On the Conversion between Binary Code and Binary-Reflected Gray Code on Boolean Cubes

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1 On the Convesion between Binay Code and BinayReflected Gay Code on Boolean Cubes The Havad community has made this aticle openly available. Please shae how this access benefits you. You stoy mattes Citation Johnsson, S. Lennat and ChingTien Ho On the Convesion between Binay Code and BinayReflected Gay Code on Boolean Cubes. Havad Compute Science Goup Technical Repot TR2091. Citable link Tems of Use This aticle was downloaded fom Havad Univesity s DASH epositoy, and is made available unde the tems and conditions applicable to Othe Posted Mateial, as set foth at ns.havad.edu/un:hul.instrepos:dash.cuent.temsofuse#laa

2 On the Convesion between Binay Code and Binay{Reected Gay Code on Boolean Cubes S. Lennat Johnsson ChingTien Ho TR2091 July 1991 Paallel Computing Reseach Goup Cente fo Reseach in Computing Technology Havad Univesity Cambidge, Massachusetts This wok has in pat been suppoted by the Ai Foce Oce of Scientic Reseach unde contact AFOSR89082, and in pat by NSF and DARPA unde contact CCR

3 On the Convesion between Binay Code and Binay{Reected Gay Code on Binay Cubes S. Lennat Johnsson Havad Univesity and Thinking Machines Cop. Cambidge, MA ChingTien Ho IBM Reseach Division Almaden Reseach Cente San Jose, CA Abstact We pesent a new algoithm fo convesion between binay code and binay{eected Gay code that equies appoximately 2K element tansfes in sequence fo K elements pe node, compaed to K element tansfes fo peviously known algoithms. Fo a binay cube of n = 2 dimensions the new algoithm degeneates to yield a complexity of K + 1 element tansfes, 2 which is optimal. The new algoithm is optimal within a facto of 1 with espect to the best known lowe bound fo any outing stategy. We show that the minimum numbe of element tansfes fo minimum path length outing is K with concuent communication on all channels of evey node of a binay cube. 1 Intoduction. Minimizing the equied data motion in memoy hieachies has been cucial in achieving high pefomance almost since the beginning of moden compute technology. In conventional memoy hieachies, minimizing data motion takes the fom of peseving tempoal and spatial locality of efeence in scheduling opeations. Massively paallel pocessos with thousands of pocessing nodes ae cuently the only altenative to exteme pefomance, i.e., a pefomance of a tillion opeations pe second and beyond. Scalability of the design, as well as pefomance, dictates that each pocessing node has its own memoy system, possibly extended with a physically shaed memoy. In such distibuted memoy hieachies, the bandwidth between a pocesso and its local memoy hieachy is often consideably highe than the bandwidth to the memoy units of othe pocessos o shaed memoy if it exists. The latency is anothe impotant pefomance issue. Ou focus is on ecient use of the communications bandwidth. The communication system in distibuted memoy hieachies often epesents a consideable faction of the total system cost. Nevetheless, due to the limited netwok bandwidth, it is often a bottleneck fo pefomance in many computations. Minimizing the demands on the communication system though pope data placement, such that data fequently used togethe eside in the same memoy unit o adjacent memoy units, is impotant. Once the data placement has been made, it is impotant to select paths and schedule the data motion such that the outing time (contention) is minimized. Placing the data among the memoy units fo optimum 1

