Efficient Algorithms for AlltoAll Communications in Multiport MessagePassing Systems


 Joy Townsend
 2 years ago
 Views:
Transcription
1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER Efficiet Algoithms fo AlltoAll Commuicatios i Multipot MessagePassig Systems Jehoshua Buc, Seio Membe, IEEE, ChigTie Ho, Membe, IEEE, Shlomo Kipis, Membe, IEEE, Eli Upfal, Seio Membe, IEEE, a Deic Weathesby Abstact We peset efficiet algoithms fo two alltoall commuicatio opeatios i messagepassig systems: iex (o alltoall pesoalize commuicatio) a cocateatio (o alltoall boacast). We assume a moel of a fully coecte messagepassig system, i which the pefomace of ay poittopoit commuicatio is iepeet of the seeeceive pai. We also assume that each pocesso has pots, though which it ca se a eceive messages i evey commuicatio ou. The complexity measues we use ae iepeet of the paticula system topology a ae base o the commuicatio statup time, a o the commuicatio bawith. I the iex opeatio amog pocessos, iitially, each pocesso has blocs of ata, a the goal is to exchage the i th bloc of pocesso j with the j th bloc of pocesso i. We peset a class of iex algoithms that is esige fo all values of a that featues a taeoff betwee the commuicatio statup time a the ata tasfe time. This class of algoithms iclues two special cases: a algoithm that is optimal with espect to the measue of the statup time, a a algoithm that is optimal with espect to the measue of the ata tasfe time. We also peset expeimetal esults featuig the pefomace tueability of ou iex algoithms o the IBM SP paallel system. I the cocateatio opeatio, amog pocessos, iitially, each pocesso has oe bloc of ata, a the goal is to cocateate the blocs of ata fom the pocessos, a to mae the cocateatio esult ow to all the pocessos. We peset a cocateatio algoithm that is optimal, fo most values of, i the umbe of commuicatio ous a i the amout of ata tasfee. Iex Tems Alltoall boacast, alltoall pesoalize commuicatio, complete exchage, cocateatio opeatio, istibutememoy system, iex opeatio, messagepassig system, multiscatte/gathe, paallel system. INTRODUCTION C ollective commuicatio opeatios [] ae commuicatio opeatios that geeally ivolve moe tha two pocessos, as oppose to the poittopoit commuicatio betwee two pocessos. Examples of collective commuicatio opeatios iclue: (oetoall) boacast, scatte, gathe, iex (alltoall pesoalize commuicatio), a cocateatio (alltoall boacast). See [3], [6] fo a suvey of collective commuicatio algoithms o vaious etwos with vaious commuicatio moels. The ee fo collective commuicatio aises fequetly i paallel computatio. Collective commuicatio opeatios simplify the pogammig of applicatios fo paallel computes, facilitate the implemetatio of efficiet commuicatio schemes o vaious machies, pomote the potability of J. Buc is with the Califoia Istitute of Techology, Mail Coe 3693, Pasaea, CA C.T. Ho a E. Upfal ae with IBM Almae Reseach Cete, 65 Hay R., Sa Jose, CA {ho, S. Kipis is with News Datacom Reseach Lt., 4 Wegewoo St., Haifa 34635, Isael. D. Weathesby is with the Depatmet of Compute Sciece a Egieeig, Uivesity of Washigto, Seattle, WA Mauscipt eceive 6 Ap. 994; evise 7 Ap Fo ifomatio o obtaiig epits of this aticle, please se to: a efeece IEEECS Log Numbe 8. applicatios acoss iffeet achitectues, a eflect coceptual goupig of pocesses. I paticula, collective commuicatio is use extesively i may scietific applicatios fo which the iteleavig of stages of local computatio with stages of global commuicatio is possible (see []). This pape stuies the esig of alltoall commuicatio algoithms, amely, collective opeatios i which evey pocesso both ses ata to a eceives ata fom evey othe pocesso. I paticula, we focus o two wiely use opeatios: iex (o alltoall pesoalize commuicatio) a cocateatio (o alltoall boacast). The algoithms escibe hee ae icopoate ito the Collective Commuicatio Libay (CCL) [], which was esige a evelope fo the ew IBM lie of scalable paallel computes. The fist compute i this lie, the IBM 976 Scalable POWERpaallel System (SP), was aouce i Febuay Defiitios a Applicatios INDEX: The system cosists of pocessos p, p, º, p . Iitially, each pocesso p i has blocs of ata B[i, ], B[i, ], º, B[i,  ], whee evey bloc B[i, j] is of size b. The goal is to exchage bloc B[i, j] (the jth ata bloc of pocesso p i ) with bloc B[j, i] (the ith ata bloc of pocesso p j ), fo all i, j . The fial 4599/97/$. 997 IEEE
2 44 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 esult is that each pocesso p i, fo i , hols blocs B[, i], B[, i], º, B[ , i]. CONCATENATION: The system cosists of pocessos p, p, º, p . Iitially, each pocesso p i has a bloc of ata B[i] of size b. The goal is to mae the cocateatio of the ata blocs, amely, B[] B[] B[  ], ow to all the pocessos. Both the iex a cocateatio opeatios ae use extesively i istibutememoy paallel computes a ae iclue i the MessagePassig Iteface (MPI) staa poposal [4]. (The iex opeatio is efee to as MPI_Alltoall i MPI, while the cocateatio is efee to as MPI_Allgathe i MPI.) Fo example, the iex opeatio ca be use fo computig the taspose of a matix, whe the matix is patitioe ito blocs of ows (o colums) with iffeet blocs esiig o iffeet pocessos. Thus, the iex opeatio ca be use to suppot the emappig of aays i HPF compiles, such as emappig the ata layout of a twoimesioal aay fom (bloc, *) to (cyclic, *), o fom (bloc, *) to (*, bloc). The iex opeatio is also use i FFT algoithms [], i Asce a Desce algoithms [6], i the Alteatig Diectio Implicit (ADI) metho [], a i the solutio of Poisso s poblem by the Fouie Aalysis Cyclic Reuctio (FACR) metho [8], [3], o the twoimesioal FFT metho [8]. The cocateatio opeatio ca be use i matix multiplicatio [9] a i basic liea algeba opeatios [].. Commuicatio Moel We assume a moel of a multipot fully coecte messagepassig system. The assumptio of full coectivity meas that each pocesso ca commuicate iectly with ay othe pocesso a that evey pai of pocessos ae equally istat. The assumptio of multiple pots meas that, i evey commuicatio step (o ou), each pocesso ca se istict messages to pocessos a simultaeously eceive messages fom othe pocessos, fo some. Thoughout the pape, we assume , whee is the umbe of pocessos i the system. The multipot moel geealizes the oepot moel that has bee wiely ivestigate. Thee ae examples of paallel systems with pot capabilities fo >, such as the CUBE/, the CM (whee is the imesio of the hypecube i both machies), a tasputebase machies. Such a fully coecte moel aesses emegig tes i may moe istibutememoy paallel computes a messagepassig commuicatio eviomets. These tes ae eviet i systems such as IBM s Vulca [6], MIT s JMachie [], NCUBE s CUBE/ [5], Thiig Machies CM5 [9], a IBM s 976 Scalable POWERpaallel System, a i eviomets such as IBM EUI [], PICL [4], PARMACS [7], Zipcoe [7], a Expess [3]. These systems a eviomets geeally igoe the specific stuctue a topology of the commuicatio etwo a assume a fully coecte collectio of pocessos, i which each pocesso ca commuicate iectly with ay othe pocesso by seig a eceivig messages. The fact that this moel oes ot assume ay sigle topology maes it geeal a flexible. Fo istace, this moel allows the evelopmet of algoithms that ae potable betwee iffeet machies, that ca opeate withi abitay a yamic subsets of pocessos, a that ca opeate i the pesece of faults (assumig coectivity is maitaie). I aitio, algoithms evelope fo this moel ca also be helpful i esigig algoithms fo specific topologies. We use the liea moel [3] to estimate the commuicatio complexity of ou algoithms. I the liea moel, the time to se a mbyte message fom oe pocesso to aothe, without cogestio, ca be moele as T = b + mt, whee b is the ovehea (statup time) associate with each se o eceive opeatio, a t is the commuicatio time fo seig each aitioal byte (o ay appopiate ata uit). Fo coveiece, we efie the followig two tems i oe to estimate the time complexities of ou commuicatio algoithms i the liea moel: C : the umbe of commuicatio steps (o ous) equie by a algoithm. C is a impotat measue whe the commuicatio statup time is high, elative to the tasfe time, of oe uit of ata, a the message size pe se/eceive opeatio is elatively small. C : the amout of ata (i the appopiate uit of commuicatio: bytes, flits, o pacets) tasfee i a sequece. Specifically, let m i be the lagest size of a message (ove all pots of all pocessos) set i ou i. The, C is the sum of all the m i s ove all ous i. C is a impotat measue whe the statup time is small compae to the message size. Thus, i ou fully coecte, liea moel, a algoithm has a estimate commuicatio time complexity of T = C b + C t. It shoul be ote that thee ae moe etaile commuicatio moels, such as the BSP moel [3], the Postal moel [3], a the LogP moel [9], which futhe tae ito accout that a eceivig pocesso geeally completes its eceive opeatio late tha the coespoig seig pocesso fiishes its se opeatio. Howeve, esigig pactical a efficiet algoithms i these moels is substatially moe complicate. Aothe impotat issue is the uifomity of the implemetatio. Fo example, i the LogP moel, the esig of collective commuicatio algoithms is base o P, the umbe of pocessos. Optimal algoithms fo two istict values of P may be vey iffeet. This pesets a challege whe the goal is to suppot collective commuicatio algoithms fo pocesso goups with vaious sizes while usig oe collective commuicatio libay..3 Mai Cotibutios a Ogaizatio We stuy the complexity of the iex a cocateatio opeatios i the pot fully coecte messagepassig moel. We eive lowe bous a evelop algoithms fo these opeatios. The followig is a esciptio of ou mai esults: Lowe bous: Sectio povies lowe bous o the complexity measues C a C fo both the cocateatio a the iex opeatios.
