvs or prormn oprtons on t M qutr n t ukt MR qutr r prsnt usn t sn mol. Alortms r sr or uln ts two t struturs tt mk us o ts prmtvs. T t-prlll lortms r ssum to mn mmory rsnt. Ty wr mplmnt on Tnkn Mns CM-5 wt 3 prossors ontnn GB o mn mmory. Introuton Sptl t onssts o ponts, lns, rons, rtnls, surs, volums, t. Sptl t rss n ppltons n mny rs nlun omputr rps, omputr vson, m prossn, n pttrn ronton. T ny o solutons to prolms n ll o ts rs s nn y t o o n pproprt rprsntton (s,.., [, ]). 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. T ky ssu s tt t volum o t t s lr. Ts s l to n ntrst n prlll prossn o su t. In ts ppr our ous s on t prmtvs tt r n to ntly onstrut t-prlll mmrs o t M qutr mly usn t sn mol o prlll omputton. 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 tprlll 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-). *Ts work ws support n prt y t Ntonl Sn Founton unr rnts IRI n BIR , n y rnt rom t Computr Rsr n Appltons Group t os Almos Ntonl ortory. T rst o ts ppr s ornz s ollows. Ston ry srs t sptl t struturs on w w ous. Ston 3 rvws t sn mol o prlll omputton. Ston susss t tprlll 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. 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. T M qutr [3] s vrtx{s mmr o t M 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 : () M qutr, () MR qutr, n () ukt MR qutr. T MR qutr (or polyonl mp rnom

2 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr 995. [0]) s n {s mmr o t M qutr mly. It mks us o prolst splttn rul wr lok s prmtt to ontn vrl numr o ln smnts. T MR 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 s k 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. T vnt o t MR qutr ovr t M qutr s tt tr s no n to suv n orr to sprt ln smnts tt r vry \los" or wos vrts r vry \los". 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 MR qutr rs wl t tm n to prorm oprtons on t nrss. Fur s n xmpl o MR qutr wt splttn trsol o two orrsponn to st o 9 s ll trou nsrt n nrsn orr. Osrv tt t sp o t MR qutr or vn tst s not unqu; nst, t pns on t orr n w t lns r nsrt nto t. Unortuntly, n t t-prlll nvronmnt, lns r nsrt smultnously urn t strutur onstruton. Tus, t orrn o t lns s unknown. Tror, t nton o t MR qutr s sltly mo to yl t ukt MR qutr wr nst o splttn n ovrown lok on, t lok (or ukt) s splt rptly untl suukt ontns no mor tn lns (wr s t mxml ukt pty). T sp s npnnt o t ln smnt nsrton orr. Not tt unlss t ukt pty s rtr tn or qul to t mxml numr o ntrston lns, t rursv omposton wll ontnu to t mxml pt llow y t ukt MR qutr. For xmpl, onsr Fur wr t rons orrsponn to t nponts o ln suv untl t mxml pt o t qutr (tr n ts s) s r. 3 Sn Mol o rlll Computton T sn mol o prlll omputton [, 3] 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 [] tks n ssotv oprtor, vtor [0 ; ; ; n? ], n rturns t vtor [ 0 ; ( 0 ); ; ( 0 n? )]. 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, tr r our smnt roups, orrsponn to smnts o sz 3,,, 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 : 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 ); ; ( 0 n? )], wl n upwr xlusv sn rturns t vtor [0; 0 ; ; ( 0 n? )]. Vrous omntons o smnt sns (wr s oun to t ton oprtor) r sown n Fur. 3. 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 3, 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 3: Exmpl lmntws ton oprton. 3.3 rmuttons 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 provs n xmpl prmutton oprton. A s t t vtor, nx s t nx vtor, n prmut(a,nx) nots t nswr vtor.