4 pefomance is a vey had poblem. Often dieent allocations ae pefeable duing dieent phases of the computations. High Pefomance Fotan [2] is an extension of Fotan90 in which diectives have been added fo use contol ove data placement among memoy units. In this pape we addess the outing issues in changing the data placement fom one common placement in binay cubes, binay{eected Gay code [8], to anothe common placement, binay code, o the convese. Computations on egula gids constitute an impotant class of computations in science and engineeing. So called explicit methods fo nite dieence appoximations of patial dieential equations and many signal and image pocessing tasks fall into this class. The computations in explicit methods fo nite dieence appoximations ae often dominated by the evaluation of the dieence appoximation, known as dieence stencil o dieence molecule. The opeation is the same as what is typically known as convolution in signal and image pocessing. Data efeences ae local in Catesian space, which is used to epesent the poblem domain fo most egula discetizations. Howeve, the fast Fouie tansfom, FFT, is a poweful algoithm that is used fequently both fo the solution of patial dieential equations and in signal and image pocessing. The CooleyTukey FFT [1] efeences data that ae adjacent in binay space. Both techniques may be used on the same data set. Thus, if a distibuted memoy hieachy allows fo an ecient placement with espect to efeences both in Catesian and binay space, such a placement may be used. If the memoy hieachy is such that eithe one o the othe efeence patten can be suppoted, but not both simultaneously, then convesion between the two placements become impotant. This is the issue addessed in this pape fo a collection of memoy units inteconnected by a binay cube netwok. Catesian gids with axes' lengths being powes of two ae subgaphs of binay cubes if the total numbe of elements is less than, o equal to, the numbe of nodes in the cube. Binay{ eected Gay codes [8] ae often used fo the embedding of aays in binay cubes, since such codes peseve adjacency in Catesian space. Fo multidimensional aays, the Gay code encoding is typically applied to the dieent axes independently, theeby peseving adjacency along each axis. In this case, distinct subsets of addess bits ae assigned to dieent axes. We efe to each such subset as an addess eld. Fo aays with axes' lengths not being powes of two, peseving adjacency foces an expansion of the numbe of equied cube nodes []. But, even if the adjacency equiement is elaxed fo multidimensional aays with abitay axes' lengths in ode to limit the expansion, binay{eected Gay codes can be used fo the embedding of subsections of aays [5]. A conventional binay encoding of aay axes is suitable fo many divide{and{conque algoithms, such as the FFT. In the binay code, adjacency is peseved in the binay space. The binay cube nodes can be labeled such that adjacent nodes die in pecisely one bit. This pape focuses on the optimal outing of data fo convesion between data placements based on binay{eected Gay codes and binay codes in binay cube netwoks, in paticula in such netwoks allowing concuent communication on all channels of evey node, all{pot communication. The Connection Machine models CM{2 and CM{200 [4, 9] ae examples of achitectues using a binay cube netwok allowing all{pot communication. Ou esults can easily be modied to systems allowing concuent communication on some, but not all, channels of evey pocesso. The esults also povide inteesting insights into the popeties of binay{ eected Gay codes. The algoithms equie vey small tempoay stoage and the contol can be made distibuted, i.e., each node can detemine its own actions based on its addess and the local histoy of events. 2

5 The main new esult is an algoithm that fo K elements pe node of a binay n{cube equies d 2K (n 2) e + (n 2) element tansfes in sequence with all{pot communication. The algoithm is based on pipelining of the element tansfes. It yields an impovement of about K element tansfes in sequence ove pevious algoithms [, 7]. The new algoithm is within a facto of 1 of the best known lowe bound. It is optimal fo n = 2. We also pesent an optimal minimum path length outing algoithm. Though the numbe of element tansfes in sequence is highe than fo the nonminimum path length outing algoithm, the minimum path length algoithm may yield fewe communication statups in a packet{switched system with packet sizes of the same ode as the local data set. We give lowe bounds fo two outing stategies: one fo using any set of outing paths, one fo using only minimum length paths. The outline of the pape is as follows. We st give some popeties of binay cubes and binay{eected Gay codes, state ou assumptions of the communication system, and ou objectives fo the outing algoithms. Then, we pesent an optimal minimum path length outing algoithm, followed by the nonminimum path length outing algoithm that is the main esult of the pape. We conclude with a summay. 2 Peliminaies. A binay n{cube has N = 2 n nodes. Two nodes ae adjacent i thei addesses die in exactly one bit. Thee exist n disjoint paths between any pai of nodes in a binay n{cube. Fo nodes at distance d, d of those paths ae of length d, and n d paths ae of length d + 2. The binay encoding of i is B n (i) = (b n 1 b n 2 b 0 ) and its binay{eected Gay code encoding is G n (i) = (g n 1 g n 2 g 0 ). The pocesso addess bits ae (a n 1 a n 2 : : : a 0 ). Z N = f0; 1; ; N 1g and (1 j ) is a sting of j instances of a bit with value one. \jj" is the concatenation symbol. Fo the complexity estimates we assume bidiectional channels and concuent communication on all channels, all{pot communication. The numbe of elements pe node is K. ^Gn is the sequence of n{bit binay{eected Gay codes fo Z N, i.e., ^G n = (G n (0); G n (1); ; G n (2 n 1)). Denition 1 [8] The binayeected Gay code is dened ecusively as follows. ^G 1 = (G 1 (0); G 1 (1)); whee G 1 (0) = 0; G 1 (1) = 1: 0 1 0jjG n (0) 0jjG n (1). 0jjG n (2 n 2) ^G n+1 = 0jjG n (2 n 1) 1jjG n (2 n 1) : 1jjG n (2 n 2). B 1jjG n (1) A 1jjG n (0) In the following, we efe to the binay{eected Gay code just dened as Gay code. The highest ode bit is the same in the binay code and in the Gay code. The emaining bits in the

