Sorting, Selection, and Routing on the Array with Reconfigurable Optical Buses

Transcription

1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 Sortig, Selectio, ad Routig o the Array with Recofigurable Optical Buses Saguthevar Raasekara, Member, IEEE Computer Society, ad Sarta Sahi, Fellow, IEEE Abstract I this paper, we preset efficiet algorithms for sortig, selectio, ad packet routig o the AROB (Array with Recofigurable Optical Buses) model. Oe of our sortig algorithms sorts geeral keys i O() time o a AROB of size e for ay costat e > 0. We also show that selectio from out of elemets ca be doe i radomized O() time employig processors. Our routig algorithm ca route ay h-relatio i radomized O(h) time. All these algorithms are clearly optimal. Idex Terms Recofigurable etworks, optical computig, mesh-coected computers, compariso problems, iterprocessor commuicatio, sortig. INTRODUCTION A N Array with Recofigurable Optical Buses (AROB) [], [7] is essetially a m recofigurable mesh [] i which the buses are implemeted usig optical techology. This model has attracted the attetio of may researchers i the recet past owig to its promise i superior practical performace. A 4 4 recofigurable mesh is show i Fig.. The switch i each processor ca be used to coect together subsets of the four bus segmets coected to the processor. Recofigurable meshes that use electroic buses have bee studied extesively. Various models such as the RN [4], RMESH [0], PARBUS [], M r [4], RMBM [9], REBSIS [5], ad DMBC [8] have bee proposed ad studied. Recofigurable meshes with optical buses have bee less extesively studied. I the AROB model of [], [7], the allowable switch settigs of the processors are the same as those i the RN model of [4]. These are show i Fig.. A bus lik coects two adacet processors x ad y ad has two associated wave guides. Oe of the wave guides permits a optical sigal to travel from x to y ad the other permits sigal movemet from y to x. By settig processor switches, bus liks are coected together to form disoit buses. O each bus, we eed to specify which orietatio of the waveguide o each lik of the bus is to be used. The resultig directed graph that represets the bus should be a directed chai. The root of this chai is the bus leader. The legth of a bus is the umber of liks o the chai represetig that bus. The positio of ay processor o a bus is its distace from the bus leader. The time eeded to trasmit a message o a bus is referred to as oe cycle. A cycle is divided ito slots of duratio t ad each slot ca carry a differet optical sigal. t is the time eeded for a optical pulse to move dow oe bus lik. I particular, t ecompasses the time to sed a b-bit message, where each bit is a light pulse with a w secod duratio (for appropriate values of b ad w). Pavel ad Akl [] have argued that for reasoable size meshes (say up to,000,000), the umber of slots i a cycle may be assumed to be for a mesh. Further, the duratio of a cycle may be assumed costat ad comparable to the time for a CPU operatio. To assist a processor i determiig which slot to use, each processor has a slot couter. These couters may be started at the begiig of a cycle. The bus leader iitiates a light pulse at this time (i.e., it writes a oe to the bus). The couter at each processor stops whe the light pulse reaches that processor. This special timig mechaism does ot require ay bus read operatio. The termial couter value is the distace of the processor from the bus leader. If more tha oe message gets writte i a slot, the last writte message remais i the slot. Note that, because a processor ca read/write from/to its bus durig oly oe slot of a cycle, it caot poll the up to light pulses movig through it i oe cycle. A additioal AROB feature that facilitates the developmet of algorithms is the delay uit at each processor. This permits a processor to itroduce a oe time slot delay i the light pulses passig through it. We refer to a AROB as a oe-dimesioal AROB. I Sectio, we provide some prelimiaries related to radomized algorithms ad survey kow results i the area of AROBs. Sectios, 4, ad 5 are devoted to the problems of sortig, selectio, ad packet routig, respectively. I Sectio 6, we provide our coclusios ad list some ope problems. The authors are with the Departmet of Computer ad Iformatio Sciece ad Egieerig, Uiversity of Florida, Gaiesville, FL 6. {ra, sahi}@cise.ufl.edu. Mauscript received Jue 995. For iformatio o obtaiig reprits of this article, please sed to: tpds@computer.org, ad referece IEEECS Log Number PRELIMINARIES I this sectio, we provide some prelimiary facts ad results that will be employed i the paper /97/$ IEEE F:\LIBRARY\TRANS\PRODUCTION\ regularpap KSM 7,6 09/0/97 :0 PM /

2 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 Fig.. A 4 4 recofigurable mesh. Fig.. Possible switch coectios.. Radomized Algorithms We say a radomized algorithm uses Of ~ (()) amout of ay resource (like time, space, etc.) if the amout of resource used is o more tha caf() with probability ( - -a ) for ay a, c > 0 beig a costat. We could also defie ~ Q(. ), ~ o (. ), etc. i a similar maer. By high probability, we mea a probability of ( - -a ) for ay costat a... Cheroff Bouds [6] These bouds ca be used to closely approximate the tail eds of a biomial distributio. A Beroulli trial has two outcomes, amely, success ad failure, the probability of success beig p. A biomial distributio with parameters ad p, deoted as B(, p), is the umber of successes i idepedet Beroulli trials. Let X be a biomial radom variable whose distributio is B(, p). If m is ay iteger > p, the the followig are true: ad for ay 0 < d <. Prob. X > m b Prob. X > + d p e g F H G pi K J m Prob. X < b - dgp e e -d p/ m m-p ; d p - ;. Problem Defiitios Give a sequece of umbers, say, k, k, º, k, the problem of sortig is to rearrage them i odecreasig order. The problem of selectio is to idetify the ith smallest umber from out of give umbers (where i is a iput ad i ). I ay fixed coectio etwork, a sigle step of iterprocessor commuicatio ca be thought of as a packet routig task. The problem of routig ca be stated as follows: There is a packet of iformatio at each ode that is destied for some other ode. Sed