3 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 45 Fo the cocateatio opeatio, we show that ay algoithm equies C log commuicatio + b a f ous a ses C uits of ata. Fo the iex opeatio, we show that ay algoithm equies C log commuicatio ous a ses C + b a f uits of ata. We also show that, whe is a powe of +, ay iex algoithm that uses the miimal umbe of commuicatio ous (i.e., C = log + ) must tasfe b C + log + uits of ata. Fially, we show that, i the oepot moel, if the umbe of commuicatio ous C is O(log ), the C must be W(b log ). Iex algoithms: Sectio 3 escibes a class of efficiet algoithms fo the iex opeatio amog pocessos. This class of algoithms is esige fo abitay values of a featues a taeoff betwee the statup time (measue C ) a the ata tasfe time (measue C ). Usig a paamete, whee, the commuicatio complexity  measues of the algoithms ae C = log  a C b log. Note that, followig ou lowe bou esults, optimal C a C caot be obtaie simultaeously. To icease the pefomace of the iex opeatio, the paamete ca be caefully chose as a fuctio of the statup time b, the ata tasfe ate t, the message size b, the umbe of pocessos, a the umbe of pots. Two special cases of this class ae of paticula iteest: Oe case exhibits the miimal umbe of commuicatio ous (i.e., C is miimize to log + by choosig = + ), a aothe case featues the miimal amout of ata tasfee (i.e., C is miimize to b  by choosig = ). The oepot vesio of the iex algoithm was implemete o the IBM s SP to cofim the existece of the taeoff betwee C a C. It shoul be ote that, whe is a powe of two, thee ae ow algoithms fo the iex opeatio which ae base o the stuctue of a hypecube (see [5], [], [8]). Howeve, oe of these algoithms ca be easily geealize to values of that ae ot powes of two without losig efficiecy. The iea of a taeoff betwee C a C is ot ew a has bee applie to hypecubes i [5], [8]. Cocateatio algoithms: Sectio 4 pesets algoithms fo the cocateatio opeatio i the pot moel. These algoithms ae optimal fo ay values of, b, a, except fo the followig age: b 3, 3, a ( + )  < < ( + ), fo some. (Thus, if b = o =, which coves most pactical cases, ou algoithm is optimal.) I this special age, we achieve eithe optimal C a suboptimal C (oe moe tha the lowe bou log + ), o optimal C a suboptimal b a f C (at most b  moe tha the lowe bou ). Pseuocoe: Appeices A a B povie pseuocoe fo the iex a cocateatio algoithms, espectively, i the oepot moel. Both the iex a cocateatio opeatios wee iclue i the Collective Commuicatio Libay [] of the Exteal Use Iteface (EUI) [] fo the 976 Scalable POWERpaallel System (SP) by IBM. I aitio, these oepot vesios of the algoithms have bee implemete o vaious aitioal softwae platfoms icluig PVM [5], a Expess [3]. LOWER BOUNDS This sectio povies lowe bous o the complexity measues C a C fo algoithms that pefom the cocateatio a iex opeatios. Popositio. was show i [3]. We iclue it hee fo completeess.. Lowe Bous fo the Cocateatio Opeatio PROPOSITION.. I the pot moel, fo, ay cocateatio algoithm equies C log + commuicatio ous. PROOF. Focus o oe paticula pocesso, say, pocesso p. The cocateatio opeatio equies, amog othe thigs, that the ata bloc B[] of pocesso p be boacast amog the pocessos. With commuicatio pots pe pocesso, ata bloc B[] ca each at most ( + ) pocessos i commuicatio ous. Fo ( + ) to be at least, we must have log+ commuicatio ous. PROPOSITION.. I the pot moel, fo, ay cocateatio algoithm tasfes C uits of b a f ata. PROOF. Each pocesso must eceive the  ata blocs of the othe  pocessos, the combie size of which is b(  ) uits of ata. Sice each pocesso ca use its iput pots simultaeously, the amout of ata tasfee though oe of the iput pots must be at least b a f.. Lowe Bous fo the Iex Opeatio PROPOSITION.3. I the pot moel, fo, ay iex algoithm equies C log commuicatio ous. + PROOF. Ay cocateatio opeatio o a aay B[i], i <, ca be euce to a iex opeatio o B[i, j], i, j <, by lettig B[i, j] = B[i] fo all i a j. Thus, the popositio follows fom Popositio.. PROPOSITION.4. I the pot moel, fo, ay iex algoithm tasfes C uits of b a f ata. PROOF. Simila to the poof of Popositio.3, the popositio follows fom Popositio..
4 46 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER Compou Lowe Bous fo the Iex Opeatio Hee, we povie aitioal lowe bous fo the iex opeatio. These lowe bous chaacteize the measue C as a fuctio of C a vice vesa. Theoems.5 a.7 show that whe C is optimize fist, the lowe bou o C becomes a oe of O(log + ) highe tha the staaloe lowe bou give i Popositio.4. The, Theoem.6 shows that whe C is optimize fist, the lowe bou o C becomes (  )/ as oppose to log +. Fially, Theoem.9 gives a moe geeal lowe bou fo the oepot case. THEOREM.5. If = ( + ), fo some itege, the ay iex algoithm that uses exactly C = log + commuicatio ous must tasfe at least C = b + log + uits of ata. PROOF. Let = ( + ). I oe to fiish the algoithm i exactly log + = ous, the umbe of pocessos havig eceive ata fom a give pocesso, say p i, must gow by a facto of + i evey ou. This efies a uique stuctue of the spaig tee T i, which is oote at p i, that is a geealize vesio of the biomial tee use to istibute the  ata blocs of pocesso p i amog the othe  pocessos. Deote by, j the umbe of pocessos at level j i tee T i oote at pocesso p i. Oe may use iuctio to show that l j j =ej j. Now, the total amout of ata D i that is ijecte ito the etwo ove the eges of the biomial tee T i oote at p i is give by Â D b j b j i lj j = = F H I K = Â j= j= j b, + whee the last equality step ca be eive by iffeetiatig both sies of j ÂF j H I = + K b g j= a the multiplyig both sies by b. Now, clealy, C  Di b b Â = = log i= THEOREM.6. Ay algoithm fo the iex opeatio that tasfes exactly C = uits of ata fom each pocesso b a f equies C  commuicatio ous. PROOF. I the iex opeatio, each pocesso has  ata blocs that it ees to se to the othe  pocessos. If each pocesso is allowe to tasfe at most b a f uits of ata pe pot ove all ous, the it must be the case that the jth ata bloc of pocesso p i is set iectly fom pocesso p i to pocesso p j. (That is, each ata bloc is set exactly oce fom its souce to its estiatio, a o pocesso ca fowa ata blocs of othe pocessos.) I this case, each pocesso must se  istict messages to the othe  pocessos. Ay such algoithm must equie C  ous. THEOREM.7. Ay iex algoithm that uses C = Èlog + commuicatio ous must tasfe at least b C = W log + + i uits of ata. PROOF. It is sufficiet to pove the theoem fo b =. Cosie ay algoithm that fiishes the iex opeatio i = C (miimum) ous. We show that the algoithm execute a total of W( log + ) ata tasmissios (ove all oes), thus, thee is a pot that tasmitte W log + + i uits of ata. We fist cocetate o the ata istibutio fom a give souce oe v to all othe  oes. Ay such algoithm ca be chaacteize by a sequece of + sets, S, S,, S, whee S i is the set of oes that have eceive thei espective ata by the e of commuicatio ou i. Thus, S = {v}, S =, a S i cotais S i, plus oes that eceive ata fom oes i S i i the ith commuicatio ous. Let x i = S i. Clealy, x i x i+ ( + )x i, because each oe i S i ca se ata to at most othe oes ue the pot moel. Next, we assig weights to the oes i the sets, S i s, whee the weight of a oe u i S i epesets the path legth (o the umbe of commuicatio ous icue) fom v to u i achievig the ata istibutio. The weights ca be assige base o the followig ule. If a oe u appeas fist i S i ue to a ata tasmissio fom oe w i S i, the the weight of u is the weight of w plus oe. Note that, oce a oe is assige a weight, it hols the same weight i all subsequet sets. By Lemma.8, we ow that thee ae at most j f ej f oes of weight f i S j. Ou goal is to give a lowe bou fo the sum of the weights of the oes i S. Without loss of geeality, we ca assume that the sum of the weights is the miimum possible. f Â ej b g Let X = f = +. By the choice of, f= X < b+ g.  Let Y f =. Sice, fo Â f= ej f f f,e j e fj, f  f  Y X  = + < + b g.