3 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr poston A nx poston prmut(a,nx) j j Fur : Exmpl o prmutton. lon l CF up-sn(cf,+,x) F w(+,,f) F prmut(,f) Sptl rmtv Oprtons In ts ston w sr t prmtv oprtons tt r n to onstrut M qutr n ukt MR qutr. Svrl o t lowr-lvl prmtvs v n sr lswr (.., [7, 9]).. Clonn Clonn (lso trm nrlz [9]) s t pross o rpltn n rtrry ollton o lmnts wtn lnr prossor orrn. Fur 5 sows n xmpl lonn oprton. Clonn my ompls usn n xlusv upwr ton sn oprton, n lmntws ton, n prmutton oprtor. lon l ' Fur 5: Exmpl o lonn oprton. Fur 6 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 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 () trmns t nw poston or lmnt n t orrn (w(+,,f)). A smpl prmutton oprton s tn us to rposton t lmnts (prmut(,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).. 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 [8] or splttn []. 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 7. Fur 6: Mns o t lonn oprton. Fur 7: Exmpl o n unsun oprton. T tul mns o t unsu oprton or t t o Fur 7 r llustrt n Fur 8. T two rnt typs w must unsu v typ ntrs n. Assum 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(=,+,n)) s us to ount t numr o 's twn n t lt n o t orrn. Smlrly, ownwr nlusv sn (own-sn(=,+,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 trmns t nw poston nx (w(-,,f)). Smlrly, or lmnt, n lmntws ton o t lult numr o ntrpos 's (F) n t ornl poston nx trmns tr nw poston ns (w(+,,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. up-sn(=,+,n) F own-sn(=,+,n) F {=} w(-,,f) F3 {=} w(+,,f) prmut(,f3) Fur 8: Mns o t unsu oprton.

4 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr Duplt Dlton Duplt lton (lso trm onntrt [9]) s t pross o rmovn uplt ntrs rom sort lnr prossor orrn. Fur 9 s n xmpl uplt lton (wt t uplt lmnts s). Duplt lton s ompls usn n upwr xlusv sn oprton, ollow y lmntws sutrton n nlly prmutton oprton. ' Fur 9: 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 0 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(-,,f)) s tn mploy to sutrt t numr o ntrpos tms to lt (F) rom t lmnt's poston nx. Ts vlu s tn us s t nw poston nx n smpl prmutton oprton (prmut(,f)) n ompltn t uplt lton oprton. uplt l DF up-sn(df,+,x) F {DF=0} w(-,,f) F {DF=0} prmut(,f) o nponts ssot wt ll lns wtn no, t s possl to trmn wtr or not som o t 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. 3 3 lns ount Fur : Exmpl o ownwr nlusv smnt sn oprton n us n no pty k. 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 []. 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. Y 3 Fur 0: Mns o t uplt lton oprton.. No Cpty Ck For sptl ompostons su s t ukt MR 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 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, t ownwr sn s sown or n xmpl tst. Follown t trmnton o t no ounts, wos ukt pty s x my mrk or suvson..5 Soul M Qutr No Splt For t M 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 W 3 Z lns Es 0 0 mn Es mx Es 0 0 Fur : Intl onurton o 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. In prlll, ln rst trmns t numr o ts nponts tt xst wtn t no; tr 0,, or. In Fur, ts numr s rprsnt y t Es (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 rprsnts ts vlus n t mn Es n mx Es ls.

