D. Etiemble J.-c. Syre (Eels.)

PARLE'92 Parallel Architectures and Languages Europe 4th International PARLE Conference Paris. France, June 1992 l'ro«e<fings

Springer-Verlag

Concurrent Data Structures for H ypercube Ma ch ine

M. R. Meybodi

Compute. Science Departmen\, Ohio Univcraity

Athens, Ohio 47501

Abstract To ellkieMly implement parallel algorithm. On ponll.1 cOm pule ... , concurrent d. t ... trodure.

(Jat., ",uct" , .. wbich "'" .imultanoou.ly upduable) au .«<led. In LI,l, pal"'r , In .... impl~ment.'­tioR! of .. priority queue on .. di.otribu ted. memor-y me .. age p,..,ing multiproc.""" with .. hype",ube topology a ... p ... ente<!. In the fi~t implementation. a lin ear chin of pmc .. oo,." i, mapped ont.<> the by!>".cube, ""d Iben .. h.ap data .truclIOn i. mapped onto ,h. chain. wl,.re each proceooor .10>"" on. 1.",,1 in the hUp. A .imihr approach i. laken for Ihe .«ond implementation, but in thi. c_, • \>anyan h •• p dal a "rud"" i. mapped onlo the lineu ch~in of proc .. oon. Agoin, each proc<ssor in th. chain beeom" rupon,ible lor one level ollh. data ,I ,.<lme. ~'or Ih. Ihu-d implement'lion, Ihe banyan heap data .truclu .. i. og.in uK<!, bul the m.pping i. nol onlo lineu chain of procooson. Inslud, tbe b.nyan hUp i, mapped ont.<> proc ..... to column by column, 10 Iht the algori lhm can m.ke better use of the <oncu,.,.enl proc ... iDg cop.bilit;"" of the hypel"Cub. t.opology in oroee to reduce ootlln..,king in the fi ... 1 P"'Xe.oor, on etT..,1 noted in the use of ~he lineu ~hin employe<! by Ihe finl lwo imrl.m.nl ~tion., Th. kq- adv~nla~e of b~nyn hoap ""u Lh. hur i. Ihl ... ith bMyan hup il i. """,ibl. to ,""iev. element. 01 dilf.",nL p.",.nlil. level •.

K cyw or dll and P hru co: Concu,.,..,,1 Dua Slrucluee, Hype","b •• Uuyan Heap. Pu.Uel AIg<>­ril hm

1 Introduction PriO>"ity 'I." .... d.to "ruct .... i.o • ve.,. importanl dolo .trueLu .. ... hoch h ... found applicalion in

nrieti .. of .;Iuallen •• ueh os : di, c,.. le event . imulal io" .,..t.m., timed· .h...,..,d eom~uting .pt.m. findin , .hort.nl path, in a ,,>ph (17). finding the minimum .panning ~ .... of. STOph (171. and iteration in num.,.ic.1 ..:heme. bued on the ideo of ... peated .. I«tion of an item wilh . malle.t le,t criteria I"!, to mention. f . ... ,

nch. cluall ruclu ... iJI a Ht of , I,menl, each of which hu an _<><;a\ed "umbe., i~ prior iLy fOT it, f or each elemenl~, p(~), the priori ly of ~ i. 0 "U",b<. from ..,me linudy order«! ..,1. Sl.nd~rd opention. on • prio'ity queue"", INSt,'HT, ... hich inse," an .Iem.nl .nd it. ",oei,ted p,iorily inlo Ibe prioril y queue, ",d XMA X, wbicb deleies the element wi lh Ihe high .. 1 priority from tho queue. LeI P denote Ihe .. 1 of all element.- priorily pain. Define

p(.)_ ((%,p(%))(p(% ) _. and (%.p(%)) E PI;

Th. ell..,1 of priority queu. ope r.lion. are .. followo:

INSERT( %,p(x))

P - PU{[%,plx))) R .. pon.., .. nuD.

XMAX, p _ P - P(p_ ... ) ... h .... p,p_~) i.o Ih. pair ... ith Ihe hig he" priority. R .. pon .. is P(P",M)'

Since moot of Lh. applint;",,, of pri,"ity qunc. or. compuLalion.,lly inten. ive, il i, ih\»OrLanl \(I h .... an effici.nt impl.mentalion , Th...,. kind. of implement.I;on, 01 priority 'lueU. h .... been ... ported in Ih. lit.ralm .. : _uenli.1 .l,orilhml 101" implementation on uniproce .. or, panllel al· ~o,il h m i< a,chit""'n, .. I", .... al;.a\;..,n in ha,.,\w ..... "nd 1'" .• \1.1 .I ...... ithm. fo ' ;ml'l.m."ta''''n on parallel conoput .... In Ih. ne~1 t hre. P"'~"l'h ....... ¥i.w II,. Ii .... ',,'" 101' t\>..., ·.Pl"oad ....

Th .... .... numbe, 01 implem.nt~ion' fO>" prio,ity quo"e on • unop .... " • ...",.. I'r i","y qne"e CA "

be mai"I';n.d .. a sorted li' t, in which c ..... dd.ting Ih •• I.m.nl .. i,h Ih. hi~h"l priority tak .. 0(1) tim" but in.ertion t~hl O(N ) l im •• or it can b. m~inl.; ncd .. an "noorted li.t, in which c.,..

insertion , .. h • .:o •• lul. time but XMAX tak .. O!N) L;m e, wheu f\' I. the numbe.- of elemont_priority pairs in the priority queue. Other implementa"on . .. hieh lend them •• I",. 10 ]og:>rilnmic lime ar. heiJht or .... i&;ht hal.ne"" IffiO parti ally ordered tru(h .. p), !oFI;,,\ 1~1 ,ug,!!; .. ted by C. A. Crane [30). paSo<h. {37 ), Ike ... hu,," !381, implicit hupo 130], and bioom;'1 queu .. 139). Th. m",,' widely used implemenlal;"o of priority queue i ... ilh a hup d~\a OI,ucln,e. A hup j. a full k-.!'ur 1.­' nch .hl the priority of th, dement AI any nod. in th. IT"" j. gr<~t 'T thn priority of II .. cl,m.,,' at each of;\.o .hikl .. n. Th". 11,. elemen' with the high ... priority i. ~Iw.y. at Ih. rool of Lh. hup. 01"' , >1;"" fNSE:RTi. pc,-fo'med by ~dding Ih. pai, 10 th. I.." lev.1 of th. he.p .. nd then con"""';"g th •• e.ullin, complete Ir« into a heap by pu.hin~ thaI p.ir"p the I ..... ro<uni .. ely. XMAX worko by ,..,pl.cing 110. p.ir 0\ 1"<>01 wil h Ih. p~ir r .. idinr in the righlm""t nodo 01 tho lui 10"01 01 of hup and .hen COn"OMi" l lhe r.,,, lling complete Iree into. hup by p".hing Ihe pair at thorool down ,10, tr"" tKu",i .. ,ly. If _ 2 tbon ... , hve binary heap wbich iI 110, mot, wildly "ud;"] .pocial .....

A mumb .. 01 mu!!.ip"""osoor d .. ig~ for maintaining priority queue hove boen propoMd in the lil,ratur, •. Th_ d .. isn. can bo d .... ifi.d into t..., main I "'"po. 1) d .. isn. with .mall number of 1''''''''-'' •• ch h .. ing ~n .m.ll amo"nl of memory, ond 2) d .. ign. ",ilh .moll number of 1'1"<1«"0" nch ha .. in, a lug, amount of. memory. Group 1 d .. ign'l5,13.n.27,33\which in lum un be c1 .... ihd into O1.tol;': tr« d .. ilns ond .yotolic ....,..y d .. ign ... e .mlabl. for imp emenlat ion by VLSI hardw .. , "' b ...... I rOUp t...., d .. i,n' 12, 4,1l,29,421 cu be 'mpl.monled eilhor in VLSI Or aJlr gonoral pur"",e li,I,llyor loooelr coupled mullip"""osoor onhit.ecture.

E~i"ing algorithm. for mullip""""""'" ...., di .. ided inL-o I...., S"'''P"': ,10"" lor .hored memory multiproc ... 1IOn and IbOM for d;"Iribuled memory muliproe...,..... When de. irning algorilhmo fOT ,hared memory m,,!tipf<l~ Ihe foeu. i. on reducing Ih. interference bet .... n conto ..... nl p""".:IU. """ ... ing lb. d ata . truc'u" ,.. h ....... in Ihe cue of d;"trib"I. d memory muh iproce.rot the locu. ;" on exploiting lb. par,lIellom in the d.t, .. ructure . ContutT.nl , 11I0rilh"", fOT manipubtin~ bin[ff: Ir« d". \.0 Kunl aDd Lehman 181, concurTenl algorithm. fOT 1)..1 ..... du, to Lebman and Yo<> 119, algorithm. for eoncurrent .. onlond in. ertion of dota in AVL-Ir .. and 2·' tr ... d". L-o Ell,. 120,21 , contuTT."I algorilhm for in .. rtion and delelion on Ih. h.ap due 10 Rao ond Kum~' [81, ""J Bl.w .. and Brow". 111 ~ eumple. 01 alsOO"ithm. for .h~_memory mullip"""OPot. E~ompl .. of algorithm. for di.lti"ul..! memory mo!tiproc....". are; balanced ,ub. du. to Dally 191, ""rted <hoin due to Omondi and B""".J13I, and binary .. onll m .. b due L-o Meybod; 14]1 .