5 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 47 Thus, the algoithm must use all the possible oes with weights less tha. To bou the sum of the weights,we ee a lowe bou o Fo f , f  Z = Â ff Hf I K f. f= f ej is mooto i f. Thus, at least i. / of the oes have weight at least  That is, Z = W(). Summig ove all oigis, the total umbe of tasmissios is at least Z = W( ). Thus, at least oe pot has a sequece of C W W log + = F H G I = KJ F HG + ata tasmissios. j f LEMMA.8. Thee ae o moe tha ej f oes of weight f i S j (efie i the poof of Theoem.7). PROOF. We pove by iuctio o j. Thee is clealy o moe tha oe oe of weight zeo a oes of weight oe i S. Assume that the hypothesis hols fo j . j f Note that S j cotais up to e f j oes of weight f that appeae with the same weight i S j, plus up to j f efj oes that eceive ata at commuicatio ou j fom oes with weight f  i S j. The claim hols fo j sice F H I j  j + f K Hf K F I I KJ j f f f f = F H I K. THEOREM.9. Whe =, ay algoithm fo the iex opeatio that uses C = O(log ) commuicatio ous must tasfe C = W(b log ) uits of ata. PROOF. Assume that thee is a algoithm with C c log fo some costat c. Cosie the biomial istibutio e j j. Let h be the miimal,, such that clog l+ clog e j j. Oe ca show that ay algoithm Âj= that fiishes i c log ous must have the followig popety. Fo evey j such that j h, thee exist clog e j j messages fom each oe that tavel at least j hops i the etwo. Notice that, i this popety, each message ca oly be coute oce fo a give j. Theefoe, the aveage umbe of hops a message has to tavel fo each oe is h/, if h log, o log /, if h log. Sice h must be W(log ) fom Lemma C. i Appeix C, we have C = W(b log ). 3 INDEX ALGORITHMS This sectio pesets a class of efficiet algoithms fo the iex opeatio. Fist, we povie a oveview of the algoithms. The, we focus o the commuicatio phase of the algoithms fo the oepot moel. Next, we escibe two special cases of this class of algoithms. The, we geealize the algoithms to the pot moel. A fially, we commet o the implemetatio a pefomace of this class of algoithms. 3. Oveview The class of algoithms fo the iex opeatio amog pocessos ca be epesete as a sequece of pocessomemoy cofiguatios. Each pocessomemoy cofiguatio has colums of blocs each. Colums ae labele fom though  (fom left to ight i the figues) a blocs ae labele fom though  (fom top to bottom i the figues). Colum i epesets pocesso p i, a bloc j epesets the jth ata bloc i the memoy offset. The objective of the iex opeatio, the, is to taspose these colums of blocs. Fig. shows a example of the pocessomemoy cofiguatios befoe a afte the iex opeatio fo = 5 pocessos. The otatio ij i each box epesets the jth ata bloc iitially allocate to pocesso p i. The label j is efee to as the bloci. All the algoithms i the class cosist of thee phases. Phases a 3 equie oly local ata eaagemet o each pocesso, while Phase ivolves itepocesso commuicatio. PHASE. Each pocesso p i iepeetly otates its ata blocs i steps upwas i a cyclical mae. PHASE. Each pocesso p i otates its jth ata bloc j steps to the ight i a cyclical mae. This otatio is im Fig.. Memoypocesso cofiguatios befoe a afte a iex opeatio o five pocessos.
6 48 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 Fig.. A example of memoypocesso cofiguatios fo the thee phases of the iex opeatio o five pocessos. plemete by itepocesso commuicatio. PHASE 3. Each pocesso p i iepeetly otates its ata blocs i steps owwas i a cyclical mae. Fig. pesets a example of these thee phases of the algoithm fo pefomig a iex opeatio amog = 5 pocessos. The implemetatio of Phases a 3 o each pocesso ivolves oly local ata movemets a is staightfowa. I the sequel, we focus oly o the implemetatio of Phase. Diffeet algoithms ae eive epeig o how the commuicatio patte of Phase is ecompose ito a sequece of poittopoit commuicatio ous. 3. The Itepocesso Commuicatio Phase We peset the ecompositio of Phase ito a sequece of poittopoit commuicatio ous, assumig the oepot moel a usig a paamete (fo aix) i the age. Fo coveiece, we say that the bloci of the jth ata bloc i each pocesso afte Phase is j. Cosie the otatio equie i Phase. Each bloc with a bloci j i pocesso i ees to be otate to pocesso (i + j) mo. The bloci j, whee j , ca be ecoe usig aix epesetatio usig w = log igits. Fo coveiece, we efe to these w igits fom zeo though w  statig with the least sigificat igit. Ou algoithm fo Phase cosists of w subphases coespoig to the w igits. Each subphase cosists of at most  steps, coespoig to the (up to)  iffeet ozeo values of a give igit. I subphase x, fo x w , we iteate Step though Step , as follows: Duig Step z of subphase x, whee z  a x w , all ata blocs, fo which the xth igit of thei bloci is z, ae otate z x steps to the ight. This is accomplishe i a commuicatio ou by a iect poittopoit commuicatios betwee pocesso i a pocesso (i + z x ) mo, fo each i . Fo example, whe is chose to be 3, the fifth bloc will be otate two steps to the ight uig Step of Subphase, a late otate agai thee steps to the ight uig Step of Subphase. This follows fom the fact that 5 is ecoe ito usig aix3 epesetatio. Note that, afte w subphases, all ata blocs have bee otate to the coect estiatio pocesso as specifie by the pocesso i. Howeve, ata blocs ae ot ecessaily i thei coect memoy locatios. Phase 3 of the algoithm fixes this poblem. The followig poits ae mae egaig the pefomace of this algoithm. Each step ca be ealize by a sigle commuicatio ou by pacig all the outgoig blocs to the same estiatio ito a tempoay aay a seig them togethe i oe message. Hece, each subphase ca be ealize i at most  commuicatio ous. The size of each message ivolve i a commuicatio ou is at most b ata. Hece, the class of the iex algoithms has complexity measues C  a f log a a f log, C b  whee is chose i the age. 3.3 Two Special Cases The class of algoithms fo the iex opeatio i the oepot moel cotais two iteestig special cases: ) Whe =, the eive algoithm equies
7 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 49 Fig. 3. A example of memoypocesso cofiguatios fo the iex algoithm o five pocessos, which has a optimal C measue. C = log commuicatio ous, which is optimal with espect to the measue C. Also, i this case, C b log, which is optimal (to withi a multiplicative facto) fo the case whe C = log. Fig. 3 shows such a example with = a = 5. The shae ata blocs ae the oes subject to otatio uig the ext subphase. ) Whe =, the eive algoithm tasfes C = b(  ) uits of ata fom each oe, which is optimal with espect to the measue C. The value of C i this case is C = , which is optimal fo the case whe C = b(  ). Hece, = shoul be chose whe the statup time of the uelyig machie is elatively sigificat, a the pouct of the bloc size b a the peelemet tasfe time is elatively small. O the othe ha, = shoul be chose whe the statup time is egligible. I geeal, ca be fietue accoig to the paametes of the uelyig machies to balace betwee the statup time a the ata tasfe time. 3.4 Geealizatio to the Pot Moel We ow peset a moificatio to the iex algoithm above fo the pot moel. Phase a Phase 3 of the algoithm emai the same. I Phase, we still have w = log subphases as befoe, coespoig to the w igits i aix epesetatio of ay bloci j, whee j . I each subphase, thee ae, at most,  iepeet poittopoit commuicatio steps that ee to be pefome. Sice these poittopoit commuicatio steps ae iepeet, they ca be pefome i paallel, subject to the costait o the umbe of paallel iput/output pots. Thus, evey of these commuicatio steps ca be goupe togethe a pefome cocuetly. Theefoe, each subphase cosists of at most  commuicatio steps. The complexity measues fo the iex algo ithm ue the pot moel, theefoe, ae   C log a C b log, whee ca be chose i the age. To miimize both C a C, oe clealy ees to choose, such that (  ) mo =. 3.5 Implemetatio We have implemete the oepot vesio ( = ) of the iex algoithm o a IBM SP paallel system. (The IBM SP is close to the oepot moel i the omai of the multipot moel.) The implemetatio is oe o top of the poittopoit messagepassig exteal use iteface (EUI), uig o the EUIH eviomet. At this level, the commuicatio statup, b, measues about 9 msec, a the