5 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr 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 must suv. Y W 3 Z 3 lns MBBs W - Z Z Z Z Z Z Fur 3: Clulton o t npont mnmum ounn oxs (MBBs) or wr t mxmum n mnmum numr o nponts r qul. T rk ry no ws prvously trmn n Fur to rqur splt, t lt ry no s urrntly trmn to rqur splt, wl t ross no os not rqur suvson. For t rmnn n t xmpl, tonl normton s nssry n orr to trmn wtr or not t no must suv. For 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 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 3, t mnmum ounn oxs r rprsnt y t ollton o npont lls (.., W,, Y, n Z) nt ln. For xmpl, t npont mnmum ounn ox or no ontns nponts n W, wl t mnmum ounn ox or no ontns only npont Z. Bs upon t lult ounn oxs, no must suv, wl no os not n to suv. Y W 3 Z 3 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, n wr trmn to rqur suvson, wl no not rqur suvson. Wn 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 M qutr no must suv..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. lns ount 5 3 Fur 5: Exmpl ntl ln to no ssoton urn no splttn pross. T no pty k ps o t pross s lt. 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 5 pts ts pross or snl no n v ssot ln smnts. I t numr o lns ssot wt t no prossor xs t prn no pty ( n ts xmpl), tn t no must splt nto our su n o t lns must rroup, orn to t t ntrsts. lns lon Fur 6: 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 ) s sown wt t lon vlu o. A lonn oprton, s sr n Ston., s tn prorm on t lns n t no tt ntrst t splt xs. Ts s sown n Fur 6. 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

6 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr lns s T B T B B T B lns s R R R R R Fur 7: Follown ln lonn, ln n prlll trmns wtr t ls n t top (T) or ottom (B) l o t two rsultn. An unsu oprton s tn ppl s upon w l t ln rss n. Fur 9: Follown ln lonn, ln n prlll trmns wtr t ls n t lt () or rt (R) l o t two rsultn. An unsu oprton s tn ppl s upon w l t ln rss n. no. In prlll, ln my mk ts trmnton us ln stors t sz n poston o t no tt t rss n. In Fur 8, t s vlu rprsnts wtr t ssot ln s n t top (T) or ottom (B) l o t splttn no. Rroupn o t lns s v wt n un-su oprton s tl n Ston.. 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 or low 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 8. lns lon Fur 8: Rsult o t vrtl no splt. T son ps ns wt ln w ntrsts t orzontl splt xs n lon. T son l o t no splttn oprton uss nloous tnqus n splttn t two rsultn n n l orzontlly. Ts orzontl splt rsults n t ornl no pt n Fur 5 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 8, t ntrstn ln (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 () 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 n prlll to rt two smnt roups or o t two splttn. 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 9. T rsult o t un-su oprton s pt n Fur 0. At ts pont, t qutr no splttn 0 0 oprton s omplt. 3 lns 3 Fur 0: Fnl rsult o t no splt oprton. 5 Dt-rlll Bul Alortms In ts ston w sow ow to ul M qutr n ukt MR qutr. T lortms r r n mk us o t prmtvs sr n Ston. 5. M Qutr Construton Buln t-prlll M -qutr ns wt ln ssn to snl qutr no s pt n Fur. T s M qutr onstruton s n trtv pross wr r suv untl tr splttn rtron (rr to Stons n.5 or tl srpton) s no lonr sts. Usn t sm tnqu s sr n Ston.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.6. lns Fur : Intl onurton. Follown t suvson o t root no, w r lt wt t stuton sown n Fur. 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.

7 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr lns 3 Fur : Rsult o t rst roun o no splttn. E susqunt trton s smlr to t rst: no s rst k to s t must suv, n tn n, suv t no usn t qutr no splttn prmtv rom Ston.6. In Fur, t nw, n, n s must suv. wr nn t nsrton orr o lns 3 n 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 MR qutr or t t-prlll nvronmnt s ts sp s npnnt o t orr n w t lns r nsrt n ts wllv ukt splttn rul (.., tr s no muty wt rspt to ow mny suvsons tk pl wn svrl lns r nsrt smultnously) lns Fur 3: Atr son roun o no splttn. 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. Bus no mor must splt, t M qutr onstruton pross s omplt. For n ln smnts, t t-prlll M 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) lns Fur : Rsult o t M qutr ul pross. 5. Bukt MR 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 MR 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 5 () 3 Fur 5: () An xmpl MR qutr (splttn trsol o ), wt t lns nsrt n numrl orr, n () t rsultn MR qutr wn t nsrton orr s sltly mo so ln s nsrt or ln 3. lns () Fur 6: Intl MR qutr prossor ssnmnts. A ukt MR qutr s ult n n trtv son, smlr to t M qutr onstruton lortm. Intlly, snl prossor s ssn to ln n t t st, n on prossor to t rsultnt ukt MR qutr s pt or t smpl t st n Fur 6 (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.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 6, 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 7. Contnun wt ts trtv pross, n Fur 7, t nw n s wll suv, rsultn n t stuton pt n Fur 8. Ts trtv suvson pross ontnus untl ll n t ukt MR qutr v ln ount lss tn or qul to t ukt pty, or t mxml rsoluton o t qutr s n r (..,