II.houl bo nol,d .ht m."t of the machine. or algorilbm, ciled in Ihi. pI pe r off .. eaplb'lili .. .. h;':h go beyond priority queue operalion •. 0110" operation •• upported by m."t of Ihe ciled algo­rithm. or machin .. ~ operotin ... SEARCH, NEAR, UPDATE, and DE~£TE on .. I of kor-ruord 1"''''. In tb, context of priority queue. rocord io Ihe elem. nl ud .he <o"""'ponding key io 110. priority ... oci,ted 10 Ihal .Iemont .

In Ihi. p.""r " e p .... n. thr« implem.ntation. of priorily que". on diotribu.ed.m.mory m .. .ag. p .... in~ mulliproce ..... ,..ilh hrPercub. topology. Th. finl implemenla.ion ;" 10-'1 on hup data OIruclure. Th. heap io mapped into a linnr chain of PToe",""", whith i. then embedded into hrper­cub •. The olher t_ impl.m,nla.ion. are b&oed on hanyan nul' data . Iruelu". They differ in tbe w~r Ihl Ihe hany. n ;, In>r''''~ inl<> the hyperen .... Th, key .rlv.,nl.,<:," "I h •• pn he>p 0." the h",p i. thl ... it~ banran h •. ", it to """ihl. L-o "tri •• , .Iemen ... t d,If ... nt ""re.nlil. l.v.lo.. A hypercube com pul ... i. brieRy d_ribed belo .....

An "- dimenlional hypercube machine con, i, .. of 2~ 1>", .... or node. intereonntete<! 10 one another 10 lorm a d dimen. ional cube. In an d dimen . ional cube Iwo nod .. are connteled if and only if Ihe binlry ",pre .. nta.ion of tbeir numbe ... diff .. by one and only one bit. ~b p""",,'or node i" 110. hypercube 10 .. a 1''''''''''''''' and I local memory fOT thot p"""._ r. Th. P""""""'" WOT. independenl!y and uy nchronoully , nd communicale by p ... inS m .... g .. 131·

The ..... 1 of 11,i. paper i. Oflonlled .. folio,... Seclion 2 giv ... h .. p b&oed implementation of priority queue on a di. tributed_memory m .... g •. p_inl. 5«lion 3 d.fin .. b.,ny .. graph. and banyan hul'". In _lion 4 "0 diK"" an implement •• ion of priority que". b"ed on b'nr~n he~p d . l o . Iruclure Socl ion S d .. i .... anal ylical formul. fOT Ih. perotntil. 1 .... 1 of a relri . .. e<! .l.m.nt. In toel;on 6, Ih. 'teond banyan 10 .. 1' b .. ed implemenl . lion of prio.ily qu.ue i. pT .. enled. Th. 1 .... 1 _tion i. the conclu.ion.

2 First Implementation Thi. implementalion i. b-'1 on hup data IIn!Cture. Th~ h O~1> i, fi" l mapped into a linear chain

01 n proc....., ... Thi. chin ,. then embedd. d inlo the hypeKu b. in . neh a ... ~y '" to pr .... "'. the p",~ imily property, i.e.,'" . h.t aJly t"" odj",ent proc. "", ... in the eh .in are mapped inlo neighbor noo .. in Ih. hpercube (3), Th. "," of Gr~r cod ... ;" on. ho"·n le<ohnique to obtain a m3pp'nt ... hich pr ••• "" Ih , pf<lx,mity p.-operly. Tbi, ttehniq"e .an be Hplained OS follow" If the binary repr".ntalion of the node number in Ih, cho", it 64_ ,61_ , .. . /00, t h. n it i. mapl"'d ;nlo the nod. ,.. 'Ih number , '~_"._' ... '"' ... here <; ~ &; + 6;t l, il (t < d - 1), ond <, .. b" if 1 _ d - I. For en mpl. , node numbo, 26 (011010) in Ihe chain,. mopped inlo nod. 23(010111) in the loypeKube. If Z· < 101 " Ihen mo .... Ihan one le .. el of 110. 10.,1' may be ..,ignod 10 0 . ingle procoosar in the chain. Thi. HIen.ion will nol. be dioco ... d here. Pro,"""", p" (0 ::; i::; n _ 1), of tbe chain ,tor .. 110. ;'~ I~el of lh. he.p. E""h nod. hos 6 field" DATA, P RIORITY, LCIIILD, RCHILD, LEMPTYNODES, and REMPTYI'ODES. For a node, Ih.

DATA fi.ld hold, an element and the PRIORITY field hold. Ih. priority .... ooialM with t ht element, LCHILD and RCHILD hold. r •• pedively pointen '0 tb. 1.1101;ld IUld rightchild ,,/ thaI node, and LEMPTYNODES iUld REMPTYNODES hold the numb.r of null nod., (nodes with no informalion) In th. 1.1, and riShl .ublrH1! of <hI node . Initially, Ih. OAT A field of .Ilth. nooo. ;v. od to nnU and th. PRIORITY field of an th. nod .. ar •• d 10 -1. The LEMPTYNODES and REMPTYNODES fidd. of all tho nO<! .. at level i a ... initi ali,"'; to 2~ -; - 1. The,. Iwo field, aT< updated .. d ••• elemen" ou in .. rled inlo or delded from lb. hur . Information .h"\1\ lb. numbu ol.mpty, nod .. i. u..a by INSERT <>p"ralion to d«ide .. hieh polh In tho hUp 010uld b< followt<! dUring Ih. ,n • .,.I;on proc.... Lack of . uch information may l ... d 10 an "verA" .. , it".\ion in Ih. I .... ' pro< ... or. Thi. hppen, if th. INSERT oper ~tion mov.o along a path in which .. II th. nod •• ar. non_empty.

Tb. prooe .. running on . ""h proo""'" ;n Ihe hYJ><reub. cono;.I. of Iwo d;.I;ncl par' •. Th. finl p~rt belong. to Ihe applica,ion lhal runo on alilh. P'Moo",,,, 01 the hYJ><reub •. Th. o«,md pari, called u" .. I; •• , io ' ''pon,ibl. for Ih. execut ion 01 Ih. priorily qu.". operalionl. An . xeculi ... Can ~.i"" ~nd proc ... many m .... g .. , imullueou,ly. Priority queu. operMio"o ..... inilialed by th. applicalion pari 01 Ih. pro<",," tIlnning on Ih. proc ... ",,,, 01 Ih. bpucub<. An operalio" wuod by • proce .. i. communical.d 10 Ihe .... culi"" of Ihl proc .... Th. <>lee""ive Ih.n ..,nd. lhal oJ><ulion 10 Ihe proc...." al Ih. head of Ih. chain (",) for u«ulion. Th. priorily qu.u. opeTllion. uc.ived by p."""",or"., w,l1 b< executed In a pipelined f.."hion along the chin of 1'.0<.'''' .... The ... pon .. 10 ,h. XMA X operation produced by '" i, fo ..... rd. d 10 Ih~ 1',,,,,,."0' "" lo i<h origin all y in itiat.d tb. oper ~' ion. An uo<uliv< • ..-;.i"", in""I<I;on, .ilh .. from ~no\h" ,~ ,""nLiv" 0' fmm th e .'pplic~L;on pal"l of Ih. proce .. 10 whi<-h il belons •. Now .. e de.cribe oJ><'"lioll' XMAX and INSEHT.

The algorithm. for XMA X and INSERT,.... diff ... nt from Lhdr .. q".nli~1 ~O,," ... pat". in IhM Ihey bOlh pe.fo,m lh. , .. Ltu~t"ri"g proc . .. from Ihe lop. Th. following algorithm .... pl~in . Ih . INSERT op<ulion .... h.n imel"lion i. J><rfo.med from Ih. lOp. Thi. new INSERT oJ><ulion t,aVfl'O" Ihe heap from Ih, lop to J><,form Ih. , .. I. oclo,in8 1'.0«" and, un lik. Ihe convenlional algorithm ;1 doe. not nec ... u ily . xp&nd th. heap levd by l. veL

INSERT operalion, Wb.n pr"",,",o' p, , (0 -S ; -S n - I), ... ceiv •• oJ>< .~tion fNSERT( p, item ) il perfo.mo Ih. follo .. iJ>g I(Olion, . It ~ ... I <.omP'Tfl Ihe PRIORITY,p] .. ilb 'he priorily of il<m, if Ih. priorily of il.m i. g .... ter Ihn lhal of DATA(p) il ,.plac .. DATAfp) by il.m .>nd Ih.n wUe' INSERT(Q,DATA(p)) 10 proceowr p;~ , . If Ih, priorily of il .... i, 1 . .. t ho Ihl of DATA(p), Ihe. proc"IIO'P; only «.do TNSER~q'ilem) 10 Pit' .The Ieller q ref",.. 10 Lh. :odd .... 01 Ihe righl child of nod. I' if LEMPTYI'ODESlp < REMPTYNODES'I) and rele .. \0 Ih. addre .. of Ihe Iefl child of nod. p if LEMPTYNODES I' > REMPTYNO DES pl. ?roc."o. p, al. o decre~. Ihe LEMP· TYNODES(p) or REMPTYNO ES(p) by on<, depending on ",· h. ,h .. Ih. item will he in«rted into Ih. righl (q .. RCIlILD(p)) 0. lell (q .. LCHILD'p)) . ublre. of nod. p. INSERToperaiion will nol be prop. g.ted ful"lh .. down if n<>d. I' i. ~ nun nod . .. hich in 'hat c ... Ih. only action perform.d by proc. ... or p; io to .Iot< Ih • .t,men' in DATA(p).