6 ?? Routing in a 2cube. 2 5 Routing in a cube. 4 Figue 1: Routing by exchanges in a 2cube and a cube. encoding of i 2 Z N=2 ae dened by G n 1 ((b n 2 b n b 0 )). The emaining bits in the encoding of i 2 Z N Z N=2 ae dened by G n 1 ((b n 2 b n b 0 )). Lemma 1 [8] b m = g n 1 g n 2 g m, m 2 Z n. Convesely, g m = b m b m+1, m 2 Z n with b n = 0. We pesent all of ou esults fo Gay{to{binay convesion, i.e., convesion fom Gay code encoding to binay code encoding. The adaptations of the esults to binay{to{gay convesion is staightfowad. We assume that each addess eld subject to the Gay{to{binay convesion is of length at least two. This assumption avoids the tivial case of a one bit code fo which the Gay code encoding is the same as the binay encoding. Bidiectional communication is assumed fo ou complexity estimates. All communication complexities ae stated in tems of the numbe of element tansfes in sequence. Given B n (i) and G n (i), an exclusive{o opeation on the two elds detemines the cube dimensions though which the data fom each node must be outed. The issues we addess ae: outing with constant stoage (pemutation outing) minimizing the congestion outing without tags A outing that can be viewed as a sequence of pemutations between neighboing nodes is highly desiable since such a outing conseves memoy equiements. Limiting the poblem size due to a demand fo bue space is often met with sevee citicism fom uses. The Gay{to{ binay convesion has the popety that the communication can be pefomed as a sequence of exchanges, theeby ealizing the st objective. This popety of the two codes is illustated in Figue 1 fo the cases n = 2 and n =. Ou minimal path length algoithm is based on Theoem 1 below. Befoe poving the theoem, we pove one citical popety of the binay{eected Gay code. Lemma 2 Let node a in a binay ncube initially contain the element of index G 1 (a). Fo any m 2 f1; 2; ; n 1g, if each node a fo which a n 1 a n 2 a m = 1 exchanges its element with its neighbo acoss dimension m 1, then node a contains the element of index G 1 (a n 1 a m )jjg 1 (a m 1 a 0 ) afte the exchange. 4

7 Poof: It suces to show that G 1 (a n 1 a m )jjg 1 G 1 (a (a m 1 a 0 ) = n 1 a m a 0 ); if a n 1 a m = 0, G 1 (a n 1 a m a 0 ); othewise. Let G 1 (a n 1 a m )jjg 1 (a m 1 a 0 ) = (a 0 n 1 a 0 0), let G 1 (a n 1 a 0 ) = (a 00 n 1 a 00 0) and let G 1 (a n 1 a m a 0 ) = (a 000 n 1 a000 0 ). It follows fom Lemma 1 that a0 j = a n 1 a j = a 00 j = a000 j fo m j n 1. Fo j being in the ange 0 j m 1, conside two cases. (i) Case 1: a n 1 a m = 0. Then, a 00 j = a n 1 a j = a m 1 a j = a 0 j. (ii) Case 2: a n 1 a m = 1. Then, a 000 j = a n 1 a m a j = 1a n 1 a j = a m 1 a j = a 0 j : Lemma 2 is easily genealized to the splitting of one out of seveal addess elds by obseving that the splitting of one addess eld is independent of othe addess elds and thei encoding. Theoem 1 The Gay{to{binay convesion can be pefomed as a sequence of exchanges in dimensions f0; 1; ; n 2g taken in abitay ode. Poof: Pefoming an exchange as in Lemma 2 ceates two independently Gay coded addess elds. The same splitting pocedue can be applied to the ceated subelds in abitay ode until thee ae n 1 addess elds of one bit each. Since the Gay code and binay code fo a one bit eld is identical, the poof is complete. The poof of Lemma 2 povides a way of deiving the exchange contol locally. Let m be the cuent exchange dimension, and let x be the next highe dimension which has aleady appeaed in the exchange sequence. If no such dimension exists, then let x = n 1. Then, the cuent data of node a should be exchanged if and only if a x 1 a m+1 = 1. Figue 2 gives an example of using an algoithm poceeding fom dimension n 2 to dimension 0. Initially, pocesso G 4 (i) contains data of index i. Afte the convesion, i is assigned to pocesso B 4 (i). A pseudocode fo the algoithm is given below. Initially, g m (i)! a m, m 2 Z n 1 and on temination b m (i)! a m, m 2 Z n 1, whee a = (a n 1 a n 2 : : :a 0 ), is the pocesso addess as befoe. /* Conveting Gay code to binay code stating fom the most signicant dimension */ fo j := n 2 downto 0 do if a j+1 = 1 then exchange content with the neighbo in dim. j endif enddo Fo the optimal minimum path length outing algoithm we pesent late, any dimension may be used as a stating dimension. Fo an abitay stating dimension m, m 2 Z n 1 with 5