8 5 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 Fig. 6. The measue times of the iex algoithm as a fuctio of aix fo vaious message sizes o a 64 oe SP. Fig. 4. The measue time of the iex algoithm as a fuctio of message sizes o a 64 oe SP. sustaie poittopoit commuicatio bawith is about 8.5 Mbytes/sec, i.e., t <. msec/byte. Fig. 4 shows the measue times of the iex algoithm as a fuctio of message size with vaious poweoftwo aices o a 64 oe SP. As ca be see, the smalle aix tes to pefom bette fo smalle message sizes, a vice vesa. Fig. 5 compaes the measue times of the iex algoithm with =, = = 64, a optimal amog all poweoftwo aices, espectively, o a 64 oe SP. The beaeve poit of the message size betwee the two special cases of the iex algoithms (i.e., = a = ) occus at about to bytes. The iex algoithm with optimal poweoftwo aix, as expecte, is the best oveall choice. Fig. 6 shows the measue times of the iex algoithm as a fuctio of aix fo thee iffeet message sizes: 3 bytes, 64 bytes, a 8 bytes. As the message size iceases, the miimal time of the cuve tes to occu at a highe aix. Whe compaig these measue times with ou peicte times base o the liea moel, we fi big iscepacies quatitatively, but elatively cosistet qualitatively. Note that we ae maily iteeste i the qualitatively behavio of the iex algoithm o a geeal messagepassig system. We believe the quatitative iffeeces betwee the measue times a the peicte times ae ue to the followig factos: ) Thee ae vaious system outies uig i the bacgou that have a highe pioity tha the use pocesses. ) We o ot moel the copy time icue by the fuctio copy, pac, a upac (see the pseuocoe i Appeix A). 3) We o ot moel the cogestio behavio of the SP. 4) Thee is a slowow facto, somewhee betwee oe a two, fom the liea moel to the se_a_eceive moel. If we moel the cogestio behavio as a fixe multiplicative facto of t c a assume the system outies have a fixe slowow facto of the oveall time, the the total time fo the iex opeatio ca be moele as T = g C t s + g C t c + g 3. Fig. 5. The measue times of the iex algoithm with =, = = 64, a optimal amog all poweoftwo aices, espectively, o a 64 oe SP. 4 CONCATENATION ALGORITHMS Thee ae two ow algoithms fo the cocateatio opeatio i the oepot moel. The fist is a simple folloe algoithm which cosists of two phases. I the fist phase, the blocs of ata fom the pocessos ae accumulate to a esigate pocesso, say pocesso p. This ca be oe usig a biomial tee (o a subtee of it whe is ot a powe of two). I the seco phase, the cocateatio esult fom pocesso p is boacast to the pocessos usig the same biomial tee. This algoithm is ot optimal sice it cosists of C = log commuicatio ous a
9 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 5 tasfes C = b(  ) uits of ata. The seco ow cocateatio algoithm is fo the case whe is a powe of two a = (see []). This algoithm is base o the stuctue of a biay hypecube a is optimal i both C a C. Fo a give, this algoithm ca be geealize to the case whee is a powe of + by usig the stuctue of a geealize hypecube [4]. Howeve, fo geeal values of, we o ot ow of ay existig cocateatio algoithm that is optimal i both C a i C, eve whe b = =. I this sectio, we peset efficiet cocateatio algoithms fo the pot commuicatio moel that, i most cases of a, ae optimal i both C a C. Thoughout this sectio, we assume that is i the age . Notice that, fo , the tivial algoithm that taes a sigle ou is optimal. The mai stuctue that we use fo eivig the algoithms is that of ciculat gaphs. We ote hee that ciculat gaphs ae also useful i costuctig faulttoleat etwos [7]. DEFINITION. A ciculat gaph G(, S) is chaacteize by two paametes: the umbe of oes, a a set of offsets S. I G(, S), the oes ae labele fom though , a each oe i is coecte to oe ((i  s) mo ) a to oe ((i+s) mo ) fo all s Œ S (see []). The cocateatio algoithm cosists of ous. Let eges {(, ), (, ),, (, )}.) I geeal, i ou, whee , we a eges with offsets i S to the cuet patial spaig tee to fom a ew lage patial spaig tee. It is easy to veify that, afte  ous, the esultig tee spas the fist oes statig fom oe, amely, oes though . Fig. 7 illustates the pocess of costuctig T fo the case of = a = 9. Next, we use tee T to costuct the spaig tees T i, fo i . We o this by taslatig each oe j i T to oe (j + i) mo i T i. Also, the ou i associate with each tee ege i T i (which epesets the ou uig which the coespoig commuicatio is pefome) is the same as that of the coespoig tee ege i T. Fig. 8 illustates tee T fo the case of = a = 9. It is easy to see that T was obtaie fom T by aig oe (moulo ie) to the labels of the oes i T. log, that is, ( + )  < ( + ). Also let = = + +, whee = ( + )  a. The ous of the algoithm ca be ivie ito two phases. The fist phase cosists of  ous, at the e of which evey oe has the cocateatio esult of the  oes that pecee it i the umbeig (i a ciculat sese). The seco phase cosists of a sigle ou a completes the cocateatio opeatio amog the oes. Fig. 7. The two ous i costuctig the spaig tee oote at oe fo = 9 a =. 4. The Fist  Rous Fo the fist  ous, we use a ciculat gaph G(, S), whee S = S < S < < S , S i = {( + ) i, ( + ) i, º, ( + ) i }. We ietify the pocessos with the oes of G(, S), which ae labele fom though . The commuicatio patte elate to boacastig the ata item of each oe ca be escibe by a spaig tee. Let T i eote the spaig tee associate with the ata item B[i] of oe i (amely, T i is oote at oe i). We escibe the spaig tee associate with each oe by specifyig the eges that ae use i evey commuicatio ou. The eges associate with ou i ae calle ouieges. Fist, we escibe the tee T, a the we show how tee T i, fo i , ca be eive fom tee T. We stat with a iitial tee T which cosists oly of oe. I ou, we a eges with offsets i S to T to fom a patial spaig tee; the ae eges ae the oueges. (That is, i ou, we a the set of Fig. 8. The two ous i costuctig the spaig tee oote at oe fo = 9 a =. They ca be eive by taslatig oe aesses of the spaig tee oote at oe i Fig. 7. The cocateatio algoithm i each oe is specifie by the tees T i, fo i , as follows: I ou i, fo i , o: Fo all j , if ata item B[j] is peset at the oe, the se it o all ouieges of tee T j. Receive the coespoig ata items o the ouieges of all the tees. THEOREM 4.. Afte  ous of the above algoithm, evey oe i, fo i , has the ata items B[j], whee i j i  + j + (mo ). Also, uig these  ous, the measue C is optimal:
10 5 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 b C = c h. PROOF. The spaig tees T i, fo T i , ae eive fom T by shiftig the iices i a cyclic mae. Hece, it suffices to focus o the spaig tee T. Notice that the algoithm ca be implemete i a pot moel, sice, i evey ou i, we use oly the set of offsets S i, which cosists of offsets. Also, the tee T is a spaig tee fo the oes p i, whee i , because evey i i this age ca be epesete usig a set of istict offsets fom S. Hece, afte  ous of the algoithm, the ata items ae istibute accoig to the claim of the theoem. Next, we ee to pove that C associate with the  ous is as claime. By iuctio o i, it follows that, befoe ou i, ay oe has at most ( + ) i istict ata items. Hece, i ou i, ay oe ses at most ( + ) i ata items o ay give ege. Thus, b g b g b g c h  b C b L + + = . Howeve, by the lowe bou agumet, we have b C c h, a the claim follows. 4. The Last Rou Befoe ou , the last ou of the algoithm, we have the followig situatio: Evey oe i ha boacast its message to the  oes succeeig it i the cicula gaph a ha eceive the boacast message fom the  oes peceig it i the cicula gaph. Cosie tee T just befoe the last ou. The fist oes (oes though  ) ae iclue i the cuet tee, a the emaiig oes still ee to be spae. We big the followig popositio. PROPOSITION 4.. The last ou ca be pefome with C b =, fo ay combiatio of, b, a, except fo the followig age: b 3, 3, a ( + )  < < ( + ), fo some. The poof of this popositio is somewhat complicate, a we oly give the mai ieas hee. The basic iea is to tasfom the scheulig poblem fo the last ou of the algoithm ito a table patitioig poblem. (I the sese that, if the table patitioig poblem ca be solve, the we have a optimal algoithm by eivig a optimal scheule fo the last ou.) The table patitioig poblem is efie as follows. Let a = b. Give a table of b ows a colums, we woul lie to patitio the table ito isjoit aeas, eote by A, A,, A, such that the columspa of A i, fo all i, is at most, whee the columspa of A i is efie as R i  L i + if R i a L i ae the ightmost a leftmost colums, espectively, touche by A i ; a the umbe of table eties i A i, fo all i, is at most a. If a solutio ca be fou to the tablepatitioig poblem, the a scheule fo the last ou ca be eive as follows. Each of the table colums coespos to oe of the oes yet to be spae, a each of the b table ows epesets oe byte. Table elemets i the same aea, say A i, will use the same offset, which is etemie by the iex of the leftmost colum touche by A i. It ca be show that a staightfowa algoithm fo patitioig the table satisfies the above two coitios fo ay combiatio of, b, a, except fo the followig age: b 3, 3 a ( + )  < < ( + ), fo some. Fo istace, Table pesets a patitioig example fo = 3, = 7, b = 3, a = 3, which fall i the optimal age of. The aea covee by A i is mae by the umbe i. Fom this table, oe ca eive the followig scheulig fo the last ou: The sum of the weighte eges with offset 3 (i aea A ) is 7. Thus, oe p 3 eceives thee bytes fom p, oe p 4 eceives thee bytes fom p, a oe p 5 eceives oe byte fom p. The sum of the weighte eges with offset 5 (i aea A ) is 7. Thus, oe p 5 eceives two bytes fom p, oe p 6 eceives thee bytes fom p, a oe p 7 eceives two bytes fom p 3. The sum of the weighte eges with offset 7 (i aea A 3 ) is 7. Thus, oe p 7 eceives oe byte fom p, oe p 8 eceives thee bytes fom p, a oe p 9 eceives thee bytes fom p. Afte otatio, to geeate spaig tees, each of which is oote at a iffeet oe, each oe i ees to se seve bytes to oes (i + 3) mo, (i + 5) mo, a (i + 7) mo, a eceive seve bytes fom oes (i  3) mo, (i  5) mo, a (i  7) mo. THEOREM 4.3. The above cocateatio algoithm attais optimal C log a C = + = a f b fo ay combiatio of, b, a, except fo the followig age: b 3, 3, a ( + )  < < ( + ), fo some itege. PROOF. By combiig Theoem 4. a Popositio 4., we b b b have C =  + =, which matches c h a f the lowe bou of C i Popositio.. Fig. 9 pesets a example of the cocateatio algoithm fo = a = 5. Note that, to simplify the pseuocoe iclue i Appeix A, we actually gow the spaig tee T i usig egative offsets. That is, i both the figue TABLE AN EXAMPLE OF THE TRANSFORMED PROBLEM FOR = 3 (p THROUGH p ), = 7 (p 3 THROUGH p 9 ), b = 3 (BYTES), AND = 3 (PORTS)
11 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 53 Fig. 9. A example of the oepot cocateatio algoithm with five pocessos. a i the pseuocoe, leftotatios ae pefome istea of ightotatios. REMARK. Fo the ooptimal age of, it is easy to achieve optimal C at the expese of iceasig C by oe ou ove the lowe bou. It is also easy to achieve optimal C a suboptimal C, whee C is at most b  moe tha the lowe bou. APPENDIX A PSEUDOCODE FOR THE INDEX ALGORITHM This appeix pesets pseuocoe fo the iex algoithm of Sectio 3 whe =. This pseuocoe setches the implemetatio of the iex opeatio i the Collective Commuicatio Libay of the EUI [] by IBM. I the pseuocoe, the fuctio iex taes six agumets: outmsg is a aay fo the outgoig message; blle is the legth i bytes of each ata bloc; imsg is a aay fo the icomig message; is the umbe of pocessos ivolve; A is the aay of the iffeet pocesso is, such that, A[i] = p i ; a is the aix use to tue the algoithm. Aays outmsg a imsg ae each of legth blle * bytes. Othe outies that appea i the coe ae as follows: Routie copy(a, B, le) copies aay A of size le bytes ito aay B. Routie geta(i,, A) etus the iex i that satisfies A[i] = i. The outie mo(x, y) etus the value x mo y i the age of though y , eve fo egative x. The fuctio se_a_ecv taes six agumets: the outgoig message; the size of the outgoig message; the estiatio of the outgoig message; the icomig message; the size of the icomig message; a the souce of the icomig message. The fuctio se_a_ecv is suppote by IBM s Message Passig Libay (MPL) [] o SP a SP, a the ecet MPI staa [4]. It ca also be implemete as a combiatio of blocig se a oblocig eceive. I the followig pseuocoe, lies 3 a 4 coespo to Phase, lies 5 though coespo to Phase, a lies though 3 coespo to Phase 3. I Phase, thee ae w subphases, which ae iexe by i. Duig each subphase, each pocesso ees to pefom the se_a_ecv opeatio  times, except fo the last subphase, whee each pocesso pefoms the se_a_ecv opeatio oly w  times. Lies 7 though tae ito accout the special case fo the last subphase. The outie pac is use to pac those blocs that ee to be otate to the same itemeiate estiatio ito a cosecutive aay. Specifically, pac(a, B, blle,,, i, j, blocs) pacs some selecte blocs of aay A ito aay B; each bloc is of size blle i bytes; those blocs, fo which the ith igit of the aix epesetatio of thei bloc is ae equal to j, ae selecte fo pacig; a value of the umbe of selecte blocs is witte to the agumet blocs. The outie upac(a, B, blle,,, i, j, blocs) is efie as the ivese fuctio of
12 54 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 pac whee B becomes the iput aay to be upace a A becomes the output aay. Fuctio iex (outmsg, blle, imsg,, A, ) () w = Èlog () my_a = geta (my_pi,, A) (3) copy (outmsg, tmp[(  my_a) * blle], my_a * blle) (4) copy (outmsg [my_a * blle], tmp, (  my_a) * blle) (5) ist = (6) fo i = to w  o (7) if (i == w  ) the (8) h = ist (9) else () h = () eif () fo j = to h  o (3) est_a = mo (my_a + j * ist, ) (4) sc_a = mo (my_a  j * ist, ) (5) pac (tmp, pace_msg, blle,,, i, j, blocs) (6) se_a_ecv (pace_msg, blle * blocs, A [est_a], pace_msg, blle * blocs, A [sc_a]) (7) upac (tmp, pace_msg, blle,,, i, j, blocs) (8) efo (9) ist = ist * () efo () fo i = to  o () copy (tmp [mo (my_a  i, ) * blle], imsg [i * blle], blle) (3) efo (4) etu APPENDIX B PSEUDOCODE FOR THE CONCATENATION ALGORITHM This appeix pesets pseuocoe fo the cocateatio algoithm of Sectio 4 whe =. This pseuocoe setches the implemetatio of the cocateatio opeatio i the Collective Commuicatio Libay of the EUI [] by IBM. I this pseuocoe, the fuctio cocat taes five agumets: outmsg is a aay fo the outgoig message; le is the legth i bytes of aay outmsg; imsg is a aay fo the icomig message; is the umbe of pocessos ivolve; a A is the aay of the iffeet pocesso is, such that, A[i] = p i. Aay imsg is of legth le * bytes. The fuctio cocat ses a eceives messages usig the se_a_ecv outie. The outies copy, geta, se_a_ecv, a mo wee efie i Appeix A. I the followig pseuocoe, each pocesso fist iitializes some vaiables a copies its outmsg aay ito a tempoay aay temp (lies though 5). The, each pocesso pefoms the fist  ous of the algoithm (lies 6 though ). The, each pocesso pefoms the last ou of the algoithm (lies 3 a 6). Fially, each pocesso pefoms a local cicula shift of the ata such that all ata blocs i its imsg aay begi with the bloc