XUA X operat,on: When the .~.e"liv. of Po ~.iv., operalion XM,I X from anolh" executive o. from Ih •• ppli<-ation pMt of il. own prO<. .... il g.n .. at .. an operation <-ailed ad) .. 1 .. hich propagal .. Ih",ugh Ih. chain of proc.e"oTO ond converl. Ih. binary I"", (after Ihe rool i. f<mov«l) inlo .. he.p. With .... pecl 10 oJ><ulioh XMAX, all th. proc.-... . XC.pl pro~ ... or '" J><.form the nm •• et of a~tion •. W. doocrib. oJ>< ratiQn XMAX .... follow",

p r "" .... or Po: Wh.n proce .. or '" ,«.iv .. operalion XMAX, il reporlo , h. elemenl wilh Ih. IIigh .. 1 prio.it y 10 Ih. pro .. ,,,,r lhal i.,u,d Ih. XMAX oJ><ralion .nd Ih,n .,nd. oJ><ulion adj~.1 10 proc • ...,r Pl.

p ,,,,, ••• or 1'; , (0 < i < ,, - 1) : Lei I' be Ih •• dd .... of Ihe nod. in Ih. local memory of proc ... or PI _, whOll' vaIn. had been movNlnp o ... ported by proc .. ,or PI_l in the previollO ,ieI'. On r«. iv_ inR operalion ~dj""(P! by Pi, il 6nd. Ih. child of nod. p, q. Ihot ~"nt,i" . th • • lem". ' wilh high .. p"orily . •• nds DATA q) 10 p,_, to replace DATA(p), "nd ".~ I ,,"d. 4dj _.t(j) 10 proce .. o. 1',+\ . PTOc< .. or 1" _1, (i:> I "f,,, fillin. up the node pwiLh DATAh). in""·,,,,", tho .F:Ml'TYNODF.-S or Ht;:"\PT YNODBS field of nod. p by ono dopending on .. heth.r q 11 .. been Ihe Mdr ... of left or right child of nod. p. If bolh child .. n of nod. p aro .mplY Ihen p,oc.,,,,r 1', wil l no\ g .... al. any mo .. ~dj_'1 oJ><ralion; il on ly ..,"d. ,. me""a~e 10 proce""". P' _1 ... kin S 10 emply node p. No pr"" ... or will receive an ~dj",1 op<nlion unlillhe mo, t .. cent ddjnl optration i,,".d by Ihal proce""". is compltled by il. neighbo.ing p.o< .. ",r.

The ... pon .. li me for XUA X i. Olios 11'), bec"n,. il t,k., O(log /1') tim. fo. XMA X oper.>lion to reach prou .. o. '" a nd it aIM> uk .. O(log ,II) I,m. 10 .. nd lh. ~ult. produced .1 pl'O< ... or 1'<> bad 10 the proccooo, which inili .t<d Ih. ol'<',.tion. The piJ><lin e period for bolll 1h. XMAX and INSERT oJ><Tahon. " 0(1), ,nd,p,.dent of Ihe lenglh of Ih. chin of pro<.<""",. Tlte uo<Ulion tim. fo •• ach of Ih. XUA X or INSERT operalion i. O(log N). Thi. i, due 10 Ih, facl lhat il moy lake O(log N ) lime for ,. new el.menl 10 find ii, correcl pi"". wilhin Ih. hup or to KIl up Ih. gap prod"~od .." a

r .. ul~ of XMAX op<nlion.

R.o:muk 1 Information "o.M in th. LEMPTYNODES ",,<I REMPTYNODES field, of ,h. Dod .. are ~oed by INSERT o""ralion lo decide .. hklt palh in the " .. p .hould b. follo.....,J dnrin, .h. in.miao proc .... Tilis uniq"e pall. un bo .100 <om put"" on Ihe Hy by lb. INSERT 01'<,&1;00 S,8). Thil requirn th. 'yolo", Lo maintain t..., pioc .. of in formalion, /"'1, Ih. index of lb. luI oo.emply node of Ih. hup and Ii"!' lb. indu of th. 1.l\mOlI node in the d«pen le¥el of tho h •• p .. hieh <on l ain. al leall o,,,e HOuempty "(XI<. LeI 1 _ 1 .. 1 - Ii .. l and p .. Io! /4.1. Th. "u .... hor J <u be expraoed al • pob" binary "umber. The biury "'p,"enlation of of I lel1, DO .. hethe. lo go t.o ,h. righl or left .. hen we Ira".tinl down Ih. h •• p. AI level i ... e D .. Ih. i'· bil r...,m the t.f, of 110 decide .. belhe. 10 I" t<J Ih right (if I) or left (if 0). ROO!. of tb. bup ;. a' I~ol 1. Thio p<o<edut< tan be utended t.o .pply ' 0 • i-ary h •• p .. her. k ., 2". If , _ I •• ! _ r",t and p .. los IA't. Then al 1"".1 d of the I,eap w. n .. (I + I)'· b," of binary "pr •• ublion of Iw eh.,.,... the nod. al the nox, 1 .... 1 of lb. h.ap. Th. adQJlhg< o( Ihit p"",.dn. o ... r Ihe one .. oed in ,bil, .. port is lhal il requireo I ... m.mory .p;ac. and al"", it upand, th. hup 1 . ... 1 by Iov.1 jun .. ill Ih. """"'ntioul oequenti.J al gorithm. Th. o .. fnln ... of informalion . boUI Ih. num!>.. of . mpty nod .. of 110. lofl and righl ... blre< b.<om .. apparonl wh on ... , lalk aboul banyan heap in 'h, rollowing OKtio ....

3 Banyan graphs and banyan heaps A b. nyan gr.ph i. a lias>< diaorram 134) of a pmial orderin, in .. hith tl,.,. i, only one pAllo fron,

any bue t.o any apex . A b-. i. d.&n~ .. aoy verla: ",ith no arcs indd.nt out of it and an ape" io d.6nod .. any ... rt", ... ilh no U'f.II incidonl int.o il. A ....,.1", lnal i. neilher an apex nor a bue yerl,"" it called .n intermedia'" yort.,..

An I..-1,y,1 banyan il a banyan in which tho p.th from bue t.o .pe,, (or apex to bue) i, of I.",th L. Then:fo .. , in an I..-I""ol banyan, Ihe" an. L+ 1 lovell of nod" ... d L 1,.,...,1. of edg ... By""n ... nl;on, apex .. an. ""n, id"red 10!>. al l.y,,1 0 and baHt a, 1 .... 1 L. In a banY3n , .. ph, 110" outde,r ... nd Ibe ind.l ..... of a nod. are called .pread and hnoul of lhal node. If Ih ... i. An edge be"'·.en I ..... node •. x al leyd i aDd Y al 1.",,1 • + I, Ih"n w. oay x i. 110" pM<nl of y. and y i. Ih. child of x.

D eBnltlon 1 .4 "",u" call,d. _nilo"" " .. ,u iJ oJl !A. 1>0'; .. ",,'Ain 110. '''m< 1 ... ,/ A" •• id".,i.oJ 'P" "'; " " ,; luo.!.

lu a uifonn b~nY&n ,Ihe fanou~ and .pread ~alu .. m.y be char.th.ri.ed by L ,ompono,,1 y«lor1l, F ., (/0, j" ... , h_ d and 5 _ ('" '" ... , 'd, Ih. rano~I y""t,or and Ip,ud y",,~r, , .. pecl;v.ly, wheu ' ; and j, d. nol~. Ih, ' r TOad ~nd fonoul of ~ node al l.y,l i.

D e6n;t ;oD l If .;~, ~ I;, (0:5 i:5 L - I), 141;' F ~ 5, !A." !A. "",,,,, i. ,,,II,d ",'o"1..J" •. 1/ . .... , '" " lor ." .... i, lIou 110. "nyu i, Mn."d""g..Jot.