8 Gay code Exchange Exchange Exchange dim. 2 dim. 1 dim. 0 data padd b data b2 data b1 data Figue 2: Convesion of a Gay code to binay code. exchanges in successive dimensions of deceasing ode, cyclicly, the st exchange equies the computation of the mth bit of G 1 (a), which is a n 1 a m+1. The subsequent steps ae simila to the algoithm above. Figue gives an example. Sequence 2 is the same as in Figue 2. The gue shows the location of i fo each step of the algoithm fo each sequence. /* Conveting Gay code to binay code stating fom dimension m. Dimensions in deceasing ode, cyclically*/ if a n 1 a n 2 a m+1 = 1 then exchange content with the neighbo in dim. m endif fo j := m 1 downto 0 do if a j+1 = 1 then exchange content with the neighbo in dim. j endif enddo fo j := n 2 downto m + 1 do if a j+1 = 1 then exchange content with the neighbo in dim. j endif enddo Lowe bounds. We st conside the case whee Gay{to{binay convesion shall be pefomed on a single pocesso addess eld. We then conside the case fo multiple pocesso addess elds, whee

9 Gay code Seq 2 Seq 1 Seq 0 assignment Exchange dim. Exchange dim. Exchange dim. Data padd Figue : Concuent convesion of a Gay code to binay code. each eld is subject to a Gay{to{binay convesion..1 A single eld. Theoem 2 With the communication fo Gay{to{binay convesion esticted to minimum length paths only, a lowe bound is max(k; n 1) element tansfes in sequence fo a code eld of n bits on an n{cube with all{pot communication. Poof: With K elements pe node eithe all o no elements must be communicated to some othe node fo code convesion. Fo any n 2 thee always exists a pai of nodes at distance one that must exchange data (nodes 2 and ). Resticting the communication to minimum length paths implies that the code convesion equies at least K element tansfes in sequence. Theoem A lowe bound fo the numbe of element tansfes in sequence equied fo Gay{ to{binay convesion on an n{cube with all{pot communication is max((1 1 n ) K 2 ; n 1). Poof: The total amount of communication equied fo Gay{to{binay convesion using only minimum length paths is 2 P n 1 n 1 i=0 i i = (n 1)2 n 1 K. (It follows fom the denition of the Gay code that the numbe of paths of length i is 2 n 1 i.) The numbe of available edges in an n{cube is n2 n. Hence, the lowe bound follows. Theoem does not give a tight lowe bound, since the only way in which the full bandwidth of the binay cube can be used is by using nonminimum length paths. Fo instance, fo n = 2 Theoem yields the bound K. A bound based on the numbe of disjoint paths is K. An uppe 4 2 bound fo n = 2 is K + 1. Simply oute K + 1 elements along the length one path and K elements ove the length path. Figue 4 illustates the outing of the data. The bound K is also a lowe bound fo K 2, since using two paths implies the use of nonminimum path length outing. 7

