Hexagonal Arrays for Fault-Tolerant Matrix Multiplication

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1 Filomt 9:9 (5), DOI.98/FIL59969M Pulished y Fulty of Sienes nd Mthemtis, University of Niš, Seri Aville t: Hexgonl Arrys for Fult-Tolernt Mtrix Multiplition Emin I. Milovnović, Igor Ž. Milovnović, Mile K. Stojčev Fulty of Eletronis Engineering, A. Medvedev 4, 8 Niš, Seri Astrt. This pper desries mthemtil proedure for designing hexgonl systoli rrys tht implement fult-tolernt mtrix multiplition. Fult-tolerne is hieved y introduing redundny t lgorithm level y defining three equivlent lgorithms with disjoint index spes. The essene of the proposed method is sed on mpping dt dependeny grph tht orresponds to the mtrix multiplition lgorithm, y n pproprite epimorphism, into grph with desired properties. Sine there is : orrespondene etween the lgorithm nd it s grph representtion, ll trnsformtions performed on the grph diretly ffet the lgorithm. Chosen epimorphism depends on the projetion diretion vetor µ [µ µ µ 3 T nd enles otining hexgonl rrys with optiml numer of proessing elements (PEs) for the given mtrix dimensions, whih relizes fult-tolernt mtrix multiplition for the shortest possile time for tht numer of PEs. The proposed proedure is formlly desried y expliit formuls nd n e used s softwre tool for utomti synthesis of fult-tolernt rrys.. Introdution Mtrix multiplition is one of the fundmentl opertion in mny sientifi nd tehnil pplitions. It ppers s mro-funtion within mny numeril nd non-numeril prolems. Typil exmples inlude solving prolems in liner lger, suh s finding inverse mtrix, determinnt, LU ftoriztion, eigenvlues, et., grph theory, suh s trnsitive losure, finding ll-pirs shortest pths, miniml spnning tree, et. Mtrix multiplition is very time onsuming sine it requires O(n 3 ) omputtionl steps. On the other hnd, it is highly regulr nd posses inherent prllelism nd is therefore suitle for implementtion on prllel rhitetures suh s proessor or systoli rrys. Nowdys, CMOS VLSI tehnology mde it possile to develop mssively prllel omputtionl systems. However, shrinking dimensions of trnsistors, whih ring lower operting voltges nd smller ritil hrges, mke these devies prone to vrious kind of fults. These fults n e tegorized into three types: trnsient, intermittent, nd permnent fults. Trnsient fults re filures in whih the iruit remins funtionl, ut the vlue it genertes is orrupted. Trnsient fults re used y strikes from energeti prtiles, suh s lph prtile strikes generted within the pkge of the hip, or neutron strikes used y osmi ry tivity in outer spe. Intermittent fults re used y the omintion of devie ging nd y stress reted y environmentl onditions during hip opertion [,. Permnent errors n Mthemtis Sujet Clssifition. Primry 68M7; Seondry 68Q35 Keywords. mtrix multiplition, fult-tolerne, systoli rrys. Reeived: 5 Septemer 3; Aepted: 3 Ferury 4 Communited y Drgn S. Djordjević Reserh supported y Serin Ministry of Edution nd Siene, Grnt No TR-3 nd TR-39. Emil ddresses: em@elfk.ni..rs (Emin I. Milovnović), igor@elfk.ni..rs (Igor Ž. Milovnović), mile.stojev@elfk.ni..rs (Mile K. Stojčev)