D " Bolt;on 3 .4 "",II" ;, ",it !o k "1..J .. • j.; .. t, {I :5 i :5 LJ, .. ...I j, '"' f, (0 :5 i:5 L - I), Jor .,,"" o"""",.! t ""t J. 0110.""". il;, '4id!0 6e i,"f..Jo •.

Ddln;t;on 4 .4 "",0" i, "" SW,M",." ,/il loll, IA. /0/1""""1 ho" Addll10ui p!"Op .. !itl , 4) T_ nod .. ", ... i"h"".d;"!. 1 .. 01 i, h ... illo .. .. 0 or ,,1/ co",,,,,,n "" .. "II .1 1 .. e1, - 1, ~) ho" "od ... , i"I.,.".,d;"l, 1 .... 1. h .... 110 .. " 0 o. oJl '"'''''''''' dild .. " .! , .... , i + I.

D eBnll ion 5 .4" SW,."n,u i, ,,,.11 I" 6e ""'."9~I"t ./il" " 10/", "n" I, '"' d, (1:5.:5 L) •• "d /; = d, (1:5 i :5 L - I), /" •• " .... ")!"'A'" d.

Havillg inlroduced .he nec ... ary nol .. ionl and d.~nition, w, d,6no banyan hoap.

DeBnltion II .4" I..-I ... I kt"1n IuGP " "" £.1 ... 1 loa",,,,, nolo !Ad tAt. prio"/J "f tlte clem,nt ", .od. I>Od • .. ,,_oJ O. ,....,/<1" tA." tIu priori!i .. 0/ II .. 01, ...... 1> .! .ocA oJ i!. diJd, ....

Figure I .ho ... an uample o( 4· !evel r«tangu13r hnY3n hup. In Ihi' ""p0rt we OIudy Ihe im­plemonta,i"" o( M~'" reclanlul .. SIV-bany~n heap wilh ,; _ 2 en ~ n d,m"n.;onal cube wh .... 2ft _ los '" + l. '" is Ihe numbe r o( apex", Th. rutriclion ~ an Mx,lf T«Ianlular S W_banyan it in Ihe inl .... 1 o( ,;mplicily of p .... nl.l;"n. In ouch banyan. Ihe RUm!>'t of l.yeb it log.lf + 1 .• ~""h nod" in Ihe banyan hup h ... six fields: DATA, PRIORITY, LCHILD, RCHILD, LEMPTYNODf,;S, and REMPTYNODES. In addilion 10 Ihe aboye ,iw. fiold., nch ap<~ h,. 3nolh .. ~old called NEXT. Thil &eld io uoed 10 link apex .. &oedh",_ Inilially, th. DATA fi.ld. o( all tho nod .. ar ... 1 10 nuo and the priorily fi"ld. of .llih. nod .. ar. '0110 -1. To iniliali •• Ih .... field., fir1l1 partilion \I,. hup inlo '" di.joinl bin.>ry t ...... and Ihen U,"" the umo rule .. fo, ,h. fi",' do.i~n 10 in;Ii.~.e Ih. LEMP_ TYNODES and n£MPTYNODES fiold. of 110. Dod .. in .aeh pmilioR. The partilioning p"",_ 11m. wilh Iho ld'm""t ape ~ and con tinuos in inc",uing Oro .. of .h. ",,",x number •. Th. Idlm",1 apex i. "umbeted l. Par';I;on; i, Iho oe' of aU nod .. which a .. "'Mhabl. from ape" • by moYin~ dnwn 'h" hup ud a~ RM pan of p .... ilion i - l. Th. r",,1 "r r"li';~n' ;, .'pu '- Th. deplh nr" 1' ...... ' ;,,0 i •• he d.p.n of Ih. ,"""",.",,ndin ~ bin...,. I..,.,. Th .... 1 of p"I,loon. "nd .he in;Ii.>I ... tling, of LEM?TYNODES and REMPTYNODES fi.lds f" r an ad S W. banY3n is gi""n in figure Z.

De fi nition 7 A ~ L-I .. cI,..Tlilion<i knyon A.~p" u L· I.wl k"yo .. n,A IA~I ,~<A "",tilio" 01 tA, k ny. " ( •• •• liMa do •• ) i. a bi"." A,op.

D efin itlon e A ""rl;lw ... d L.I ... I bonya" hoop':' •• id to .. /,,,11 ~p 10 opu d il .11 tAo nod .. in ,..,litio ... j, (j < d). ,IT, no,,·, .. 11 a"d tAo .. od .. in til ... moini"9 "",titioll.l ... " .. II.

D efini tion 9 A" ..... i" ....... 'u A.op i, ,.,d to .. ".,Adl. b, ""rllllo" 'ro"" ."". , if it> ,.." M ;, .... <hdlo by ..... 11/,: .... lrom 01' .... Nod, 1.1 1 ... .1 i +l " ... ddl, ~, "",tillo" lrom n<><1< t d' I,.d i if "od. I .i,Il" Au "o~· ... . REMPTYNODES .nd ;1 i , ,A •• igAI .h,ld of "od. 1,0' h,. non·UM LEMPTYNODES ... d <I i. tAo 1.lt , Aii.d of nod. I. An .pe. i. ".,Adl. b.w "".tition Irnm dulf.

Remark 2 Th. null nod ..... hi<.h a~ ",,,,,habl. by partition lrom "siv. n .pu ... ill be fillod up by in • .,.tion. initiUod . t Ib.1 apt" unl ... Ih • • uchblny oIlh, nod,. will ,hug. by a laler d.lelion o""'&lion ioilialod at oom. olh ... ptu •. R.&thablity do .. n~ imply "'&th.b1ity by parlilion.

4 Se con d Implementation Th. ,[ective impl.m.ntation 01 bany~n h.ap on hypt«ube requ i,.." .ffici.nl mapping 01 th, banyan

It""tU'" among the proc.18OfI 01 Ih. hyp'l"eub •. W •• umin. 1_ ouch mopping'. Th. finl m.pping is obtaint<! by tint m.

h .. ! ~II. b~")'l<" into a linu/" chain .nd Ih.n .mbetlding Ih~ chain inlO Ih.

hy"" ... uh. (~ 5ju,"" 3 , T ~ second mapping is obhinod by ,ol!a""ing column. of Ih. hany.n inlo .ingl. p.o,~ .. 3( 1_ al is .a<h b ....... ap.~ palh in Ih. bany",,- wilh id.nli<ally labelt<! v.,,;' •• is m. pped into on. proceooor in Ih. hyp.n:ub. wilh Ih. nm. label. A«ording 10 Ih;" mapping, if lwo nod .. art adj .... nl in Ih. banyan Ih.n Ih.y are map.p<d inlo two adj"".nl p.oc .. oon in Ih. hyptrcub., IhO!. hying label. Ihl diff.r ,~&ttly in Ih. , ' digit. ~ •• 6gure 6 for a:n .umpl._ In this seclion w. <on.idcr Ih. fi",1 m. pping.

a.1_ w. ,i ... a d.lailod d .... riplion of Ih. ope ratiou. XMAX and INSERT"", .ho .. how to <om· put. Ih. add'--f:I of th children of a gi ... n nod., The nod .. al a liv.n I.yd a", ,to.od ><qu.ntiallr in an arTaY of .i. t M in the loc.1 m.mory 01 Ih. co ..... 'ponding proc...."., 1..1 "c; to d.not. Ih. i' nod. on Ih.l'a lev.1 of Ih. b.nyan ... h.re 0:':::;:'::: M ,O:'::: (log M + 1). Th,n "e. il ,onnecl.d 10 t_ nod .. n1' '' 'j; and "l'HI .. , where m i. Ih. intog .. lound by in,..,.ting th~ i,a moot .ignifi,ant bit in Ih. binary re preun lalion of;, W. col! nod •• n(H' I; . nd nj, .. 'I .. Ih. righl ~nd 1.11 child ... " of nod. "';, r .. p",li ... ly. Th"".fo"" Ih. I.ft child and nghl . hild 0 nod. i ,I lev.1 k a .... tored ""peclivoly in Ih •• '" lo<ation and m,a loution of Ih. array r .. idinS in tho local m.mory 01 procesoor k + l.

An the XMAX and INSERT op".lioo' inil;alod at di[ . ... nl proc....."." . re .. nl to Ih. head or Ih, embetldod chain and ..,.. OJ<ecutod in a pi""linod manner. Wh.n optra~ion INSERT i. r",.ived. by proceuor Po, it fi"'l find. Ih. 1.llmool pmil;on whi.h h ... at 1 ... 1 on •• mply nodo. Thi. ,an be don. u.in , inlormat;"" OIOrt<! in Ih. REMPTYNODES ud LEMPTYNO DES field. oIlho aptx<I in O(M} lim •. II Ihon pu , h .. Ih •• Iom.nl " ''lu •• I. d to b. in""rlt<! down Ih. hanpn hup u.ing op<ralion ."".I_.d, .. ,1. Th. optrati"n iU"' -odj'" pu"lI .. the .km'nl ,]""," (~lon8 Ih. p~lh, lrom Ih. rool of Lh. parlition 10 Ih. b .... . ) unta it find. ito 'OTrec' pooit ion Thi, ~"i ... 0(10' M) lim •. Th n.INSERT ;, an O(M) ol><'fation.