all the packets to their correct destiatios as quickly as possible, makig sure that, at most, oe packet crosses ay edge at ay time. Packet routig is equivalet to the radom access write operatio first defied by Nassimi ad Sahi [9]. The ru time of ay packet routig algorithm is defied to be the time take by the last packet to reach its destiatio. The queue size is the maximum umber of packets that ay processor will have to store durig the algorithm. The problem of partial permutatio routig is the task of routig where, at most, oe packet origiates from ay ode ad, at most, oe packet is destied for ay ode. Ay routig problem where, at most, h packets origiate from ay ode ad, at most, h packets are destied for ay ode will be called h - h routig or h-relatios [0].. Previous Results ad Extesios I [], the AROB model has bee defied. Similar models have bee employed before as well (see, e.g., []). A related model kow as the Optical Commuicatio Parallel Computer (OCPC) has also bee defied i the literature (see, e.g., [], [8], [0], [9]). I a OCPC, ay processor ca commuicate with ay other processor i oe uit of time, provided there are o coflicts. If more tha oe processors tries to sed a message to the same processor, o message reaches the iteded destiatio. I [], algorithms for such problems as prefix computatio, routig o a liear array, matrix multiplicatio, etc. have bee give for the AROB. O the other had, [] cosiders the problem of selectio o a mesh with optical buses. LEMMA.. Cosider a processor AROB. If each processor has a bit, the the prefix sums of these bits ca be computed i O() cycles []. The idea behid the above the algorithm is as follows: Processor iitiates a light pulse i time slot oe of a cycle if its bit is zero, ad i slot two, otherwise. All processors start their couters at the start of the cycle ad also set their delay uits to itroduce a oe slot delay i case the processor s bit is oe. A processor s couter is tured off whe the light pulse iitiated by processor reaches it. By usig the termial couter value, its data bit ad its distace from processor, each processor ca compute its prefix sum value. The above algorithm ca be exteded to show the followig []: LEMMA.. The additio of log -bit umbers ca be performed i O() cycles o a log AROB. A costat time algorithm for prefix sums o a D AROB ca be foud i []. I particular, they show: F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM / 0

3 RAJASEKARAN AND SAHNI: SORTING, SELECTION, AND ROUTING ON THE ARRAY WITH RECONFIGURABLE OPTICAL BUSES LEMMA.. If there is a bit at each ode of a AROB, we ca compute the prefix sums of these bits i O() cycles. This algorithm uses the algorithm of [] to compute the prefix sums of a iteger sequece. The maximum bus legth employed by this algorithm is. We ext show that oe ca reduce the bus legth for the prefix sums problem to (which is a factor of three improvemet over [] s algorithm). Furthermore, our algorithm is simpler. LEMMA.4. Prefix sums of bits o a AROB ca be computed i O() cycles, keepig the maximum bus legth as. PROOF. We proceed as i []. Say we are iterested i computig the prefix sums i row maor order. Usig the algorithm of Lemma., we ca compute the prefix sums alog each row i O() time. At the ed of this step, each processor i the last colum has the sum of all s i the correspodig row. Now the origial problem of computig prefix sums reduces to computig prefix sums of umbers where each umber is at most (i.e., each umber is a O(log )- bit umber). Next we describe how to compute the prefix sums of O(log )-bit umbers i O() time o a AROB. (I [], this is doe usig the algorithm of [].) Step : Group the umbers with log umbers i each group. Allocatig a subarray of size log log, compute the sum of umbers i each group as doe i []. This step takes O() time ad the maximum bus legth is O(log ). log Step : Compute prefix sums of the group sums. This ca be doe usig Lemma. as follows. For each prefix sum (there are of them), allocate a log subarray of size log. Broadcast the appropriate umbers to each subarray ad, withi each log subarray, add the umbers, usig Lemma.. Here, also, the maximum bus legth is. Step : Compute prefix sums local to each group of log umbers. This step is similar to Step. Each group will get a subarray of size log log. Maximum bus legth is O(log ). Clearly, the above algorithm rus i O() time. LEMMA.5. I a AROB of size, ay permutatio ca be routed i O() cycles []. PROOF. The time it takes for a packet to move from oe processor to the ext is assumed to be t. Sice there are two wave guides (oe for left-to-right trasmissio ad the other for right-to-left trasmissio), the flow of packets from left to right is ot affected by the flow of packets from right to left. Cosider ay permutatio to be routed. Let the processors be umbered,, º, startig from left. I the followig discussio, cosider oly packets that have to travel from left to right (sice the other packets ca be aalyzed i a aalogous maer). There is a time slot assiged to each processor for readig from (ad writig ito) the bus. Let the readig time slot for processor i be i. Processor creates a time slot for each packet that moves oe edge per t time. The time slots created will be i the order of the processors, i.e., the first time slot is meat for processor, time slot is meat for processor, ad so o. If processor p has a message for processor q, p will write this message at time t + (p + q)t, where t is the start time. Clearly, this algorithm termiates i two cycles (or t time). The above lemma ca be stregtheed as follows: LEMMA.6. Let / be a AROB of size. Cosider a routig problem where O() packets origiate from ay ode ad O() packets are destied for ay ode. This problem ca also be solved i O() cycles. PROOF. Let c (resp. d) be a upper boud o the umber of packets destied for (origiatig from) ay ode. It suffices to cosider the case d =, sice that algorithm ca be repeated d times to take care of the geeral case. There will be c rus of the algorithm give above for Lemma.5. The above algorithm has the property that the message read by ay processor q is the last message writte i time slot q. If more tha oe processors wrote i time slot q, they ca determie