B[] (lies 7 a 8). Fuctio cocat (outmsg, le, imsg,, A) () = Èlog () my_a = geta (my_pi,, A) (3) copy (outmsg, temp, le) (4) bl = (5) cuet_le = le (6) fo = to  o (7) est_a = mo (my_a  bl, ) (8) sc_a = mo (my_a + bl, ) (9) se_a_ecv (temp, cuet_le, A [est_a],temp [cuet_le], cuet_le, A [sc_a]) () bl = bl * () cuet_le = cuet_le * () efo (3) cuet_le = le * (  bl) (4) est_a = mo (my_a  bl, ) (5) sc_a = mo (my_a + bl, ) (6) se_a_ecv (temp, cuet_le, A [est_a], temp [cuet_le], cuet_le, A [sc_a]) (7) copy (temp, imsg [le * my_a], le * (  my_a)) (8) copy (temp [le * (  my_a)], imsg, le * my_a) (9) etu APPENDIX C PROOF OF A LEMMA LEMMA C.. Let c a m be iteges such that c m. The, if h cm m Â e j j j, the h mi(m/64, m/8 log c). = PROOF. Assume, fo the sae of cotaictio, that the lemma oes ot hol. Fist, ote that the lemma hols if h m/64, so it must be the case that h < m/64. Also, cm m ote that e j = <, so h a m > 64. Theefoe, h + m cm. Because h < m/64 cm/8, the tems h i the summatio cm m Â e j j j ae mootoically = iceasig, so h Â b g b ga f! m cm cm h F I H j K h + F I H j K h + cm h j= Note that h! h h+/ /e h, so m log (h + ) + h log (cm) + h log e  (h + /) log h (h + ) log (cm) + h log e  h log h. Because h < m/64 m/( log e), m/ h (log (cm)  log h) + log (cm). Because log (cm) log m m/4, it follows that m/4 h (log (cm)  log h). Let h = m/x a ote that x > 64, so
13 BRUCK ET AL.: EFFICIENT ALGORIITHMS FOR ALLTOALL COMMUNICATIONS IN MULTIPORT MESSAGEPASSING SYSTEMS 55 m/4 (m/x)(log c + log x), which implies that x 4 log c + 4 log x a x  4 log x 4 log c. Note that x 8log x, so x/ x  4 log x 4 log c. Theefoe, x 8log c a h = m/x m/8 log c, which is a cotaictio. ACKNOWLEDGMENTS We tha Robet Cyphe fo his help i eivig Lemma C.. Jehoshua Buc was suppote i pat by U.S. Natioal Sciece Fouatio Youg Ivestigato Awa CCR , by the Sloa Reseach Fellowship, a by DARPA a BMDO though a ageemet with NASA/OSAT. REFERENCES [] V. Bala, J. Buc, R. Byat, R. Cyphe, P. ejog, P. Elustoo, D. Fye, A. Ho, C.T. Ho, G. Iwi, S. Kipis, R. Lawece, a M. Si, The IBM Exteal Use Iteface fo Scalable Paallel Systems, Paallel Computig, vol., o. 4, pp , Ap [] V. Bala, J. Buc, R. Cyphe, P. Elustoo, A. Ho, C.T. Ho, S. Kipis, a M. Si, CCL: A Potable a Tuable Collective Commuicatio Libay fo Scalable Paallel Computes, IEEE Tas. Paallel a Distibute Systems, vol. 6, o., pp , Feb [3] A. BaNoy a S. Kipis, Desigig Boacastig Algoithms i the Postal Moel fo MessagePassig Systems, Mathematical Systems Theoy, vol. 7, o. 5, pp , Sept./Oct [4] L. Bhuya a D. Agawal, Geealize Hypecube a Hypebus Stuctues fo a Compute Netwo, IEEE Tas. Computes, vol. 33, o. 4, pp , Ap [5] S. Bohai, Multiphase Complete Exchage o a Cicuit Switche Hypecube, Poc. 99 It l Cof. Paallel Pocessig, vol. I, pp , Aug. 99. [6] J. Buc, R. Cyphe, L. Gavao, A. Ho, C.T. Ho, S. Kipis, S. Kostatiiou, M. Si, a E. Upfal, Suvey of Routig Issues fo the Vulca Paallel Compute, IBM Reseach Repot, RJ8839, Jue 99. [7] J. Buc, R. Cyphe, a C.T. Ho, FaultToleat Meshes a Hypecubes with Miimal Numbes of Spaes, IEEE Tas. Computes, vol. 4, o. 9, pp.,89,4, Sept [8] C.Y. Chu, Compaiso of Twoimesioal FFT Methos o the Hypecubes, Poc. Thi Cof. Hypecube Cocuet Computes a Applicatios, pp.,43,437, 988. [9] D. Culle, R. Kap, D. Patteso, A. Sahay, K.E. Schause, E. Satos, R. Subamoia, a T. vo Eice, LogP: Towas a Realistic Moel of Paallel Computatio, Poc. Fouth SIGPLAN Symp. Piciples a Pactices Paallel Pogammig, ACM, May 993. [] W.J. Dally, A. Chie, S. Fise, W. Howat, J. Kee, M. Laivee, R. Lethi, P. Nuth, S. Wills, P. Caic, a G. Fyle, The JMachie: a FieGai Cocuet Compute, Poc. Ifomatio Pocessig 89, pp.,47,53, 989. [] B. Elspas a J. Tue, Gaphs with Ciculat Ajacecy Matices, J. Combiatoial Theoy, o. 9, pp , 97. [] G. Fox, M. Johsso, G. Lyzega, S. Otto, J. Salmo, a D. Wale, Solvig Poblems o Cocuet Pocessos, Vol. I. Petice Hall, 988. [3] P. Faigiau a E. Laza, Methos a Poblems of Commuicatio i Usual Netwos, Discete Applie Math., vol. 53, pp , 994. [4] G.A. Geist, M.T. Heath, B.W. Peyto, a P.H. Woley, A Use s Guie to PICL: A Potable Istumete Commuicatio Libay, ORNL Techical Repot o. ORNL/TM66, Oct. 99. [5] G.A. Geist a V.S. Sueam, Netwo Base Cocuet Computig o the PVM System, ORNL Techical Repot o. ORNL/TM76, Jue 99. [6] S.M. Heetiemi, S.T. Heetiemi, a A.L. Liestma, A Suvey of Gossipig a Boacastig i Commuicatio Netwos, Netwos, vol. 8, pp , 988. [7] R. Hempel, The ANL/GMD Macos (PARMACS) i FORTRAN fo Potable Paallel Pogammig Usig the Message Passig Pogammig Moel, Use s Guie a Refeece Maual, techical memoaum, Gesellschaft fü Mathemati u Dateveabeitug mbh, West Gemay. [8] C.T. Ho a M.T. Raghuath, Efficiet Commuicatio Pimitives o Hypecubes, Cocuecy: Pactice a Expeiece, vol. 4, o. 6, pp , Sept. 99. [9] S.L Johsso a C.T. Ho, Matix Multiplicatio o Boolea Cubes Usig Geeic Commuicatio Pimitives, Paallel Pocessig a MeiumScale Multipocessos, A. Wou, e., pp SIAM, 989. [] S.L. Johsso a C.T. Ho, Spaig Gaphs fo Optimum Boacastig a Pesoalize Commuicatio i Hypecubes, IEEE Tas. Computes, vol. 38, o. 9, pp.,49,68, Sept [] S.L. Johsso a C.T. Ho, Optimizig Tiiagoal Solves fo Alteatig Diectio Methos o Boolea Cube Multipocessos, SIAM J. Scietific a Statistical Computig, vol., o. 3, pp , 99. [] S.L. Johsso, C.T. Ho, M. Jacquemi, a A. Ruttebeg, Computig Fast Fouie Tasfoms o Boolea Cubes a Relate Netwos, Avace Algoithms a Achitectues fo Sigal Pocessig II, vol. 86, pp Soc. PhotoOptical Istumetatio Egiees, 987. [3] O.A. McBya a E.F. Va e Vele, Hypecube Algoithms a Implemetatios, SIAM J. Scietific a Statistical Computig, vol. 8, o., pp. 7 87, Ma [4] Message Passig Iteface Foum, MPI: A MessagePassig Iteface Staa, May 994. [5] J.F. Palme The NCUBE Family of Paallel Supecomputes, Poc. It l Cof. Compute Desig, 986. [6] F.P. Pepaata a J.E. Vuillemi, The Cube Coecte Cycles: A Vesatile Netwo fo Paallel Computatio, Comm. ACM, vol. 4, o. 5, pp. 3 39, May 98. [7] A. Sjellum a A.P. Leug, Zipcoe: A Potable Multicompute Commuicatio Libay Atop the Reactive Keel, Poc. Fifth Distibute Memoy Computig Cof., pp , Ap. 99. [8] P.N. Swaztaube, The Methos of Cyclic Reuctio, Fouie Aalysis, a the FACR Algoithm fo the Discete Solutio of Poisso s Equatio o a Rectagle, SIAM Rev., vol. 9, pp. 49 5, 977. [9] Coectio Machie CM5 Techical Summay. Thiig Machies Copoatio, 99. [3] L.G. Valiat, A Bigig Moel fo Paallel Computatio, Comm. ACM, vol. 33, o. 8, pp. 3, Aug. 99. [3] Expess 3. Itouctoy Guie. Paasoft Copoatio, 99. Jehoshua Buc eceive the BSc a MSc egees i electical egieeig fom the Techio, Isael Istitute of Techology, i 98 a 985, espectively, a the PhD egee i electical egieeig fom Stafo Uivesity i 989. He is a associate pofesso of computatio a eual systems a electical egieeig at the Califoia Istitute of Techology. His eseach iteests iclue paallel a istibute computig, faulttoleat computig, eocoectig coes, computatio theoy, a eual a biological systems. D. Buc has extesive iustial expeiece icluig, sevig as maage of the Fouatios of Massively Paallel Computig Goup at the IBM Almae Reseach Cete fom , a eseach staff membe at the IBM Almae Reseach Cete fom , a a eseache at the IBM Haifa Sciece cete fom D. Buc is the ecipiet of a 995 Sloa Reseach Fellowship, a 994 Natioal Sciece Fouatio Youg Ivestigato Awa, six IBM Plateau Ivetio Achievemet Awas, a 99 IBM Outstaig Iovatio Awa fo his wo o Hamoic Aalysis of Neual Netwos, a a 994 IBM Outstaig Techical Achievemet Awa fo his cotibutios to the esig a implemetatio of the SP, the fist IBM scalable paallel compute. He has publishe moe tha joual a cofeece papes i his aeas of iteests a he hols patets. D. Buc is a seio membe of the IEEE a a membe of the eitoial boa of the IEEE Tasactios o Paallel a Distibute Systems.