Tn Ih~ cod •• Ci,",. brl<>w w. u ... h. lollowi"c .ynl;u ~nd um~nl;' 1<>< .. n~ .'nd "",.i", in.ln«· liono. The in.l.uetion Sond«pro, ... or>,'<in.lrutlion>'j .. nd. in.truehon <in.lrudion> to pro--""or < proc._.> for ",«ulion. Th. execulion of ..... ive( <p.oc<ooor>, (i.formalion) cau .... h. informa'ion .pecifi ed by Ih, "",ond argu m.nt be ob.ained from proc ....... <pro, ... o/"> and fo ..... ard.d 10 .h. ~u .. ling proc,,,,,,r (Ih. proc"".",. ",ecu li ng .1.. ,,,,.i,,. in.truclion). A «<,ive inslru,tion u",ul.d hy proto ..... p; i. nol compl.l. unlil a m .... g. i • .-..:.iv.d lrom p.oc.-..or p;.,.

Upon . rec;.ivini INSERT(p,(iltm,priorily}) by proc • ..". pO, it execul •• Ih. follo ... ing cod ... Th . letter P II Ih. addt ... 01 Ih. l.rlmOfl al'<X, and (item, priority) i. the pair l"t'lu,,"l.d to be i • ..,.lod.

lound ~ fal .. While (no. found) do

if DATA(p) -I null . hon il priority> PRIORITY(p)

btg;n if LEMPTYNODES(p)1' 0 0. REMPTYl'iODES(p)'l'" 0 Ih ••

begin if LEMPTYl'iODES(p) > REMPTYNODES(p} Ihon

begin p' _ RCHlLD(p); REMPTYl'iODES(p) ~ REMPTYNODF.S(p) -1 .. , ....

begin p' _ LCHTLD(p); LEMPTYNODES (p} - LEMPTYNODES{p) - 1

,., ...... d(Po • 'in ... rl-adju.t(p', DATA(p))'); DATA(p) _item; PRIORITY(p)_ pnority; found _ lrue ,., , ..

p - NEXT(p)

'"' .10. hOi;n ( priority.:: PRIORITY(p) I

if LEMPTYNODES(r) '" 0 or REMPTYNODES(p);i 0 the ... booSi ...

if LEMPTYNODES(p) > REMPTYNODES(r) then b<gin

p' _ LCHlLD(p)i

LEMPTYNODES(p) _ LEMPTYNODES(p) - 1 ,., ,'-

1,..~ ;" p' - nCHILD(pli REMPTYNODES(p)_ REMPTYNODES(p) -1 ,.,

.. nd( f'O, 'inoert_adju. «p·. (item,priority) }'}, found _ true ,.,

,'" r - NEXT!p)

,'" begin DATA(p] _ item; PRIORITY(p) _ priority

end;

Proceuo. P;. (2 $ i :;: L), upon receiving in.<r!-Gdi~'I(p,(iI.m ,priori1.)")) M"'U~' the following <od ... ,

ifDATA(p) 'I' null then if priority> PRIORITY(?) then

b< gin if LEMPTYNODES{r) .,. 0 or REMPTYNODES(p) ;i 0 then

bogin if LEMPTYNODES(p) > REMPTYNODES(p) then

begin p' _ LCIIILD( p) ; LEMPTYNODES(I') ..... LEMPTYIWDES(I')-I

,.d .1 ••

b.~jn P' ..... RCHlLD(p); REMPTYNODES (p) +--- LEMPTYNODES(p) - \ ,., ,.,

..,,,d(p,+!, 'in",r\.- _>dim'! p' ,(D A TA(p).PRIOR ITY( p))) '): DATA (p) ..... it.",; PRIORITY(p) ..... priority ,.,

.1 •• begin

if LEMPTYNODES(p) .,. 0 or REMPTYNODES(p) {o 0 , hen begin

if LEMPTYNODES(p) > REMPTYNODES (p) then begin

p' ..... RCHILD(p]; REMPTYNODES(p) _ REMPTYIWDESh,) -I ,.,

,'" begin p' _ LCIIlLD( p);

LEMPTYNODES(p) _ Lt>MPTYNODES(p)-1 •• d

.. ftd(p;." 'in .. r\oodju.l{p' ,(item,prioril,))'); .. , .... ~Ji.l>

DATA(p) _ il~mi PRIORITY(p) _ priorily .. , .," bel;n

DATA( p) - item; PRIORIty(p) - priority

endi

XMAX oper~lioD ~lloc~\e' tho ape x .. bich contain. Ihe .Iom,nt wilh .h, highe" priorily, up.,..., thai ,l,monl \0 tk, oa"ido world, and Ih~n fill. up Ihu .p .... with Ih~ tlnntnl ia oM 01 it.o .hild",a. rmu·u"j .. , it _pon,ibl, for ""'1r.cl.rinllh~ banr""" il m""a d.-a Ih~ h •• p. Whtn XMAX( p) it reui~ by proc_ Plo il uecna II.. followi_. cod .. , wh.", ,. i. the add.,... oIth, loftmool apex. Th .. add ....... kn own \0 II.. omuid. world (f,..,nl .nd com pUler).

p' - p .. hile NEXT(p) # nil and OATA(NEXT(p») # null do

belin if PRIORITY(p) > PRIORITY( NEXT( p)) then p' _ p

•• ! - NEXT(p)

.. nd('ou,"id~ .....,..Id' DATA(p'II ' _.iyo (p" ((PRIORITY(RCI LO{P'11,DATAIRCItII,D1PI"'

(PRIO RITYILCIIILD p'il,DATAI CIIILD(p'JII) I if OATA(RCIIILO(p·n.;. null or DAtA(LCII1 .Dlp'))),! n.lIthen