if their message was read by q or ot by reversig the routig process. This way, i every two executios of the above routig algorithm, a processor receives oe message destied for it. A further extesio of the above ideas leads to the followig []: LEMMA.7. Say there are k elemets arbitrarily distributed (at most oe per processor) i a two-dimesioal AROB of size. We would like to compact them i the first k rows. This problem ca be solved i O() cycles. The algorithm for the above problem figures out a uique address for each elemet (usig the prefix algorithm of Lemma.) ad the routes the elemets usig greedy paths. There is o possibility of a collisio. The followig lemma pertais to selectio o a AROB ad is due to []: LEMMA.8. The selectio problem (from out of elemets) ca be solved i a expected time of Oe p log cycles o a AROB of size p. The worst case ru time is O() cycles. The basic idea behid the above algorithm is to pick a radom elemet as the pivot elemet ad partitio the iput ito two aroud the pivot; decide which partitio the ith elemet is i; throw away the irrelevat partitio ad perform a appropriate selectio i the relevat partitio. May O() time algorithms are kow for sortig o the recofigurable mesh (e.g., [], [0], [6]): LEMMA.9. Sortig of umbers ca be performed i O() time o a recofigurable mesh of size. F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM / 0

4 4 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 The same algorithm rus o the AROB, preservig the ru time... Routig o the OCPC Several packet routig algorithms for the OCPC model ca be foud i the literature. Aderso ad Miller have show that a special case of log -relatios o a -ode OCPC ca be routed i O ~ (log ) time []. Also, [0] ad [8] have preseted efficiet algorithms for h-relatios. A algorithm for arbitrary h-relatios with a ru time of Oh ~ ( + loglog ) has bee give by Goldberg et al. [9]..4 New Results I this paper, we preset a sortig algorithm that ca sort geeral keys i O() time o a AROB of size e for ay costat e > 0. We also poit out that this algorithm is optimal. We also preset a sortig algorithm that ca sort k-bit umbers i O(k) time o a AROB of size. Notice that such a algorithm caot be devised eve o the CRCW PRAM. Our selectio algorithm is radomized ad ca perform selectio from out of elemets i O ~ () time usig a AROB. This algorithm is clearly optimal. I cotrast, the selectio algorithm of Pa [] has a expected ru time of O(log ) ad a worst case ru time of O(). We cosider several variats of the packet routig problem i this paper. Oe of the mai theorems we prove shows that ay h-relatio ca be routed i Oh ~ ( ) time o a oedimesioal AROB. I cotrast, the best kow algorithm for the OCPC model has a ru time of Oh ~ ( + loglog ) [9] ad is much more complicated tha our algorithm. Clearly, our algorithm is optimal. We also preset a algorithm for h- relatio routig o a AROB with a ru time of ~ Oh ( + loglog ). O the other had, the best kow algorithm for the same problem o the OCPC model has a ru time of Oh ~ log + e log log [8]. Our determiistic algorithm for h- relatios rus i time O(h log ) o a AROB, as well as o a AROB. SORTING ON THE AROB I this sectio, we preset optimal algorithms for sortig both geeral ad iteger keys o the two-dimesioal AROB. The geeral sortig algorithm sorts umbers i a AROB of size e for ay costat e > 0. The ru time is O() cycles ad, hece, the algorithm is optimal i view of the followig lower boud: LEMMA.. Sortig of umbers eeds W p parallel compariso tree processors [5]. F HG I e K J log p log + time usig This lower boud implies that if sortig of umbers has to be doe i O() time, the there must be W( +e ) processors, for some costat e > 0. I [5], the lower boud has bee prove for the parallel compariso tree model of Valiat. Sice a parallel compariso tree ca simulate a AROB step per step, the same lower boud applies to the AROB as well. Our iteger sortig algorithm rus o a AROB ad ca sort k-bit umbers i O(k) time. Notice that, eve o the CRCW PRAM model, such a algorithm caot be devised i view of the lower boud result of Beame ad Hastad [].. Geeral Sortig Let k, k, º, k be the give umbers. Thik of these umbers as formig a matrix M with r = / rows ad s = / colums. We employ the colum sort algorithm of Leighto [4]. There are seve steps i the algorithm: Algorithm Sort ) Sort the colums of M i icreasig order; ) Traspose the matrix preservig the dimesio as r s. I particular, pick the elemets i colum maor order ad fill the rows i row maor order; ) Sort each colum i icreasig order; 4) Rearrage the umbers applyig the reverse of the permutatio employed i step ; 5) Sort the colums i a way that adacet colums are sorted i reverse order; 6) Apply two steps of odd-eve traspositio sort to the rows. Specifically, i the first step, perform a compariso-exchage betwee processors i + ad i, for i = 0,, º ad, i the secod step, perform a compariso-exchage betwee processors i ad i +, for i =,, º; ad 7) Sort each colum i icreasig order. At the ed of this step, it ca be show that, the umbers will be sorted i colum maor order... Implemetatio o the AROB The give umbers will be stored i the first row of the e AROB, oe key per processor. At ay give time, each key will kow which row ad which colum of the matrix M it belogs to. Wheever we eed to sort the colums, we will make sure that the umbers belogig to the same colum will be foud i successive processors. O a AROB, ote that ay permutatio ca be routed i O() time. This meas that steps ad 4 ca be performed i O() time. Step 6 ca be performed i O() time as well as follows: Rearrage the umbers such that elemets i the same row are i successive processors ad apply two steps of the odd-eve traspositio sort. After this, move the keys to where they came from. Next we describe how we implemet Steps,, 5, ad 7. We first assume that we have a AROB of size /. Later, we will idicate how to reduce the size to e for ay e > 0. Partitio the AROB ito / parts, each of size / /, each part correspodig to a colum of M. Rearrage the give umbers such that the first colum of M is i the first / processors of row ; the secod colum is i the ext / processors of the first row; ad so o. Now, sort the umbers i each part (i.e., each colum of M) usig Lemma.9. This ca be doe i O() time. This implies that Steps,, 5, ad 7 of colum-sort ca be performed i O() time. Therefore, it follows that umbers ca be sorted i O() time o a AROB of size /. F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 4 / 0