14 56 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 ChigTie Ho eceive a BS egee i electical egieeig fom Natioal Taiwa Uivesity i 979 a the MS, MPhil, a PhD egees i compute sciece fom Yale Uivesity i 985, 986, a 99, espectively. He joie IBM Almae Reseach Cete as a eseach staff membe i 989. He was maage of the Fouatios of Massively Paallel Computig goup fom , whee he le the evelopmet of collective commuicatio, as pat of IBM MPL a MPI, fo IBM SP a SP paallel systems. His pimay eseach iteests iclue commuicatio issues fo itecoectio etwos, algoithms fo collective commuicatios, gaph embeigs, fault toleace, a paallel algoithms a achitectues. His cuet iteests ae ata miig a olie aalytical pocessig. He has publishe moe tha 8 joual a cofeece papes i these aeas. D. Ho is a coecipiet of the 986 Outstaig Pape Awa of the Iteatioal Cofeece o Paallel Pocessig. He has eceive a IBM Outstaig Iovatio Awa, two IBM Outstaig Techical Achievemet Awas, a fou IBM Plateau Ivetio Achievemet Awas. He has patets gate o peig. He is o the eitoial boa of the IEEE Tasactios o Paallel a Distibute Systems. He will be oe of the pogam vice chais fo the 998 Iteatioal Cofeece o Paallel Pocessig. He has seve o pogam committees of may paallel pocessig cofeeces a woshops. He is a membe of the ACM, the IEEE, a the IEEE Compute Society. Eli Upfal eceive a BSc i mathematics fom the Hebew Uivesity i 978, a MSc i compute sciece fom the Weizma Istitute i 98, a a PhD i compute sciece fom the Hebew Uivesity i 983. Duig , he was a eseach fellow at the Uivesity of Califoia at Beeley, a, i , a postoctoal fellow at Stafo Uivesity. I 985, D. Upfal joie the IBM Almae Reseach Cete, whee he is cuetly a eseach staff membe i the Fouatios of Compute Sciece Goup. I 988, he also joie the Faculty of Applie Mathematics a Compute Sciece at the Weizma Istitute, whee he is cuetly the Noma D. Cohe Pofesso of Compute Sciece. D. Upfal s eseach iteest iclue theoy of algoithms, aomize computig, pobabilistic aalysis of algoithms, commuicatio etwos, a paallel a istibute computig. He is a seio membe of the IEEE. W. Deic Weathesby is a PhD caiate i the Depatmet of Compute Sciece at the Uivesity of Washigto, Seattle, Washigto. His cuet eseach ivolves compile optimizatios fo collective commuicatio pimitives, potable softwae suppot fo efficiet collective commuicatio libaies, a paallel pogammig laguage esig. Shlomo Kipis (M 87) eceive a BSc i mathematics a physics i 983 a a MSc i compute sciece i 985, both fom the Hebew Uivesity of Jeusalem, Isael. He eceive a PhD i electical egieeig a compute sciece i 99 fom the Massachusetts Istitute of Techology. Fom , he woe as a eseach staff membe at the IBM T. J. Watso Reseach Cete i Yotow Heights, New Yo. Fom , he woe as a eseach staff membe at the IBM Haifa Reseach Laboatoy i Isael. Cuetly, he is woig as maage of ew techologies at NDS Techologies Isael. I aitio, sice 994, D. Kipis has bee a ajuct pofesso of compute sciece at Ba Ila Uivesity a at Tel Aviv Uivesity. His eseach iteests iclue paallel a istibute pocessig, efficiet commuicatio stuctues a algoithms, a system secuity. D. Kipis is a membe of the IEEE, the IEEE Compute Society, ACM, a ILA. He has publishe i umeous jouals a pesete his wo i may cofeeces a woshops. He is also a iveto a coiveto of two U.S. patets.
The dinner table problem: the rectangular case
The ie table poblem: the ectagula case axiv:math/009v [mathco] Jul 00 Itouctio Robeto Tauaso Dipatimeto i Matematica Uivesità i Roma To Vegata 00 Roma, Italy tauaso@matuiomait Decembe, 0 Assume that people
More informationPeriodic Review Probabilistic MultiItem Inventory System with Zero Lead Time under Constraints and Varying Order Cost
Ameica Joual of Applied Scieces (8: 37, 005 ISS 546939 005 Sciece Publicatios Peiodic Review Pobabilistic MultiItem Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue
More informationTwo degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system
wo degee of feedom systems Equatios of motio fo foced vibatio Fee vibatio aalysis of a udamped system Itoductio Systems that equie two idepedet d coodiates to descibe thei motio ae called two degee of
More informationOn the Optimality and Interconnection of Valiant LoadBalancing Networks
O the Optimality ad Itecoectio of Valiat LoadBalacig Netwoks Moshe Babaioff ad Joh Chuag School of Ifomatio Uivesity of Califoia at Bekeley Bekeley, Califoia 94720 4600 {moshe,chuag}@sims.bekeley.edu
More informationUnderstanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions
Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isketu tadeoff ad time value of
More informationBINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand
BINOMIAL THEOREM Itoductio Whe we epad ( + ) ad ( + ), we get ad ( + ) = ( + )( + ) = + + + = + + ( + ) = ( + )( + ) = ( + )( + + ) = + + + + + = + + + 4 5 espectively Howeve, whe we ty to epad ( + ) ad
More informationANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... 2 WHAT IS AN ANNUITY?... 2 EXAMPLE 1... 2 QUESTIONS...
ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... WHAT IS AN ANNUITY?... EXAMPLE 1... QUESTIONS... EXAMPLE BRANDON S
More informationSTATISTICS: MODULE 12122. Chapter 3  Bivariate or joint probability distributions
STATISTICS: MODULE Chapte  Bivaiate o joit pobabilit distibutios I this chapte we coside the distibutio of two adom vaiables whee both adom vaiables ae discete (cosideed fist) ad pobabl moe impotatl whee
More informationMoney Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version
Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio
More informationbetween Modern Degree Model Logistics Industry in Gansu Province 2. Measurement Model 1. Introduction 2.1 Synergetic Degree
www.ijcsi.og 385 Calculatio adaalysis alysis of the Syegetic Degee Model betwee Mode Logistics ad Taspotatio Idusty i Gasu Povice Ya Ya 1, Yogsheg Qia, Yogzhog Yag 3,Juwei Zeg 4 ad Mi Wag 5 1 School of
More informationLearning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)
Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 21 22 Leaig
More informationAnnuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments
8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise
More informationMaximum Entropy, Parallel Computation and Lotteries
Maximum Etopy, Paallel Computatio ad Lotteies S.J. Cox Depatmet of Electoics ad Compute Sciece, Uivesity of Southampto, UK. G.J. Daiell Depatmet of Physics ad Astoomy, Uivesity of Southampto, UK. D.A.
More informationLogistic Regression, AdaBoost and Bregman Distances
A exteded abstact of this joual submissio appeaed ipoceedigs of the Thiteeth Aual Cofeece o ComputatioalLeaig Theoy, 2000 Logistic Regessio, Adaoost ad egma istaces Michael Collis AT&T Labs Reseach Shao
More informationFinance Practice Problems
Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.
More informationChisquared goodnessoffit test.
Sectio 1 Chisquaed goodessoffit test. Example. Let us stat with a Matlab example. Let us geeate a vecto X of 1 i.i.d. uifom adom vaiables o [, 1] : X=ad(1,1). Paametes (1, 1) hee mea that we geeate
More informationOPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT EELECTION SCHEME
OPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT EELECTION SCHEME G. Aja Babu, 2 D. M. Padmavathamma Lectue i Compute Sciece, S.V. Ats College fo Me, Tiupati, Idia 2 Head, Depatmet of Compute Applicatio.