if DAtA(RCIIILD(p')) > DATA(J.ClllLDlr'))lhen b'liA

DATA(p') - OATA lnC11!LDlr:)l;

~~~?:4~~~6.;!~~~RJi~j!li.rJ~'?,~I~') + I; ,n.rnd (/>:l, ':.m ... · odjuf\(RCH ILD(p 'll') .," begin

DATAlp'} - OATA(LCHILD(p')l; PRIORlty/p') _ PRIORITY/1;(;HlLD/p'll; LE~IPTYNODES(p') _ LEMPTYNOD~rp') + I i ... d (po, '.m .... _a.!j • • I(LCHILD(p'))' .. , ....

ho,in OATA( p') _ nml~ PRIORITY(p') __ 1

•• d

~eiyt(p;+" ((PRIORITY{RCHlLD! .. n,DATA (RCHILO/ Pl)!,

[

PRIORrtY(LCHittl(P)l,DATA{LCHltD p)))) I; i! DATA/RCHILD pIlI- n~lIor DATA{LCHILO{p» "I- nO I ~b

i! OAtA(RCHI 0 pI) > DATA(LCHILD(p)) Ihu begin

DATA{p} - DATA/RCH !LO(~.))i PRIORITY( p - PRIORITY RCHILD p ); REMPTYN OhF..5(p) - REM TYNOD~l(p) + ,: .. nd(p, ... " ' o:mu •• dJulI(RCHlLD( p)) 'J .. ,

.'M belin

DATA{p] _ OATA/LCHILD/I>l); PRIORITY(p) _ PRIORITy(LCHlLO(p));

LEMPTYNODES(pl- PRIORITY!!» + 1, ... d(P.H, 'xmu-a<!ju.I(LCHlLD(p) 'J

.. d .1 ••

MSi" DATA(pl _ ""II: PRIORITY(!» _ - L

•• d

R.emark 3 Th •• Iemenl, ,"or«! in Ih •• ~" nod .. o.u no\ nnk...t in . ny par \;<uIAfordet. Th;'.p~. up the in .. ,tion ."" ... , b", Iud. \0 0 M) lime ro. del.';"n, II i. p" .. ibl. to in .. n. lb •• Iemonl, in ,uch a ... y lnat S,t . Iemen' "'ith Ih. hi~ ~t prio. ily i • • Iway. av.il.bl •• 1 Ih. l.hm.,.' .pu in whkh .a •• loe.ling Ih. e"",,,,<1 'pe" to initiate Ihe in ... tion I.k .. OfM) ti", •. Th'. melhod ... m. to b< more effi,ienl doe to Ih. rut I porlion of I h. time 'penllo find tit. corr«1 posilion . In be "".,Iapped .. illt l it. lim • • ponllo ]oc.le In apu ,. ith .or" REMPTYNODES or .HO LEM?TYNODES.ln Ih. algonlhm. p..-nled .bo ... we h ... u.& Ih. 5,..,\ approKh. Th. O<'(:ond appro""h wi ll b. roport..! in another po"" •.

From Ih. proputi •• of SlV.hny~n gT~ph. and th. ~bov< ~Igorithm., w. <~n .Iat. th. folia..- in g !'flult •.

T heorem 1 OpcrAlion XMAX ",q.i ... O(M) lim. 10 .ompl.I •.

Theorem 7 OpcrM;o .. INSERT .. </_i .. . O( M ) I;",. 10 .0mpl'I • .

L"mm. I .J TA. , .... ,,-nd'.'1 op .... t.o .. .. eo .. u.oul", ~ .. 0<1. u·A ',A " Mn·". 11 GOld Ao, ~.ro LEMPTYNODt:5 ud .... H.f::MPTYNODt:S.~) TA. i ... .. /.Adj •• 1 opc ro/io n Alwo~. fi .. d, A ... Ii .. od. 10 i ..... 1 ,e. d.mul.

P roo f a) H op<Tal ion zmu .• djul .n.ount.~ a non· null nod. with LEMPTYNODES and REMP· TY NODES fi.ld. "'I o~1 to .. ro Ihen il mo.1 have 1><.01 inil;'IM from an apu ... ith both REMP· TY NODES iUld LEMPTYNODES "'lui 1o IHO. Thi. is im_ible ac.ordin~ to Ih, algorilhm for INSERT. b) Proof i. immediate from Ih. proof of pari . and the definilion. of LEMPTYNO DES and RE MPTYNO DES.

R"lnATk ~ Dd<lion of ~n d.",enl fron, parlilio" • "' 'Y <au .. on< of 11, •• 1,,,,onU in olher pulilion • .. h_ nod .. ~ ...... l.Ch3bl. from .pe~' to k<ome nuU. Thi. h->ppen, if. d.lele operation 03 .... the .m<U_Adi .. ~ on i1.o .... y down the hup, 10 m"". up the <onlenl of on. of Ih. Inf nod .. of pu!.ilion • to &11 up i1.o parenl ... hieh ho.. bee n emptit<! by .mu'Gdj •• 1 op<T.tion al the previou. "ep. The .mptin ... of .hi. node now will be reftectt<! in Ihe REMPTYNODES or LEMPTYKOD ES of ape x i. Thio node i. na..- ruch.ble by rarlilion from apex i and ... i]] be filled by an in .... lion inilialfll at ' pex i. The m ... imum numl>< r 0 nod .. thai m.y I>«ome "l.Chabl, by parlition from apex; as a !'flull of a d. letion i. "'1",110 (L+ 1) - D, whe .. D i. the depth of pn L.ilion i.

Lewm~ ~ Z~ro REMP T YNODSS BM .~m LEMPTYNODES 'or Bn BI"" do .. no l impl, lhal 1111 lA, ,,0<1 .. ... ,A, <o . .... po .. di .. , p''''i';o .. 01"< ..... ·".11.

Proof From remark 4.

L~U1m. 3 Apcz;, ( I 5: i 5: M ), 01 .... ,. ,o"'oi,.. 1M <I.monl ",A><A hal Ih' Aigh." p" on'l, 01"'0'" ,A, .1.", ... /.0 "o ... d , .. IA. nod .. 0/ ""rLiLio" i .

Proof From algorilhm. for XMAX, zmoz.odj..", INSERT, and in"d.odj.jl.

L~U1m8 4 TA, d ..... '" "",A ,A, A;gAut p";"ri,y i. 01 ... ,. "p.r',d~, opc .... /'o .. XMAX .

Proof: From lemma 4 ond Ihe 6 ... 1 part of algorilhm for XMAX.

Defin it ion 10 A por'ilio .. in.J",u ~, LEMPTYNODES nd REMPTYNODES {oold. %pu i i. /A, .. , 0/011 "od .. ",AicA M' ".,Adl. ~, p4 .1;,i."/.-.,,, 0PC2 •.

5 Retrieval at Percentile Levels On< of the mo.t imporlanl iOdvantage. 01 banyan hea p """ Iho binary hoap i. lhal it ;. _iblo

10 "I,iov, .lem.nl, ~\ d iff ... nl percenlile 1 • ...,1 •. In Ihi. ,..:tion wo derivo lormul ... lor Iho porcont il. lev.l 01 Ihe . lem.nl r.porled by opaalion XMAX lor diR".r<nl c .... .

D efinition 11 A~ d<"'<~1 ",mo •• d /'0'" 4 I>o.~yo~ ~<4P i, 01 /><",u /il, , 'f 0,1,,,,,. p .. «~1 0/ I~' ,I,,,,,"h ./o .. d ;11 I~. A<o~ A~ •• prioril, I, .. IA.~ 0' ,q~.1 /0 lA, "non/y 0/ lA, dd.,.d d.m.nl.

W. d.fine REM PTY NODES, 3nd LE M PTY N ODES, 10 donol, T"pediv<ly Ih. value of REMP· TYNODES field and LEMPTYNODES fi.ld of apex i. Th. proof 01 th. following ( lemm ... Me immodi a~ from Ih. d. 6nilion. of REMPTYNODES and LEMPTYNODES.

Lemma L T~. 10/.1 n~m'" of nidi nod .. wAi,~ 0'" ",.,Adl. b~ I'M/ilion/rom .pu i i, REM PTY NO DES, ... LEMPTYNODES,

Lemma 6 1/.n M.M ".,.lIlIo".d ""'."gul •• SW.60" ,." I>o."~.,, h,.p ,,/~II up 10 01''' d tA. "

., L)lIEMI'TYNODES; + LEMPTYNODI,'S,) = 0 .

; B'

Lemma 7 [" on M:rJ.l .. ,t."g~lo. S W.~~n,o~, th, 101.1 n~mb<r 0/ null "od .. ... ,hbl. b~ "."lllioll from .p.r .. 1 'A 'O~9A d. w, it/.n N ULLNODES(M,d}, i, Vi.,,, by:

< N ULLNOD ES(M,dj .. L(REMI'TYNODES, + LEM I'TY NODES, ) + K.

; * ,

",A ... K i, IA. ~~m'" of ~"/I ~p<%" i, (i$ d) .

Lem .. ", 8 TA. lo!~l ~.m 6 .. of Mn· n.1I Md .. i~ d M:tJ.f l'<',tilioMd ,.."tUg"ld' SW.I>o.~ydn, ""il!<n NONNULLNODES{M,M}, i. M(logM + 1) - NULLNODES (M , M).

Lemma 9 TA. lolal "umo., o/po.titio,," 0' d.pU. k in d f .. 11 Pd,/ilioMd 6o.nyaR ~'dP up 10 01''' d, "",,1.11 N p., ;, gi.en 'y: d - ,,~ -, NP

NI'.=l .:..., . , 'J ,

Pr<>of. W, proY< Ihi. lomm~ by induction on d.

B~,io: For d ~ 1. NP; = 0 for alii $,. $IOiN + I .

lnduclion: Allume Ihu

Con.id .. d+ 1. Eilher d i. even or il ;. odd . If d ;. ev. n th. " ape x d+ 1 ;. 'h~ root of 3 pMI;lion with depth 1. Th ... fo ....

d+ l - L;: ~ NP;- (,vI', + 1) ,'11' . .. [ 2 J,

For Iho c= Iht d i. odd w. con.ider lwo .uh<;~: Apex d + 1 i. cil h .. the roo~ 01 ~ partilion of depth k OT il i. the root of a partilion ",ilh depth A. (h < k) . If d + 1 i, the TOol of a partition of depth d Ihen w, ... ill hove

Sin,e d i. even Ih.n .... have

01- ~h' N P NP. _[ £..,., fJ. ,

If .po ~ d-+- I io Ih ......... of & puloiliou wit~ dep\h " ... ho •• I :s " < 100: N -'- 1 ud "-;' k .hen ... e h ...

,\'/\ = l d + I - L: 'Si' •• - I.,;" II' P, - IN P,. + I) J.

r..,mtn~ 10 Th. 10141 um~" of h<I~ - ... 1I ~oJ .. ;~ • /~11 MzAf "".1;110 .. '.1 r«14ft , _I". S W" A .. ,U ~p lOOp" d, ",.w ... ,i •• (M,d). ,. , i •• " ' "

M........ . n'z« M,d)_ 2: 2"+ L NP,Z;

•• 0 ,..,