5 RAJASEKARAN AND SAHNI: SORTING, SELECTION, AND ROUTING ON THE ARRAY WITH RECONFIGURABLE OPTICAL BUSES 5 We ca reduce the size of the AROB to 4/9 as follows: We still use Leighto s sort, with r = / ad s = /. I steps,, 5, ad 7, each part of / umbers will be sorted usig a AROB of size 4/9 /. This is doe usig the AROB algorithm above. I a similar way we ca reduce the size to 8/7, 6/8, ad so o. Thus, we get the followig theorem: THEOREM.. We ca sort umbers i O() cycles usig a AROB of size e, where e is ay costat > 0.. Iteger Sortig I this subsectio, we preset a algorithm for sortig k-bit umbers i O() time o a AROB. This algorithm makes use of the idea of radix sortig ad Lemmas. ad.5... Radix Sortig The idea is captured by the followig lemma: LEMMA.. If umbers i the rage [0, R] ca be stable sorted usig P processors i time T, the we ca also stable sort umbers i the rage [0, R c ] i O(T) time usig P processors, c beig ay costat. A sortig algorithm is said to be stable if equal keys remai i the same relative order i the output as they were i the iput. The algorithm proceeds as follows: There are k stages. I stage i, we sort the umbers with respect to their ith LSBs. To be more specific, i the first stage, we sort the umbers with respect to their LSBs. I the ext stage, we apply a sort i the resultat sequece with respect to the ext LSBs, ad so o. Thus, there will be k stages i the algorithm. Each stage ca be performed i O() time as follows: Notice that each stage is othig but sortig oe-bit umbers. Perform a prefix sums operatio for the zeros i the iput. Do the same for the oes i the iput. Usig these two sums, each processor ca determie the positio of its data i the sorted list. Permute the data to complete the sort for the stage. Sice the prefix sums, as well as the permutatio, take O() time each, each stage takes O() time as well (c.f. Lemmas. ad.5.) Thus, we have prove the followig: THEOREM.. A AROB ca sort k-bit umbers i O(k) cycles. Realize that o PRAM algorithm ca achieve the above performace, sice the lower boud theorem of [] implies that, for sortig of bits o the CRCW, PRAM will eed W log time, give oly a polyomial umber of processors. e log log 4 SELECTION Give a sequece of umbers k, k, º, k ad a i, the problem of selectio is to idetify the ith smallest of the umbers. A elegat liear time sequetial algorithm is kow for selectio (see, e.g., []). Floyd ad Rivest have give a simple liear time radomized algorithm for sequetial selectio [7]. Optimal parallel algorithms are also kow for selectio o various models of computig (see, e.g., [5]). Most of the parallel selectio algorithms (both determiistic ad radomized) make use of the techique of samplig. I this sectio, we show that selectio from out of umbers ca be doe i O ~ () time o a AROB of size. There is a umber iput at each processor. The basic idea is the followig: ) Pick a radom sample S of size q = o(); ) Choose two elemets, ad, from the sample whose raks i S are i q - d ad i q + d for some appropriate d. Oe ca show that these elemets bracket the elemet to be selected with high probability; ) Elimiate all keys whose values are outside the rage [,,, ]; 4) Perform a appropriate selectio from out of the remaiig keys. A Samplig Lemma. Let Y be a sequece of umbers from a liear order ad let S = {k, k, º, k s } be a radom sample from Y. Also, let k, k, K, k s be the sorted order of this sample. If r i is the rak of k i i Y, the followig lemma provides a high probability cofidece iterval for r i. (The rak of ay elemet k i Y is oe plus the umber of elemets < k i Y.) LEMMA 4.. For every a > 0, a e s s. Prob. r i - i > a log < - M P N M P N A proof of the above lemma ca be foud i [6]. This lemma ca be used to aalyze may of the selectio ad sortig algorithms based o radom samplig. Our selectio algorithm makes use of radom samplig. Now, we are ready to preset our selectio algorithm. The algorithm selects the ith smallest key from the iput. N deotes the umber of alive keys at the begiig of the th iteratio of the repeat loop. To begi with, the umber of alive keys, N, is the same as. The umber of alive keys that do ot get deleted i the th iteratio is deoted by s. S is the set of sample keys employed i the th iteratio. Algorithm Select = ; N = ; i = i; repeat ) Each alive key decides to iclude itself i the sample S with probability 06.. There will be ~ 04. Q( N ) N keys i the sample. ) Compact the sample keys i the first row. This is doe usig Lemma.7. With high probability, oly the first row will be oempty. ) Sort the umbers i the first row usig Theorem.. Processors which do ot get a sample key will have a key valued. Let q be the umber of keys i the sample. Choose keys, ad, from S with raks L iq O iq - d q log N ad L O + d q log N, respectively, d beig a costat > a. (a is as i Lemma 4..) F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 5 / 0

6 6 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 This takes O() time. 4) Elimiate keys that fall outside the rage [,,, ]. Cout the umber, s, of survivig keys. It ca be show that this umber is ~ 08. ON e log N. 5) Cout the umber del of keys deleted that are <,. If the key to be selected is ot oe of the remaiig keys (i.e., if del i or i > del + s ), start all over agai (i.e., go to Step with =, N =, ad i = i). Set i + = i - del ; N + = s ; ad = +. util N 6) Compact the survivig keys i the first row ust like i Step. Sort the first row ad output the i th smallest key from out of the remaiig keys. THEOREM 4.. The above algorithm rus i O ~ () time o a AROB. PROOF. Step takes O() time. The distributio of the umber of keys i the sample S is B N F HG, 06. N I K J. Usig Cheroff bouds, S is ~. Q( N 04 ). Step takes O() time (c.f. Lemma.7). Step rus i O() time i accordace with Theorem.. Coutig i steps 4 ad 5 ca be performed i time O() as well (cf. Lemma.). I step 6, cocetratio ad sortig take O() time each. Now, it suffices to show that the