10 ?? Routing fom node 2 2 Routing fom node.2 Multiple independent elds. Figue 4: Routing paths in a 2cube. Fo the embedding of multidimensional aays a sepaate encoding is often used fo each addess eld. With d sepaately encoded addess elds and a total of n dimensions used fo the encoding, Theoem 2 is modied as follows. Theoem 4 With the communication fo Gay{to{binay convesion esticted to paths of minimum length only, a lowe bound fo the numbe of element tansfes in sequence is max(k; n d) fo all{pot communication with a code eld of n bits distibuted among d axes. Poof: Thee exists a pai of nodes (say the nodes with indices 2 o fo one axis, and all othe axis indices 0) at distance one that must exchange data. Thus, a time K is equied using only minimum length paths. Theoem 5 A lowe bound fo the numbe of element tansfes in sequence fo Gay{to{binay convesion on an n{cube with all{pot communication is max((1 d ) K ; n d) fo d addess n 2 elds sepaately encoded in n bits. Poof: The outing equiements fo axis i encoded in n i bits is (n i 1)2 n i 1 K. Thee ae 2 n n i such subcubes. Hence, the total outing need is P d 1 i=0 (n i 1)2 n 1 K = (n d)2 n 1 K. 4 Algoithms using only minimum length paths. In this section we use the popety that the code convesion can stat in an abitay dimension to geneate seveal concuent exchange sequences fo all{pot communication. The outing algoithm below uses only minimum length paths. It may be advantageous compaed to the nonminimum path length algoithm in the next section when thee ae othe simultaneous outing needs. The total load on the communications netwok is minimal fo the minimum path length outing, but the contention is not. The minimum path length outing may also be pefeable when thee is a signicant communications ovehead in a packet{switched communications system and the maximum allowable packet size is elatively lage. The minimum path length outing descibed below may yield up to 40% fewe statups than the nonminimum path length K outing fo cubes of high dimension and a maximum packet size of at least elements. n 1 8

11 4.1 Convesion of a single addess eld. We will st conside code convesion of a single addess eld. allocation [] of one{dimensional aays satisfy this constaint. depicted as follows in tems of the addess eld: Consecutive and cyclic data The data allocation can be (g p 1 g p 2 g p n b p n 1 b p n 2 b 0 ) Consecutive {z } {z } padd madd (b p 1 b p 2 b n g n 1 g n 2 g 0 ) Cyclic {z } {z } madd padd In the consecutive allocation, the pocesso addess eld is assigned to the most signicant bits of the aay index domain. In the cyclic allocation, the pocesso addess eld is instead assigned to the least signicant bits of the aay index domain. In the illustation, it is in both cases assumed that the local memoy addesses ae encoded in binay code and that the pocesso addess eld is encoded in Gay code. Theoem The Gay{to{binay convesion fo a eld of n bits can be attained with max(n 1; K) element tansfes in sequence using only minimum length paths and (n 1)pot communication. Poof: The main idea is to geneate seveal independent concuent exchange sequences that ceates a unifom load on all channels used by a minimum path length outing stategy. Fo the poof we conside thee dieent sets of values of K, the numbe of local elements. Case 1 is used fo most elements fo lage values of K and is the only elevant case when K is a multiple of n 1. Case 2 is used to assue optimal tansmission fo the emainde of elements when K is lage but not a multiple of n 1. Case 1. K mod n 1 = 0: Ceate n 1 exchange sequences that ae dieent otations of the dimensions n 2; n ; ; 0. Patition the data set in each pocesso into n 1 sets of the same size and assign one sequence to each data set. If the data set in a node does not equie communication in a dimension, then the communication is simply not pefomed. The numbe of element tansfes in sequence is K. Case 2. n 1 < K < 2n 2: Let x = K (n 1). Ceate K exchange sequences that ae distinct otations of the dimensions n 2; n ; ; 0; 1 ; 2 ; ; x, whee i 's ae dummy dimensions. No exchange is pefomed in a dummy dimension. Duing each step, no dimension in Z n 1 is used by moe than one sequence. The numbe of element exchanges in sequence is K. Case. K < n 1: It is easy to see that n 1 element tansfes in sequence ae necessay and sucient. Fo abitay K n 1, dene the outing by patitioning the data set such that cases 1 and 2 apply. In implementing the algoithm, a table can be set up in each pocesso such that fo each memoy patition thee is a table enty fo each dimension. The enty indicates whethe o not a communication shall be pefomed. 9