LMnm. II II u Mz).{ ruto"f'/o . SW· .... 'u po . hhOft.,J A •• p" 1.11 x, to .,,01 ~ IAn U .. d,mu! d"f'td.' .,..z J II 0' ,.. .. Uli[. Iud

[1M-I) . \00 .... (M ,d)

P r oof " .. {M ,d) it the tot.l number of "od .. in an M x M ro<un~"lu S IY. banyan hup .. hich it fall up 10 &I'U d and 2M - I it the Iotal number of nod .. in partition I.

Lewma I Z 1 .. u Mill "".titioned ."lu,.I • • S W"u,u h. op .,A"h i.l_1I ~p 10 . pu d, .f "".._ otion XMAX iUII/i, .... i,(i<d), Mn· ... I1.,..r .. lAu lA, " .. « n'ilo elthe "'I"','ed d,mn! "

P roof fum le",ma 10.

WmID& u 1/ ",..,..hu XMAX 'umi ..... "" ... 1 IA",.,A J •• Oil MzM ",<lng.lo, .... 'u .'0, IAn 110. ,. ..... hI • • / IA. ""o,'uI tI .... ul u ""oJl .. lAu . r .,.oJ IQ

P r oof It a buraa kup io full up <0 apu J l b ... by I,mma 12 lb. per«nli!. of lit. ol<mtnl ...,por'Ud by o~ation XMA X ;'

I

By lemma 7, Ihil u n boo writlen ""

ou.(M, J) • 100 ( .. · •• (M, Mj- NU LLNODES(M, All)'

I)~. I ~' banyan i. " '" '~II ~p I .. apox J ... d .~u.r.".. oomo 01' Ih. "<>d~. whid. ar. r .. ch. bl. 1m", ape><u I 10 J an nil Lel F be Ih. 101.1 "Umber of .",h lIodu_ n...-.I_ Ihe !>'"f<e .. "l. 01' Ih, elemonl "'porl«l by X Mil X opeu lion ,.ill be

(.ue(M,d) - F) . 100

(om (M , M) N ULLNOD£S(M, M ))'

110;. pf'OYU ,h, lemma.

rr",,,,"rk 5 A priority <IU,", c~n ~l", I>< impl,men\od ... banyan he.p. " paMition,d banyu hup c.>n I>< conv<rt.d into a banyan h • • p by an o~rAtion cilled ~dj~,t. M log .lf - 2 .dj..,t o~r.\ion' ",e b,oade.""1 by Ihe XMAX o~,.tion .. hen it inurl, an element into.n . mpty p..,-tilion. Th.,. ~dj.8I o~ratlOn. ca" ••• om. of th . element, in the nodu of Ihose p.>rliliono which are r .. chable from a~x ito mOV. up and 6II up the nod •• of PMtil ion i . A" a r.,"II , all the node . .. h"". wnt.nl> (empty or non. ,mpty) h ... b, .. m.,...od by Ih. ~dl.,t o~ratlon be<on,. re . <hable by part ition from a~x •. It .hould b. not.d Ih < ..,m. of Ih • • di~" o~ration . iniliated at proe • ...,r pO by XMA X o~ral ion may not hav. any elf«1 On Ih •• truclur. of the hUp ,

The w vantage of banyan hup o ... r p..-tition.d banyan hup i. tht it allow. a more uniform di. tribution of dal~ elemen" among Ih. pari ilion. in Ih. he.p and lead. to a mo~ uniform inc", .. e in th. ~l'<entil.l.v.1 of Ih e reportod .lement .. the numl><r of eu min..! a~n' inc~ .. e. ,

Algo";lhm, for XMA X, %m.%.~di.' t and in",t .• dj'..,I are Ihe n me for Ihe banyan heap , The o~ntion INSERT and th. ne .. o~ .. tion ~di • • t ",e d .. c";be-d in 141.

6 Th ird implementation In the 6 ... 1 Iwo implementalion. , proc",.." '" b"""meA a botll.n«k becu •• all Ih. o~ .. tion.

mn,t p ... Ihrough th;" proceMOr. Th;' limits the potential to"<urr.ncy of th ... t..., implemenlalion. 3nd p,-.,venl! them from fully utili. ing the hypercube t.opolou. For the th ird impl.m. nlalion, 110. hnyan hup d at~ . Irueture i • • gain u.ed, but the mapping", not onto a linur chain of proc.osors. In, tead, 110. banyan heap io mapped onlO proee"",rs column by column. S •• figure 4. Thi, Iud. 10 an implem.ntation that t.nd, to diotr ibute the load amon, th. proc •• ",rs more ev.nly .nd achieve ,reoter con~urreney, Thi. mapping . 11""" 110. algorithm 10 make 1><11 .. u •• of concurrenl proce .. in~ capabilit;". of Ihe hy~",ube topology in order 10 '''''uee bot\lneeh"g in th. fi ... t proce"or , an .!fecl not..! in t he u.e of Ih. linur <hain .mploy"" in Ih. fi,.,t t .. o implementation •.

In Ihi. impl'ment al ion, each column of Ih. h.nyan heap i ..... igned to a unique proeu,or .• proc • ...,r .. hOi. numl><r io the .ame .. the numl><r of thai column, Thu., if a node i. ",. ign ed 10 pmceowr % then Ih. lefl child of Iht node i. ,tot.d in preee""r % and Ih. righl ehild io .Ior..! in • proe • ...,r which", Wj...,.nt to proce.",r % and .. h",. add"",.;. m ", h. re m i. the int.ger comput..! "I inv. rting tb. i" ",,,,,t ,ignifiunt bit in ,h. binary rep,e.ent . lion ofi. Th. noo .. in 0 given <olumn o the b.nyan hup ", •• to~ •• quentiilly in an &rray of Ii .. log M + I in Ih. local m.mory of it. proeeswr.

Th. al gorithmic principl •• of XM A X and INSERT oper3Iio" . are ,h ... m ... the seeond imple-m. ntation , Bnl . in« column, of Ihe h .. p are .to~ in cli.tinc' proc .... o ... in Ih. hy~rcul><. th. o~ntion of computing Ihe leflm"'l partil;on that 10M at 1 .... 1 one empt y node ( .. plITt of INSERT oPfl'a\ion) "nd Ih. operation of locating the ~~x with the h;~h"l prio,.;"y (." pa" of XMA X op­. . .. ion) un b. ['<rform..! in O(lo~ M) ralher than O(M), Thi . I~ , ,,b 10 .> lolal TO.pon.e lime of O(log M) for XMA X ope"lio" M "' ill 1.>. diocu ... d I,ter in ,hi. >«I;on ,

To imp.,m.n1 the priorily qu.". o~ .. Lion . ... n • ..! tw<> o~r.,tion >: Glol>«l O"",d,", I.>n n GlOMI Mi~. GloW Bro.d,.d i. u.ed 10 di, I,ibul. a m ... ag. t.o all the nod. in the hy~",ube, . nd Clol>«i Min i. u • ..! \0 comrute the minimum of a lin of il.m . .. hich are di , tribu te-d among Ihe pro""",,,," of Ih. hy~r<ube 16. Belo .. we de. cribe algorilhm. for Ih . .. two o~ .. tio" •.

GloW Broad,.,t The proc . dure i. b ...... d on Ih. idu of a fanOUI teff in .. hich each node >end. Ih. mes.og. to all of ito neighbo ... .. hicb Ita ... nol alre.dy rK.i .. d il. The fanoul ~ree root..! al nod. o for a 4_dimen. ional hy~r<ul>< i. gi ... " in fig"~ 5. In Ihe fanout lree each nod ... nd. th, data to .11 n. ighbors on Ihe li.1 of ne;~hbon of that node .fter Ih. node Ihey , ,,,.i,,, il from. Node. in •• ch of Ih. 1i.1 of neighbo ... . re ord.red by incr .... ing node number. The Ian o" t " .. rooled al .>ny olhor node r c1m be compul..! by lak ing Ih •• xclu.ive OR of any nod. in <b. t,..,. with r. See fi~ure 6 for the fanoullre. roolod .1 nod. 13. The following th""rem from Brandenburg and Scoll 171"ugg •• t, iUl efficient implem. ntalion for ,hi. algorithm.

Theor~m 4 I'I~. rool • ,ud, !o ... . yi.>dy .nd <Q'~ oth •• .. oo, %, .'hi,h ""'"" ~ ",,,,.g. ''''''' z, Iud. it 0" I. oil ... i9M." Y _ ~ + 2; ."'~ th., 2; > % + ', th." Ih. " .. /ling ,"Mol I." i. Ih, '4mo ., !hM obl.i",d ' .om .. d.,i • • OR wilh r ., tre. r",,'ed .1 "TO,

GI.1>o1 Ati,, : Th. algo. ithm for ' hio o~nlion ;. al.., b .... d on the idea of a fanoul Ir ••. In Ih. fanoul I,..,., eath noo. comp"t .. the minimnm of it. cur~n ' item and Ih",. of it. children .nd thon p."'" it to it. parenl . The ex«ution of Ih. cod . for Glol>«l Min . t differenl proe ... o .. i •• tarted when a m.,ng. i. re«ived from Ihe root of Ih. corre.ponding fanoul I ... r"iu"' ting the comput.lion of Ihe minimum ,

OperMion. X MA X .nd INSERT for Ihi, impl.mentalion are d •• crib.d I><low,

(NSERT, An INSERT operation iniliate-d al nooe % firOl <om put .. the odd .... of the 1.ftm",' a~x y whose parlition b30 at 1.",t one . mpty nod. (",i"g Glol>«l Min o~ralion). Th, el.ment.prioriIY pair i. th. n >eM from nod. % \0 the node which conl.in. a~x y. Th. p.ir willI>< in • .,I.d into Ih.