umber of times the repeat loop will be executed is O ~ (). The umber of alive keys at the ed of th iteratio is ~ 08. ON e log N, as per Lemma 4.. This, i tur, meas that the umber of alive keys at the ed of the first k iteratios is ~ (. O 08 k (log ) ). Thus, with high probability, there will be oly 4 iteratios of the repeat loop. 5 PACKET ROUTING Packet routig is a fudametal problem of parallel computtig, sice algorithms for packet routig ca be used as mechaisms for iterprocessor commuicatio. I this sectio, we preset efficiet algorithms for packet routig o the AROB. For the OCPC model several routig algorithms are kow: ) Aderso ad Miller have show that a special case of log -relatios o a -ode OCPC ca be routed i O ~ (log ) time []; ) Valiat exteded their algorithm to show that ay h- relatio ca be achieved i time Oh ~ ( + log ) [0]; ) Geréb-Graus ad Tsatilas have give a simple algorithm that ca route ay h-relatio i time ~ Oh ( + log log log ) [8]; 4) A ~ more complicated algorithm, with a ru time of Oh ( + loglog ), has bee give by Goldberg et al. [9]. There is a crucial differece betwee the OCPC model ad the AROB model. O the OCPC model, if more tha oe message is set to some processor p at the same time, oe of them reaches p. O the other had, uder the same sceario, oe of the messages will reach p o the AROB model. Also, operatios such as prefix sums (limited to itegers of certai magitude) ad compactio ca be performed i O() time o the AROB model ad ot o the OCPC model. Realize that a sigle step of a -ode OCPC ca be simulated o a AROB i O() cycles. Therefore, all the OCPC packet routig algorithms metioed above ca be used o a AROB without ay chage i the asymptotic ru times: LEMMA ~ 5.. Ay h-relatio ca be routed o a AROB i Oh ( + loglog ) cycles. A iterestig questio will be if there exists a better algorithm for the oe-dimesioal AROB. Cosider the simple problem, where there are two processors (o a -ode machie) that wat to sed a message to the same processor. If both of them attempt to trasmit at the same time, the o message will ever reach the destiatio o the OCPC. A radomized strategy could break the deadlock. For example, each processor ca attempt a trasmissio with some probability less tha oe (say, /). Such a algorithm ca be see to termiate i O ~ (log ) time. The same problem ca be solved i three cycles o a oe-dimesioal AROB. I this sectio, we demostrate the power of a oedimesioal AROB with efficiet algorithms for h-relatios. We also cosider the same problem o a two-dimesioal AROB. I particular, we preset the followig mai results: ) A Oh ~ ( ) time algorithm for h-relatios o a AROB; ) A Oh ~ ( + loglog ) time algorithm for a AROB that ca route arbitrary h-relatios; ad ) A O(h log ) time determiistic algorithm for ay h- relatios o a oe-dimesioal AROB, as well as a two-dimesioal AROB. 5. Routig o a AROB Let / be a AROB. We are iterested i routig a arbitrary h-relatio. Here we preset a algorithm that rus i time Oh. ~ ( ) Notice that a special case, where h = O(), has already bee cosidered i Lemma.6. We look at some special cases of routig before dealig with the geeral case. Problem [Load_Balacig(, k, N)]: I a -ode etwork, there are at most k packets at ay ode. Let N be the total umber of packets. The problem is to do a load balacig, i.e., to rearrage the packets, such that each ode has, at N most, packets. LEMMA 5.. Load_Balacig(, k, N) ca be solved i O(k) cycles o a AROB. The same problem ca also be solved i O(k) cycles o a AROB. F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 6 / 0

7 RAJASEKARAN AND SAHNI: SORTING, SELECTION, AND ROUTING ON THE ARRAY WITH RECONFIGURABLE OPTICAL BUSES 7 PROOF. We give the proof for a oe-dimesioal AROB. The same proof ca be exteded to a D AROB also. Let the processors order their packets from to k. Perform a prefix sums computatio for the first packets of all the processors (usig Lemma.) ad compute a uique address for each such packet. Route the first packets. Prefix takes O() cycles ad so does the routig. Likewise, process the secod packets, the third packets, ad so o. Oe should make sure, for example, that, if q is the umber of first packets, the the secod packets will be routed to odes startig from q +. Total time is clearly O(k). Problem [Route(,, k, )]: There are at most, packets origiatig from ay ode of a etwork. Also, at most k packets are destied for ay ode. Route the packets. LEMMA 5.. Route(,, k,) ca be solved o a AROB i O(k,) cycles. PROOF. The packets are routed i cycles. I ay cycle, a processor will choose oe of its remaiig packets (if ay) ad try to sed it. It may ot be successful i oe attempt. If it succeeds, it takes up the ext packet; otherwise, it will try to sed the same packet. A packet will ot reach its destiatio oly if there is a coflict. Thus, a packet ca meet with failure i at most k - cycles. This, i tur, meas that every processor will be able to trasmit all of its, packets i,k cycles or less. Now, we state ad prove our mai theorem: THEOREM 5.. Ay h-relatio ca be routed i Oh ~ ( ) time o a AROB. PROOF. We preset a radomized algorithm. The idea is for every processor to radomly choose oe of its packets ad sed it. At ay time step, there is some costat probability that a processor will succeed. The algorithm ca be thought of as ruig i stages. At the ed of every stage, we perform load balacig. Let k i be the maximum umber of packets i ay ode at the begiig of stage i, with k = h. Also, let N i be the umber of packets that have ot yet bee routed at the begiig of stage i, where N h. We ll show that, after every stage of routig, the value of k i decreases by a costat factor, with high probability. Thus, there will be O ~ (log h ) stages of routig. After ~ O(log h ) stages of routig, we also prove that there will be O ~ ( ) packets left. They ca be routed by load balacig, followed by a applicatio of Lemma.6. More details follow: Algorithm Route i = ; k i = h; N i = h; repeat for = to.5k i do Step. Each processor p does the followig: It picks oe of the remaiig packets uiformly, at radom, ad seds it i the bus. Thus, there is a probability of that ay packet from p will be q set, q beig the umber of remaiig packets. The packet set may or