12 4.2 Convesion of multiple addess elds. Fo d addess elds encoding a total of n bits thee ae n d dimensions fo which an exchange is equied. Fo K mod (n d) = 0, n d exchange sequences can be geneated by applying n d dieent otations to some basic sequence in a manne analogous to the single addess eld case. The local data set can be divided into n d patitions with each patition assigned one of the exchange sequences. The cases fo K mod (n d) = 0 can also be handled in a way simila to the case with a single addess eld to attain the lowe bound of max(k; n d) element tansfes in sequence. 5 Algoithms using nonminimum length paths. 5.1 A single addess eld. In ode to achieve a time complexity of less than K element tansfes in sequence, it is necessay to use nonminimum length paths fo the convesion. In the following, we efe to the two subcubes with espect to the most signicant dimension as the zeo and one subcube, depending upon whethe o not the leading dimension is zeo o one. Fo n = 2 a lowe bound is K + 1, and we have aleady given an optimal algoithm, Figue 2 4. The citical obsevation fo the algoithm was that of the two edges in dimension 0, only one being used fo the minimum path length outing and none of the edges in dimension 1 being used. Hence, outing pat of the data to the subcube containing the unused edge in dimension 0 (the zeo subcube), pefoming the code convesion fo pat of the data in that subcube, then outing the conveted data back allows the time to be educed, by educing the contention. By consideing the outing paths in Figue 1, it can be obseved that if the {cube is collapsed into a 2{cube along dimension 2 by identifying nodes 0 and 7, 1 and, 2 and 5 and and 4, then all bidiectional links in the esulting 2{cube would be used evenly. Hence, by exchanging data between the zeo and one subcubes, all edges in both subcubes can be used fo the code convesion. Fo the case n = 2 thee is no exchange between the zeo and one subcubes, since thee is no outing equiement in the zeo subcube. Fo n = the edges between nodes 2 and 5, and and 4, espectively, ae used in both diections fo data fom both the zeo and the one subcubes. All othe edges ae used fo \single" exchanges. We will now pove that the outing paths fo Gay{to{binay convesion always can be detemined such that if an n{cube is collapsed into a (n 1){cube along the most signicant dimension, all edges ae used evenly in the collapsed cube. Theoem 7 If the outing paths fo the Gay{to{binay convesion tavese the cube dimensions in ascending ode, then an edge is used fo an exchange in subcube zeo, i the coesponding edge in subcube one is not used. Poof: Let the exchange sequence fo the Gay{to{binay convesion poceed though the cube dimensions in the ode 0; 1; ; n 2. Let the st exchange step be step 0. Fom Lemma 2, an exchange in step 0 is pefomed fo all nodes such that a n 1 a 1 = 1. It is easy to show that the exchange in step i, 0 i n 2, is detemined by a n 1 a i+1. Since 10

13 ?? Figue 5: Routing fo code convesion in a 4{cube. a n 1 is always in the expession detemining whethe o not a node shall exchange data, only one node in a pai dieing in thei most signicant bit exchange data in any of the dimensions 0; 1; : : :; n 2. Thus, an edge is used fo an exchange in subcube zeo i the coesponding edge in subcube one is not used. Figue 5 shows the edges used fo Gay{to{binay convesion with outing paths dened by outing in dimensions of ascending ode. Note that by folding the 4{cube onto a {cube, all edges in the {cube ae used evenly in the same way as all edges ae used evenly if a {cube is folded onto a 2{cube. Fo binay to Gay code convesion, the same popety holds fo an exchange sequence using the cube dimensions in descending ode. The basic idea of ou algoithm equiing appoximately 2K as follows. element tansfes in sequence is 1. Divide the local data set into two sets: the st set S 1 of size M 2K and the second set S 2 of size M 0 = K M K. The lage data sets S 1 ae outed ove minimum length outes, shot outes. The smalle data sets S 2 ae outed ove nonminimum length paths, long outes, by st outing the data to the node obtained by complementing the most signicant dimension. 2. Pefom the following two algoithms concuently fo each node i in communicating the data sets S 1 and S 2 to thei destination node G 1 (i). (a) Route each element in S 1 along an ascending ode of cube dimensions 0; 1; ; n 2, as needed. Pipelining the M elements esults in a total of M +n 2 element tansfes in sequence. (b) Route each element in S 2 along cube dimensions n 1; 0; 1; ; n 2; n 1. Note that dimension n 1 needs to be outed twice fo each element since outing in this 11