8 ro. o Computr Artturs or Mn rpton '95, Como, Itly, Sptmr lns 3 Fur 7: Rsult o t rst no suvson lns Fur 8: Rsult o t son no suvsons. 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 MR qutr lortms o not ssum n uppr oun on t numr o lns ssot wt vn no lns Fur 9: Rsult o t MR qutr ul pross. Sn no 7's ukt pty s x (s Fur 8), n t mxml rsoluton s not yt n r, notr roun o suvson s n. T rsult o t tr n nl suvson or our xmpl t st s sown n Fur 9. Not tt on o t qutr (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 MR 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). 6 Conluson A numr o t-prlll prmtv oprtons us n uln sptl t struturs su s t M qutr, ukt MR qutr, n t R-tr wr sr s wll s t lortms. Ts prmtvs v n us n t mplmntton o otr tprlll sptl oprtons su s polyonzton n sptl jon [, 5, 6]. It woul ntrstn to s wtr ts prmtvs r sunt or otr sptl oprtons n wtr mnml sust o oprtons n n. Ts s sujt or utur rsr. Rrns [] T. Bstul. rlll rms n rts or Sptl Dt. D tss, Unvrsty o Mryln, Coll rk, MD, Apr. 99. [] G. E. Blllo. Sns s prmtv prlll oprtons. IEEE Trns. on Computrs, 38():56{ 538, Nov [3] G. E. Blllo n J. J. ttl. rlll solutons to omtr prolms on t sn mol o omputton. In ro. o t 988 Intl. Con. on rlll rossn, volum 3, ps 8{, St. Crls, I, Au [] E. G. Hol n H. Smt. Dt-prlll R-tr lortms. In ro. o t 993 Intl. Con. on rlll rossn, volum 3, ps 9{53, St. Crls, I, Au [5] E. G. Hol n H. Smt. Dt-prlll sptl jon lortms. In ro. o t 99 Intl. Con. on rlll rossn, volum 3, ps 7{3, St. Crls, I, Au. 99. [6] E. G. Hol n H. Smt. rormn o tprlll sptl oprtons. In ro. o t 0t Intl. Con. on Vry r Dt Bss, ps 56{ 67, Snto, Cl, Spt. 99. [7] Y. Hun n A. Rosnl. rlll prossn o lnr qutrs on ms-onnt omputr. Jour. o rlll n Dstrut Computn, 7():{7, Au [8] C.. Kruskl,. Rnolp, n M. Snr. T powr o prlll prx. IEEE Trns. on Computrs, 3(0):965{968, Nov [9] D. Nssm n S. Sn. Dt rostn n SIMD omputrs. IEEE Trns. on Computrs, C-30():0{07, F. 98. [0] R. C. Nlson n H. Smt. A onsstnt rrl rprsntton or vtor t. Computr Grps, 0():97{06, Au [] H. Smt. Appltons o Sptl Dt Struturs: Computr Grps, Im rossn, n GIS. Ason{Wsly, Rn, MA, 990. [] H. Smt. T Dsn n Anlyss o Sptl Dt Struturs. Ason{Wsly, Rn, MA, 990. [3] H. Smt n R. E. Wr. Storn ollton o polyons usn qutrs. ACM Trns. on Grps, (3):8{, July 985. [] J. T. Swrtz. Ultromputrs. ACM Trns. on rormmn nus n Systms, ():8{ 5, Ot. 980.