may ot successfully reach its destiatio (depedig o coflicts). Step. Reverse the directio of the above routig to iform the processors as to whether or ot their packets have bee successfully set. Perform load balacig such that every processor gets very early the same umber of remaiig packets; compute k i+ ad N i+ ; i = i + ; util N i c (c beig a costat). Aalysis. We ll prove by iductio that, at the begiig of stage i +, for almost all the processors, the umber of remaiig packets destied for ay processor is h i ad, also, that k i+ is o more tha h i. The base case is easy. Assume the hypothesis for all stages up to i. We e shall prove it for stage i +. Let p be ay processor. If at ay time durig stage i, the umber (call it m) of packets yet to be routed to p falls below k i, we are fie. Thus, assume that m is, at most, k i ad, at least, k i. At ay time durig stage i, the probability that at least oe of the packets destied for p will be set is -. This ca be proved as follows: Let the umber of packets left at processor q ad destied for p be x q, q =,, º,. Clearly, Â xq = m. Probability q= that oe of the x q packets at q will be set is x q. ki Therefore, probability that oe of these x q packets will be set is F xq H - I K - xq ki e, usig the fact that ki y e - y for ay y. Ad, hece, probability that e oe of the m packets destied for p will be set is c h. Thus, probability that at - x + x + K + x k - mk i i e = e least oe packet destied for p will be set is - - e mk i -. e The fact that at least oe of these m packets has bee set esures that oe packet will successfully reach p. Call this evet (of sedig at least oe packet destied for p) a success. Therefore, the expected umber of successes i.5k i steps of routig is =.5k i e- e. This umber is k i F H with probability - 4 k i I K (usig Cheroff bouds). Aother way of lookig at this is that the expected umber of processors that have ot received k i packets i this stage is k i 4. Expected total umber of such packets is k i. We could also show that the umber of such packets is ~ O F H k i I K. F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 7 / 0

8 8 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 I summary, for almost all the processors, the umber of remaiig packets destied for ay processor at the ed of stage i is k i. Sice the total umber of packets that ca ot be delivered (as expected) is O ~ F I H k i K, after load balacig, k i+ will be k i. Sice the value of k i decreases by a factor of two i every stage, there will be O ~ (log h ) stages. As we have see, the umber of packets that caot be routed i stage i is ~ O F I H k i K log h log ~ ~ ~ O d h F I F I Â O O i k i HG i KJ = Â i H G e h K J = af. = = d. The total of this over all the stages is Thus, it follows that, after O ~ (log h ) stages, the umber of packets remaiig to be routed is O. ~ ( ) They ca be routed usig a load balacig (i time O()), followed by a applicatio of Lemma 5.. The additioal time eeded is O(h). The amout of time spet i each stage ca be computed as the sum of routig time ad load balacig time. Routig takes O(k i ) time ad load balacig also takes O(k i ) time (c.f. Lemma 5.). Therefore, the total time spet i all the stages is dlog h dlog h ch F Â i = Â i i= i= H G I KJ = bg ~ Ok O h ~ Oh. 5. Routig o a Two-Dimesioal AROB Now, we cosider the problem of routig h-relatios o a two-dimesioal AROB of size. Whe it comes to off-lie routig (i this, the h-relatio is kow i advace ad we may precompute fuctios that ca be used for free whe realizig the h-relatio), a optimal algorithm is immediate from Hall s theorem (that says that ay h-relatio ca be decomposed ito h permutatios) ad the algorithm of Baumslag ad Aexstei []. LEMMA 5.4. Off-lie routig of h-relatios ca be performed i O(h) cycles o a AROB. PROOF. The off-lie permutatio routig algorithm of [] has three phases: Phase : Perform colum permutatios; Phase : Perform row permutatios; Phase : Perform colum permutatios. Sice each row ad colum of a two-dimesioal AROB is a oe-dimesioal AROB with processors, each of the above phases ca be doe i O() cycles (see Lemma.5). Hece, if the specific permutatio eeded for each phase is precomputed (actually, oly the phase permutatio eeds to be precomputed []), the permutatio ca be realized i O() cycles. Sice each h-relatio ca be decomposed ito h permutatios, the time eeded to realize the h-relatio is O(h). This does ot iclude the time to decompose the h-relatio or that eeded to precompute the h phase permutatios. The colum permutatios of phase are computed usig a bipartite graph costructio ad Hall s theorem for complete matchigs. These colum permutatios esure that, followig Phase, o two packets i the same row have the same destiatio colum. As a result, routig packets to their destiatio colums ca be doe usig row permutatios (Phase ). Followig Phase, all packets are i their destiatio colums. Hece, o two packets i the same colum ca have the same row as their destiatio. The colum permutatios of Phase, therefore, suffice to complete the packet routig. A (o-lie) algorithm that will realize ay h-relatio i ~ log Oh e + log log time has bee give i [7] for the OCPC model. It is ot clear if the algorithm of [9] ca be exteded to get a ru time of Oh ~ ( + loglog ) for the D OCPC. We show i this sectio that a arbitrary h-relatio ca be routed o a AROB i time Oh ~ ( + loglog ). Before cosiderig the geeral case, we look at the case of h =. LEMMA 5.5. O()-relatios ca be routed i O ~ (log log ) cycles o a D AROB. PROOF. Rao ad Tsatilas [7] provide a radomized routig algorithm with this complexity for the D OCPC model. This algorithm is a radomized versio of the off-lie algorithm of Lemma 5.4 ad ca be ru o a D AROB with o chage. Now we are ready to prove the mai result: THEOREM 5.. Ay h-relatio ca be routed i Oh ~ ( + loglog ) cycles o a AROB. PROOF. The algorithm to be used is similar to that of Route give for the oe-dimesioal AROB. The chages are: a) I Step, seds it o the bus is replaced by: Step.. Each processor selects a radom row idex. The processors i each colum attempt to route their selected packets to the radomly selected rows, usig the algorithm of Lemma.5. Sice the routes i each colum do ot ecessarily defie a permutatio, oly some of the packets get through. Step.. The packets that get through i Step. are routed alog rows to their destiatio colums, usig the permutatio routig scheme of Lemma.5. Agai, sice the destiatio colums i a row may ot form a partial permutatio, some packets may ot get through. Step.. The destiatio rows of the packets i each colum that got through i Step. form a partial permutatio. The partial permutatio i each colum may be realized usig the method of Lemma.5. b) The load balacig step is doe usig Lemma 5., as applied to a D AROB. We ote that Steps.,., ad. are similar to the three phase algorithm of []. F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 8 / 0