14 dimension is not pat of the code convesion. Pipelining is applied in the following manne, assuming M 0 n. Pipelining the M 0 elements along dimensions n 1; 0; 1; ; n 2 esults in a total of M 0 + n 1 element tansfes in sequence. Initiating the nal exchange along dimension n 1 fo the M 0 elements afte M 0 time steps of the pevious pocedue esults in M 0 time steps with n 1 steps ovelapped with the pevious pocedue. Note that the set of shot outes is edge{disjoint fom the set of long outes, thus allowing fo the element tansfes to be pipelined without contention. Befoe detemining the optimum size of the small and lage data sets, we state the esult. Theoem 8 The Gay{to{binay convesion equies at most d 2K (n 2) e + (n 2) element tansfes in sequence with all{pot communication fo K > n +. The numbe of element tansfes in sequence fo K n + 2 is n. Fo a pecise optimization of the numbe of element tansfes in sequence, we note that with M elements assigned to each shot oute, the time fo these outes, T s (M), satises the elation T s (M) M + n 2: One simple scheduling fo the long outes, which ae subject to contention in dimension n 1, is to pipeline the tansfes of the elements acoss dimension n 1 with the outing equied fo code convesion in the complemented subcube. The conveted data emains in the complemented subcube until the edges in dimension n 1 become fee, when the conveted data is bought back to the oiginating subcube. Thus, fo M 0 elements assigned to long outes, T l (M 0 ) M 0 + max(m 0 ; n) The optimal patition of the data set K, is detemined by T s (M) = T l (K M). This equality yields M = d 2K (n 2) e fo K > n + 2, and M = 2 fo K n + 2. T min = ( d 2K (n 2) e + (n 2) K > n + 2 n K n Multiple addess elds. With Gay{to{binay convesion equied on multiple addess elds, the numbe of element tansfes in sequence is almost identical to code convesion fo a single addess eld encoded in the same numbe of bits. Fo the outing, the local data sets can be divided into as many patitions as thee ae axes. The code convesion fo the dieent patitions can be pefomed concuently. All patitions must undego code convesion fo each axis. 5. Binay{to{Gay convesion. A binay{to{gay convesion is obtained by unning ou algoithms backwads. Thus, 12

15 Theoem 9 The binay{to{gay convesion equies at most d 2K (n 2) e + (n 2) element tansfes in sequence with all{pot communication fo K > n +. The numbe of element tansfes in sequence fo K n + 2 is n. The theoem follows fom Theoem 8 by using descending ode outing fo the code convesion and combining outing along shot and long outes as in the Gay{to{binay convesion case. Summay. We have given an algoithm fo the convesion between binay{eected Gay code and binay code that equies appoximately 2K element tansfes in sequence. The algoithm oes a eduction in the numbe of element tansfes in sequence by about K compaed to peviously known algoithms. Fo n = 2, the new algoithm is optimal fo any outing stategy. Fo lage values of n, it is nonoptimal by at most a facto of 1 compaed to the best known lowe bound, K We have also given an optimal minimum path length outing algoithm that fo K elements pe node and concuent communication on all channels of evey pocesso in a binay ncube equies max(k; n d) element tansfes in sequence. The algoithm has n 2 element tansfes less than a pipelined algoithm [], when K n 1. The Connection Machine models CM{2 and CM{200 ae distibuted memoy achitectues which allow concuent communication on all channels of evey node, with the nodes inteconnected as a binay cube. The code convesion outine on the Connection Machine was implemented befoe the nonminimum path length outing algoithm was discoveed. The existing outine use the pipelined algoithm in []. The esults hee show that the code convesion time on the Connection Machine can be sped up by as much as 50%. Fo a packet{switched communication system with a maximum packet size B, the minimum K path length outing equies d e(n 1) statups. The nonminimum length path outing (n 1)B equies max(d 2K B e + n 2; 2d K K e + n 1) statups. Fo B = d e, the minimum length path B n 1 outing equies n 1 statups, while the nonminimum path length outing equies appoximately 2d 2(n 1) e + n 1 statups. Fo lage n, the atio of the numbe of statups between minimum path length outing and nonminimum path length outing appoaches 0.. Fo K B 1, the atio instead appoaches 1.5. Acknowledgment This wok has in pat been suppoted by the Ai Foce Oce of Scientic Reseach unde contact AFOSR89082, and in pat by NSF and ARPA unde contact CCR Refeences [1] James C. Cooley and J.W. Tukey. An algoithm fo the machine computation of complex fouie seies. Math. Comp, 19:291{01,

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