pMtit;on roolM d ~p<x y iOCoor<ling to the procedure described for ~he .<'<ond impleme nt ation ,

Th~ prohlem wilh th. INSERT operalion .. given above i. th t an in ... lion oper~tion may fi nd> parl il;on (reporlod to haveemply pooilion,) full when;t ~ri •• 10 in'e<1 a pair inlo ~hM pUl.ition, and therefore blockod ond unable to proceod. t hio i. co u.ed by allCM'ing .. veral INSERT ope .. lio •• to ... ",h the lill of apex .. concurrently for Iheir poinl of 'n,.r~;on . , which, ... . re.ul~ more lhan one INSERT operuion, may 1"« ..... Ih. um. apex whoo. corn.ponding partilion h .. on ly one . mply I>"'ilion &/I Ih. poinl of in ... I;on . In thi • • iloalion & n. w INSERT operation m.y ~ •• wned al th. nod. ",hich block..d the in~rlion .

XMAX: An XMAX ope.alion iniliatod at node ~ firsl dele.min .. Ih. ape x which containo the element wiU, Ihe hi~ h .. 1 priority (u.in g Global Min openlion) and then .. nd . an operation, called ~di." , to the nod. cont:Uning that a pex. Operalion ~dj"" fi"t reporl. the el .men~ to m.de r, and then adju. t the hny." he.,p .. d .. crib<d in Ihe previo". -=,ion.

The problem with XMA X operalion .. described above io that due 10 concu",,"1 ",cOM 10 Ihe liol of apeu. by .ev .. al proc ... o,", the nme apex may b< reported to •• verol XMAX ope"lio", .. the holder of Ih • • l. m.nt wil h the higheot priorily, T hio may lo~ 10 Iho , ilu ' lion in which .I. menl. retum..d and .ub.equ.n~ly delel~ by lOme of the XMAX operation do nol have Ih. highe.1 priority in lb. hnyan h e~p at the time of r.moval. On. IOlution 10 Ihis problem is to .. nd all the XMAX open~ions (i"uod al differenl node. ) t~ nod. "" and let node"" uocute them .. qu onti.lly. Thi, limit. lhe ~mounl of conculToncy obtainabl. in the .y. lem. Tlte amOun~ of obl~inable concurrency .hrl. to deer .... due to thi, ot .. tegy when the numb<r of olemenl' in Ih. banyan hup uc • ..d . 2'oc M+O _ 1 for t h. fi .. llimo,

Tb w relJ1 ~ OI'<",lio" XMAX "'q~i,. .. O(log M ) lim. 10 compl., •.

Proof: Operation XMA X con, i'l of 3 parto: 1. findi ng the apex cont . ini .. ~ Il,e elemenl wi lh the high.,t el.m.nl . 2. repor~in g Iho element 10 the nod. initi aled Ihe operation . 3. ;adjusting Ihe h Jlyan h<ap

E""h of Iho., .tep. uquire. O(loS M) lime .nd h,nce Ih, tot . 1 time of 0(108 M) for XMAX ,

Th.orem II Op,,,,'i,, ,, INSERT uq.i .. , O(/ogM) tim. to <"mpl.t •.

Proof: Similar to Iheore'" 5.

Theor em 7 A " XMA X "p, rn li"" n«d, n o mo .. IAon 2(M - 1) + 2 · 10gM <ommn;cal",,,, ,

Proof: The Ihird impl.mentAtion can p.rlonn M opeulion. nl a time and uch operation r<qu"e. O(loS Al) time, and h.nc. O(M f log U ) Ihroughpul.

In ",h' follow. we 6"" define a fe'" t.rms ond Ihen inlroduce Iho con<opl 01 con,i,t.nOf for ConcurTO"t dal •• Iruelme,

D e Hnit ion 12 TA. h"''' I"'''~ 0/ M' or.",li,,", i, Ih, lim< 41 ~'h" h Ih. 0/,<'0';"" ,. iniliol.d 41 • ,,<><I • ."d ,h. 1,,,,,,lomp of <> 'ommnicol;"n ;, lA, 1; ... "lomp 0/ lh. o/,<,.Iion which gen .. ol.d Ihol ,ommonl<4Iio n.

D e fin it ion 13 Op<ral'o~. 0"02,0,, ... , a"d 0 , ",iln hm"lamp' I" I,.t" ' .. ud t, .rc .aid 10 b. ~ .. q~ . .... of op.rolion. ;fl, < t2 < " .. < t, ud "p.rntion, 0" 1 <,,:5 q.rc Ih. only op.rnli"", ;" •• d b.1",.u I, ud I ,

D efin ition 14 A ,c".~"."1 d~lo .Iruch,,, ... said 10 b. con,i,l<nl 'f lA , concu,,,,,1 utcul;o" of ~"1 •• qll.n • • 0/ ol'<",li"",, 01> tho d.t~ ' Iruchl'< gi ... Ih. $Orne ... n it . , ,,<cuting tlt. op,,,,lion. "qu'I>li4i1r.

Proof: Con,id.r ., '<quenee of XMA X ope .. l;o", wailing in nodo "" for n etulion. Duo 10 the f<o<1 tht Ih ... XMA X operation. aro exetuled •• q~onti.lly in incre""ing ord.r of Ih.ir tim "lamp', il i. quite poo. ible for . n INSERT operuio" which have ., lorger time. lamp than an}, of th~ XMAX open~ion. 10 b< u ecul..d b.fore.1l the XMA X operalion. are compld..d.

Theorem 10 n. lAird impl.m.n!alion" ,on,i,t,n! if U 0«." mod. to on apu by U optration i, no! .h,,!«1 "nJ". oil !~. op.m!ion. ",i!~ !~.Io"<r !i",,,!.,,,p. ~a •• • ompl.l.d all !h.i, M+l .... ,,. , 10 IA. a",.".

& 10" we dncribe a procedure for making the third implementalion con. i'lenl.

An oporalion is fiNI r"""rded by all the proeeMOrs in Ihe hypercub<. The proc ... or ... hich inili~l<>d th. operalion broadc ... t. l hot operalion lO~elher with itllim, slamp 10 ~lIlh. proc.,.",., u.ing ilt fan OUI tr .... The oporation.."d ito tim "lamp .. added 10 th. 1i.1 of incomplete oper>lion' m""\ained by re,id.nt .~eculive of oach proc • ...,r. After Ih. ope,alion is add.d 10 th. Ii,t of incomplete ope'alioR' by a proco...", Ihat proco""," •• nd. a notific~tion ,ignallo Ih. p.oc., ,,,,, that initiatM the operaliou. The nOlifi"lion . ignal. i .. ued by Ihe proce .. o", are combin.d accord ing 10 Ih. fanoull,..., .ool.d al Ihe proce .. o, .. hidl initiated Ih. op< .. lion, no procego, .. nd. a nolific,lion .i ~n ~l to it. parenl in Ih. f~n oullre. unl . .. . 11 ilo .hildren have nol.d l\tal op<.alion. Arl .. Ih. '001 proc ... or ««ived All Ih. nolificalion , ign al., the ",,«ulion of Ih. op<r&lion .Iul,. During Ih . exe<ulion of Ih. op<ution no ""co" is mad. t.o an ap< ~ unl ... allihe oporalion . ... ilh lower lim .. l;\mP' have eompl.l<>d Iheir act ... 10 lhat apex. Wh.n an op<ralion compl.l" it. l. ' t ""<eM to Ihe 'pen. , il .."h all Ih. execnliv", 10 um",". Ihal operalion from Ih.ic li.1 of incomplet. opeulion • . Th. removal opention i. p<rfoml.d in Ihe sam. f .. hion .. Ihe op<,alion of recording an ope,alion. After an operalion i. removed from Ih. Iill of incomplele operaliono of:ut execulive, ' he ntxl p.nding operalion ... ill be allow.d t.o perform il. acc ... 10 Ih. corr<sponding ap<x.

7 Conclusion We have propoud Ihree eoncu'Tenl d.>I • • 1 ... <1 "r .. roc impl.m.nlin~ priorily queu .. on .> di , lribu'o<i·

m.mory m .. nge p ... ,iog m,lItipro<e .. or wilh hypercub e "'"1"'1081. TI, . .. ,on"".,..", d~'a . tcudur .. can be uoed by any appli~~lion running on ,h. hypercuhe ... ilhoul worrying abo"l all Ih. n«e", ary ,ommun "OIions ~nd .ynehroninlions . PTionly queue open lion' . """ require O(log M) I,me t.o eom· plde, bUl,inco M operalion' may be initialed ,imu lla.eou , ly at diff".nl proc'",ON, O(MjIQgM) Ihroughpul is ""hi.vabl .... compued 100(1) Ih rou,hpnllor .. qu.,li.1 dala ,'.uelur.,.

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4\. M. R. Moybodi, 'Binary 50areh M .. h: A Coneu"...n! Dua Structur. for lIyp<rube,' Compuler &ien« TKhn;"al Repon, Ohio Univ ... ity, Alh.n>, Ohio, Jan. 1992.

42 . M. R. M.ybodi "Banyan Heap Machin.: Proceeding. of Suth Internalional PMal"'l Pro<_ing Sy".,pooium, Universily of Soulhern Coliforni., Lot Angel .. , CA, Mareh 1992.

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