9 RAJASEKARAN AND SAHNI: SORTING, SELECTION, AND ROUTING ON THE ARRAY WITH RECONFIGURABLE OPTICAL BUSES 9 The aalysis of the algorithm is similar to that of algorithm Route. The value of k i decreases by a factor of two per iteratio of the for loop with high probability. Also, the total umber of packets that remai followig the adapted algorithm Route is O. ~ ( ) These packets ca be balaced at the ed (i time O(h)) ad routed i Oh ~ ( + loglog ) cycles (c.f. Lemma 5.5). The total umber of cycles spet i routig i the adapted algorithm Route is Oh ~ ( + loglog ). So, the overall umber of cycles is Oh ~ ( + loglog ). Determiistic Routig. We ca also perform determiistic routig i a efficiet maer o the AROB: LEMMA 5.6. Ay partial permutatio ca be routed i O(log ) time o a AROB. PROOF. Sort the packets ito ascedig order of destiatios. For this, the destiatios are mapped ito a sigle umber, usig the row maor mappig scheme. The sorted packets are i processors,, º (i row maor order). This sort ca be accomplished i O(log ) time usig a biary radix sort ad two applicatios of Lemma 5. to accomplish the sort o each bit (otice that the load balacig scheme of Lemma 5. is equivalet to a stable sort of bits). Followig the sort, o two packets i the same colum have the same row as their destiatio (as there are - packets betwee them). So, we may use Lemma.5 to route packets i each colum to their destiatio rows. Followig this, o two packets i the same row have the same colum as their destiatio. So, Lemma.5 may be used agai to route packets i the same row to their destiatio colums. The work doe followig the sort takes O() cycles. So, the overall umber of cycles is O(log ). A extesio of the above result ca also be prove: LEMMA 5.6. Ay h-relatio ca be routed i O(h log ) cycles o a D AROB of size. PROOF. Sort the packets ito odescedig order of destiatio. Followig the sort, the packets are i the first few processors (i row maor order), h packets to a processor. This is accomplished usig a biary radix sort o the row maor idex of the packet s destiatio processor. Whe sortig o bit k of this idex, we first cocetrate the packets with bit k equal to 0, h packets to a processor ad the cocetrate those with bit k equal to. The process for each bit value is similar. Cosider the case of packets with bit k equal to 0. Call these packets selected packets. A processor may have up to h selected packets. The selected packets i each processor are combied to form a superpacket of size at most h. The superpackets are compacted ito processors,, º (i row maor order) usig the D compactio algorithm of []. Sice the superpacket size is O(h), this takes O(h) time. The superpackets are ow decomposed ito the origial packets. The origial packets are to be further compacted so that we have h packets to a processor. Each packet i a processor is assiged a level umber correspodig to its order i the processor. Level umbers are i the rage to h. Prefix sums for the level i packets, i h are computed usig the D prefix sum algorithm, give i Sectio. The rak r(i, ) of a level i packet i processor is  ps( k, - ) + ( i -), where ps(k, - ) is h k= the prefix sum of the level k packet i processor -. The processor P(i, ) to which this packet is to be routed is Îr(i, )/h. Furthermore, this packet will be the roud(i, ) = r(i, ) mod h + -th packet i this processor. Sice the umber of packets i each row is at most h, o two packets (i, ) ad (k, l), where ad l are processors i the same row, have colum(p(i, )) = colum(p(k, l)) ad (row(p(i, )) π row(p(k, l)) or roud(i, ) = roud(k, l)). As a result, the compactio may be completed as below: Step : Perform h rouds of row permutatio routig o each row. I roud k, packets (i, ) with roud(i, ) = k are routed to the processor i colum colum(p(i, )). Step : Perform h rouds of colum permutatio routig. I roud k, packets (i, ) with roud(i, ) = k are routed to the processor i row row(p(i, )). The radix sort described above takes O(h log ) time. To complete the h-relatio, we perform h rouds of colum ad row permutatios. I roud i, the level i packets i each colum are first routed to the correct row usig a colum permutatio. There ca be o collisio as, for a collisio, the umber of packets destied to the same row eeds to be > h. Next, the level i packets are routed to the correct colum usig row permutatios. Agai, collisios are ot possible. 6 CONCLUSIONS I this paper, we have preseted efficiet algorithms for sortig, selectio, ad packet routig o the AROB. We have cosidered both iteger sortig ad geeral sortig problems. Our geeral sortig ad selectio algorithms are optimal. So is the h-relatios algorithm for the oedimesioal AROB. A iterestig ope problem is if there exists a O(h) routig algorithm (determiistic or radomized) for the D AROB. Also, ca our iteger sortig algorithm be improved? ACKNOWLEDGMENTS The research of Saguthevar Raasekara is supported i part by U.S. Natioal Sciece Foudatio Grat CCR The research of Sarta Sahi is supported i part by the U.S. Army Research Office uder Grat DAA H F:\LIBRARY\TRANS\PRODUCTION\TPDS\-INPROD\0456\0456_.DOC regularpaper97.dot KSM 9,968 09/0/97 :0 PM 9 / 0