More informationTHE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen
THE PRINCIPLE OF THE ACTIVE JC SCATTERER Seppo Uoukaie VTT Buildig ad Tapot Ai Hadlig Techology ad Acoutic P. O. Bo 1803, FIN 02044 VTT, Filad Seppo.Uoukaie@vtt.fi ABSTRACT The piciple of fomulatig the
More informationBreakeven Holding Periods for Tax Advantaged Savings Accounts with Early Withdrawal Penalties
Beakeve Holdig Peiods fo Tax Advataged Savigs Accouts with Ealy Withdawal Pealties Stephe M. Hoa Depatmet of Fiace St. Boavetue Uivesity St. Boavetue, New Yok 4778 Phoe: 76375209 Fax: 7637529 email:
More informationLongTerm Trend Analysis of Online Trading A Stochastic Order Switching Model
Asia Pacific Maagemet Review (24) 9(5), 893924 LogTem Ted Aalysis of Olie Tadig A Stochastic Ode Switchig Model Shalig Li * ad Zili Ouyag ** Abstact Olie bokeages ae eplacig bokes ad telephoes with
More informationHighPerformance Computing and Quantum Processing
HPCUA (Україна, Київ,  жовтня року HighPefomace Computig ad Quatum Pocessig Segey Edwad Lyshevski Depatmet of Electical ad Micoelectoic Egieeig, Rocheste Istitute of Techology, Rocheste, NY 3, USA Email:
More informationDerivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)
Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of
More informationProject Request & Project Plan
Poject Request & Poject Pla ITS Platfoms Cofiguatio Maagemet Pla Vesio: 0.3 Last Updated: 2009/01/07 Date Submitted: 2008/11/20 Submitted by: Stephe Smooge Executive Sposo: Gil Gozales/Moia Geety Expected
More informationPRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK
PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK Vaanya Vaanyuwatana Chutikan Anunyavanit Manoat Pinthong Puthapon Jaupash Aussaavut Dumongsii Siinhon Intenational Institute
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationNegotiation Programs
Negotiatio Pogams Javie Espaza 1 ad Jög Desel 2 1 Fakultät fü Ifomatik, Techische Uivesität Müche, Gemay espaza@tum.de 2 Fakultät fü Mathematik ud Ifomatik, FeUivesität i Hage, Gemay joeg.desel@feuihage.de
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationVolume 1: Distribution and Recovery of Petroleum Hydrocarbon Liquids in Porous Media
LNAPL Distibutio ad Recovey Model (LDRM) Volume 1: Distibutio ad Recovey of Petoleum Hydocabo Liquids i Poous Media Regulatoy ad Scietific Affais Depatmet API PUBLICATION 4760 JANUARY 007 3 Satuatio, Relative
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationEstimating Surface Normals in Noisy Point Cloud Data
Estiatig Suface Noals i Noisy Poit Cloud Data Niloy J. Mita Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 iloy@stafod.edu A Nguye Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 aguye@cs.stafod.edu
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationLearning Algorithm and Application of Quantum Neural Networks with Quantum Weights
Iteatioal Joual of Comute heoy ad Egieeig, Vol. 5, No. 5, Octobe 03 Leaig Algoithm ad Alicatio of Quatum Neual Netwoks with Quatum Weights Diabao Mu, Zuyou Gua, ad Hog Zhag Abstact A ovel eual etwoks model
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationStrategic Remanufacturing Decision in a Supply Chain with an External Local Remanufacturer
Assoiatio fo Ifomatio Systems AIS Eletoi Libay (AISeL) WHICEB 013 Poeedigs Wuha Iteatioal Cofeee o ebusiess 55013 Stategi Remaufatuig Deisio i a Supply Chai with a Exteal Loal Remaufatue Xu Tiatia Shool
More informationQuality Provision in TwoSided Markets: the Case of. Managed Care
Quality ovisio i TwoSie Maets: the ase of Maage ae Micea I. Macu * Deatmet of Ecoomics, Uivesity of Floia, 33 Mathely Hall.O. ox 74, Gaiesville, FL 36, 35 394, micea@.ufl.eu Job Maet ae Octobe 8 STRT
More informationAsian Development Bank Institute. ADBI Working Paper Series
DI Wokig Pape Seies Estimatig Dual Deposit Isuace Pemium Rates ad oecastig Nopefomig Loas: Two New Models Naoyuki Yoshio, ahad TaghizadehHesay, ad ahad Nili No. 5 Jauay 5 sia Developmet ak Istitute Naoyuki
More informationPaper SD07. Key words: upper tolerance limit, macros, order statistics, sample size, confidence, coverage, binomial
SESUG 212 Pae SD7 Samle Size Detemiatio fo a Noaametic Ue Toleace Limit fo ay Ode Statistic D. Deis Beal, Sciece Alicatios Iteatioal Cooatio, Oak Ridge, Teessee ABSTRACT A oaametic ue toleace limit (UTL)
More informationECE 340 Lecture 13 : Optical Absorption and Luminescence 2/19/14 ( ) Class Outline: Band Bending Optical Absorption
/9/4 ECE 34 Lectue 3 : Optical Absoptio ad Lumiescece Class Outlie: Thigs you should kow whe you leave Key Questios How do I calculate kietic ad potetial eegy fom the bads? What is diect ecombiatio? How
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationDevelopment of Customer Value Model for Healthcare Services
96 Developmet of Custome Value Model fo Healthcae Sevices Developmet of Custome Value Model fo Healthcae Sevices WaI Lee ad BihYaw Shih Depatmet of Maetig ad Distibutio Maagemet, Natioal Kaohsiug Fist,
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationCLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS  APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT
CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS  APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment
More informationNotes on Power System Load Flow Analysis using an Excel Workbook
Notes o owe System Load Flow Aalysis usig a Excel Woboo Abstact These otes descibe the featues of a MSExcel Woboo which illustates fou methods of powe system load flow aalysis. Iteative techiques ae epeseted
More informationResearch Report 2012/13 International Graduate School for Dynamics in Logistics
Reseach Repot Reseach Repot 2012/13 Iteatioal Gauate School fo Dyaics i Logistics Volue 3 2013 Reseach Repot Kuztitel es Atiels Cotet Dyaics i Logistics 2 Reseach Beyo Bouaies 4 Iteatioal Doctoal Taiig
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationPerformance evaluation of Callcenter with call redirection. Ocena performanc klicnega centra s preusmerjanjem klicev
Elektotehiški vestik 74(): 7983 7 Electotechical Review: Lublaa Sloveia Pefomace evaluatio of Callcete with call ediectio Vladimi Efimushki Dago Žepi Cetal Sciece Reseach elecommuicatios Istitute(ZNIIS)
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationCONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest
CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually
More informationCloud Service Reliability: Modeling and Analysis
Cloud Sevice eliability: Modeling and Analysis YuanShun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationUncertain Version Control in Open Collaborative Editing of TreeStructured Documents
Uncetain Vesion Contol in Open Collaboative Editing of TeeStuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecompaistech.f Talel Abdessalem
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationEfficient Redundancy Techniques for Latency Reduction in Cloud Systems
Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationFast FPTalgorithms for cleaning grids
Fast FPTalgoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information580.439 Course Notes: Nonlinear Dynamics and HodgkinHuxley Equations
58.439 Couse Notes: Noliea Dyamics ad HodgkiHuxley Equatios Readig: Hille (3 d ed.), chapts 2,3; Koch ad Segev (2 d ed.), chapt 7 (by Rizel ad Emetout). Fo uthe eadig, S.H. Stogatz, Noliea Dyamics ad
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationStreamline Compositional Simulation of Gas Injections Dacun Li, University of Texas of the Permian Basin
Abstact Steamlie Comositioal Simulatio of Gas jectios Dacu L Uivesity of Texas of the Pemia Basi Whe esevoi temeatues ae lowe tha o F ad essue is lowe tha 5sia, gas ijectios, esecially whe ijectats iclude
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationModeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN
Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationU.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirani and Professor Rao Last revised. Lecture 9
U.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirai a Professor Rao Scribe: Aupam Last revise Lecture 9 1 Sparse cuts a Cheeger s iequality Cosier the problem of partitioig a give graph G(V, E
More informationNetwork Theorems  J. R. Lucas. Z(jω) = jω L
Netwo Theoems  J.. Lucas The fudametal laws that gove electic cicuits ae the Ohm s Law ad the Kichoff s Laws. Ohm s Law Ohm s Law states that the voltage vt acoss a esisto is diectly ootioal to the cuet
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationTime Value of Money: The case of Arithmetic and Geometric growth and their Applications
CHAPTER TE SPECIAL TOPICS I FIACE Time Value of Moey: The cae of Aithmetic a Geometic owth a thei Applicatio I. Itouctio Kowlee of how iteet compou i a coetoe of fiace a i iteal i fiacial eciio at the
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More informationClustering Process to Solve Euclidean TSP
Cluteig Poce to Solve Euclidea TSP Abdulah Faja *Ifomatic Depatmet, Faculty of Egieeig Uiveita Widyatama Badug Idoeia # Faculty of Ifomatio ad Commuicatio Techology Uiveiti Tekikal Malayia Melaka #*abd.faja@gmail.com,
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationInstituto Superior Técnico Av. Rovisco Pais, 1 1049001 Lisboa Email: virginia.infante@ist.utl.pt
FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB30 AIRCRAFT  PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationFramework for Computation Offloading in Mobile Cloud Computing
Famewok fo Computatio Offloadig i Mobile Cloud Computig Deja Kovachev ad Ralf Klamma Depatmet of Ifomatio Sytem ad Databae RWTH Aache Uiveity Abtact The iheetly limited poceig powe ad battey lifetime of
More information