10 0 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO., NOVEMBER 997 REFERENCES [] R.J. Aderso ad G.L. Miller, Optical Commuicatio for Poiter Based Algorithms, Techical Report CRI-88-4, Computer Sciece Dept., Uiv. of Souther Califoria, 988. [] M. Baumslag ad F. Aexstei, A Uified Framework for Off- Lie Permutatio Routig i Parallel Networks, Mathematical Systems Theory, vol. 4, pp. -5, 99. [] P. Beame ad J. Hastad, Optimal Bouds for Decisio Problems o the CRCW PRAM, J. ACM, vol. 6, o., pp , 989. [4] Y. Be-Asher, D. Peleg, R. Ramaswami, ad A. Schuster, The Power of Recofiguratio, J. Parallel ad Distributed Computig, pp. 9-5, 99. [5] R. Bopaa, A Lower Boud for Sortig o the Parallel Compariso Tree, Iformatio Processig Letters, 989. [6] H. Cheroff, A Measure of Asymptotic Efficiecy for Tests of a Hypothesis Based o the Sum of Observatios, Aals Math. Statistics, vol., pp. 4-56, 95. [7] R.W. Floyd ad R.L. Rivest, Expected Time Bouds for Selectio, Comm. ACM, vol. 8, o., pp. 65-7, 975. [8] M. Geréb-Graus ad T. Tsatilas, Efficiet Optical Commuicatio i Parallel Computers, Proc. Symp. Parallel Algorithms ad Architectures, pp. 4-48, 99. [9] L. Goldberg, M. Jerrum, T. Leighto, ad S. Rao, A Doubly- Logarithmic Commuicatio Algorithm for the Completely Coected Optical Commuicatio Parallel Computer, Proc. Symp. Parallel Algorithms ad Architectures, pp , 99. [0] E. Hao, P.D. McKezie, ad Q.F. Stout, Selectio o the Recofigurable Mesh, Proc. Frotiers of Massively Parallel Computatio, pp. 8-45, 99. [] E. Horowitz ad S. Sahi, Fudametals of Computer Algorithms. Computer Sciece Press, 978. [] J. Jag ad V.K. Prasaa, A Optimal Sortig Algorithm o Recofigurable Mesh, Proc. It l Parallel Processig Symp., pp. 0-7, 99. [] J. Jeq ad S. Sahi, Recofigurable Mesh Algorithms for Image Shrikig, Expadig, Clusterig, ad Template Matchig, Proc. It l Parallel Processig Symp., pp. 08-5, 99. [4] T. Leighto, Tight Bouds o the Complexity of Parallel Sortig, IEEE Tras. Computers, vol. 4, o. 4, pp , Apr [5] R. Li ad S. Olariu, Recofigurable Buses with Shift Switchig: Cocepts ad Applicatios, IEEE Tras. Parallel ad Distributed Systems, vol. 6, o., pp. 9-0, Ja [6] R. Li, S. Olariu, J.L. Schwie, ad J. Zhag, Sortig i O() Time o a Recofigurable Mesh of Size N N, Proc. Europea Workshop Parallel Computig, pp. 6-7, 99. [7] R.G. Melhem, D.M. Chiarulli, ad S.P. Levita, Space Multiplexig of Waveguides i Optically Itercoected Multiprocessor Systems, Computer J., vol., o. 4, pp. 6-69, 989. [8] R. Miller, V.K. Prasaa-Kumar, D. Reisis, ad Q.F. Stout, Meshes with Recofigurable Buses, IEEE Tras. Computers, vol. 4, pp , 99. [9] D. Nassimi ad S. Sahi, A Self-Routig Bees Network ad Parallel Permutatio Algorithms, IEEE Tras. Computers, vol. 0, o. 5, pp. -40, May 98. [0] M. Nigam ad S. Sahi, Sortig Numbers o Recofigurable Meshes with Buses, Proc. It l Parallel Processig Symp., pp. 74-8, 99. [] S. Olariu, J.L. Schwig, ad J. Zhag, Iteger Problems o Recofigurable Meshes, with Applicatios, Proc. 99 Allerto Cof., vol. 4, pp. 8-80, 99. [] Y. Pa, Order Statistics o Optically Itercoected Multiprocessor Systems, Proc. First It l Workshop Massively Parallel Processig Usig Optical Itercoectios, pp. 6-69, 994. [] S. Pavel ad S.G. Akl, Matrix Operatios Usig Arrays with Recofigurable Optical Buses, mauscript, 995. [4] S. Raasekara, Meshes with Fixed ad Recofigurable Buses: Packet Routig, Sortig ad Selectio, Proc. First A. Europea Symp. Algorithms, Lecture Notes i Computer Sciece, vol. 76, pp Spriger-Verlag, 99. [5] S. Raasekara, Sortig ad Selectio o Itercoectio Networks, Proc. DIMACS Workshop Itercoectio Networks ad Mappig ad Schedulig Parallel Computatio, 995. [6] S. Raasekara ad J.H. Reif, Derivatio of Radomized Sortig ad Selectio Algorithms, Parallel Algorithm Derivatio ad Program Trasformatio, R. Paige, J.H. Reif, ad R. Wachter, eds., pp Kluwer Academic, 99. [7] S. Rao ad T. Tsatilas, Optical Iterprocessor Commuicatio Protocols, Proc. Workshop Massively Parallel Processig Usig Optical Itercoectios, pp , 994. [8] S. Sahi, Data Maipulatio o the Distributed Memory Bus Computer, Parallel Processig Letters, 995. [9] R.K. Thiruchelva, J.L. Traha, ad R. Vaidyaatha, O the Power of Segmetig ad Fusig Buses, Proc. It l Parallel Processig Symp., pp. 79-8, 99. [0] L.G. Valiat, Geeral Purpose Parallel Architectures, Hadbook of Theoretical Computer Sciece, vol. A, J. va Leeuwe, ed. North Hollad, 990. [] L.G. Valiat ad G.J. Breber, Uiversal Schemes for Parallel Commuicatio, Proc. th A.ACM Symp. Theory of Computig, pp. 6-77, 98. Saguthevar Raasekara received the BE degree i electrical egieerig from the Idia Istitute of Sciece, Bagalore, Idia, i 98, ad the PhD degree i computer sciece from Harvard Uiversity i 988. He is curretly a associate professor i the Departmet of Computer ad Iformatio Sciece ad Egieerig at the Uiversity of Florida. His research iterests iclude parallel algorithms, radomized algorithms, computer simulatios, combiatorial optimizatio, ad computatioal geometry. He is a coauthor of the texts Computer Algorithms/C++ ad Computer Algorithms. He is a member of the IEEE Computer Society. Sarta Sahi received his BTech (electrical egieerig) degree from the Idia Istitute of Techology, Kapur, ad the MS ad PhD degrees i computer sciece from Corell Uiversity. He is a Uiversity of Florida Research Foudatio Professor of Computer ad Iformatio Sciece ad Egieerig. Dr. Sahi has published more tha 50 research papers ad has writte several texts. His research publicatios are o the desig ad aalysis of efficiet algorithms, parallel computig, itercoectio etworks, ad desig automatio. Dr. Sahi is a coauthor of the texts Fudametals of Data Structures, Fudametals of Data Structures i Pascal, Fudametals of Data Structures i C, Fudametals of Data Structures i C++, Fudametals of Computer Algorithms ad Hypercube Algorithms: With Applicatios to Image Processig ad Patter Recogitio ad the author of the texts Cocepts i Discrete Mathematics ad Software Developmet i Pascal. I 997, he was awarded the IEEE Taylor L. Booth Educatio Award for cotributios to computer sciece ad egieerig educatio i the areas of data structures, algorithms, ad parallel algorithms. Dr. Sahi is a coeditor of the Joural of Parallel ad Distributed Computig ad is o the editorial boards of IEEE Parallel ad Distributed Techology ad Computer Systems: Sciece ad Egieerig. He has served as program committee chair ad geeral chair ad bee a keyote speaker for may cofereces. He is a fellow of the IEEE, ACM, AAAS, ad Miesota Supercomputer Istitute. 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