2 E. I. Milovnović et l. / Filomt 9:9 (5), our euse of mnufturing impreision during hip frition. When permnent fult is deteted, the logi lok must e isolted nd disled. Here we re interested in the tolertion of trnsient nd intermittent fults. To protet ginst these fults, some form of error detetion nd orretion pilities hve to e involved. A vriety of tehniques hve een proposed for hndling trnsient nd intermittent fults. Most of them re sed on some kind of redundny, ether informtion (suh s ABFT), hrdwre (suh s TMR), or time (suh s RESO). In this proposl, fult-tolerne is hieved through triplited omputtion followed y mjority voting, ut insted of using full hrdwre redundny (i.e. triplition) to hieve fult-tolerne, we generte redundnt results prtilly y mens of hrdwre (y dding two extr olumns of proessing elements in the systoli rry) nd prtilly y mens of time redundny y slightly inresing omputtion time. We re interested in the designing of hexgonl systoli rrys with the optiml numer of PEs with respet to the prolem size whih implement fult-tolernt mtrix multiplition. In order to redue the numer of PEs nd the exeution time, we use the ft tht during eh omputtionl step, mtrix multiplition n e performed in ny order without modifying the finl result. By involving two dditionl olumns of PEs nd the reordering of omputtionl steps, we hieve tht our hexgonl rrys hve Ω N 3 (min{n, N } + ) PEs nd perform fult-tolernt mtrix multiplition for T tot 3 mx{n, N } + min{n, N } + N 3 omputtionl steps (N, N nd N 3 re dimensions of mtries, nmely A ( ik ) N N 3, B ( kj ) N3 N nd C ( ij ) N N ). The rest of the pper is orgnized s follows. Setion desries the prolem of interest whih will e onsidered. Mthemtil kground tht serves s the sis for the synthesis of fult-tolernt systoli rrys is presented in Setion 3. Setion 4 dels with the proedure for the synthesis of systoli rrys with optiml numer of proessing elements. For tht numer of PEs the exeution time is minimized. Setion 5 onentrtes on determining initil dt rrngement tht provides orret implementtion of fult-tolernt mtrix multiplition lgorithm. Performne evlution of the synthesized rrys whih reltes to the numer of PEs nd exeution time re given in Setion 6. Some onluding remrks re given in Setion 7.. Prolem Definition Suppose A ( ik ) nd B ( kj ) re retngulr mtries of order N N 3 nd N 3 N, respetively. Their produt, C A B, C ( ij ), n e omputed ording to the following reurrent reltions () ij : (k) ij : (k ) ij + ik kj, k,,..., N 3 () ij : (N 3) ij for i,,..., N nd j,,..., N. The following systoli lgorithm n e used to ompute () Algorithm for k : to N 3 do for j : to N do for i : to N do (i, j, k) : (i, j, k); (i, j, k) : (i, j, k); (i, j, k) (i, j, k ) + (i, j, k) (i, j, k); with the following identities (i,, k) ik, (, j, k) kj, (i, j, k) (k), (i, j, ). () ij Note tht omputtions in Algorithm, i.e. eqn. (), with respet to indies i, j nd k, re performed t si permuttions of the sets {,,..., N }, {,,..., N }, nd {,,..., N 3 }, respetively. However, this is

3 E. I. Milovnović et l. / Filomt 9:9 (5), not neessry. The omputtions n e performed t ritrry permuttions of the forementioned sets (see [8). We ll this property permutility. Direted oordinted lttie grph G (P int P in, D) n e joined to the Algorithm. This grph is pled in three-dimensionl Crtesin spe generted y the unity vetors e [ T, e [ T, e 3 [ T, (3) Set P int { p [i j k T } defines verties of grph G where tive omputtions from Algorithm re performed, while P in P in () P in () P in (), where P in () { p [i k T }, P in () { p [ j k T } nd P in () { p [i j T } re verties where initil elements of mtries A, B nd C () re pled, respetively. Direted edges in grph G re defined y the olumn vetors of mtrix D D [ e 3 e 3 e 3 (4) These vetors lso define propgtion diretion of elements of mtries B, A nd C (), respetively. Systoli rry synthesis proedure, employed in this pper, is sed on mpping the grph G on the plin orthogonl to the projetion vetor µ [µ µ µ 3 T (see [5). By this mpping direted pseudo-grph in two-dimensionl Crtesin spe is otined. This grph orresponds to the desired SA. Here we re interested in the synthesis of two-dimensionl hexgonl systoli rrys (D HSA), suitle for fult-tolernt mtrix multiplition. The first ondition for SA to e suitle for fult-tolernt omputtions is tht it relizes Algorithm with pipeline period λ 3. Pipeline period is defined s time intervl etween two suessive omputtions in proessing element (see [9,, ). Hexgonl rry otined ording to the diretion µ [ T stisfies this ondition nd it is well studied in the literture (see [7, ). Here we re interested in hexgonl rrys otined for the diretions µ [ T, µ [ T nd µ [ T. These rrys relize Algorithm with pipeline period λ, nd therefore were not onsidered s suitle ndidte rhitetures for hieving fult-tolerne. However, these rrys hve shorter exeution time of mtrix multiplition lgorithm thn the rry otined y the projetion vetor µ [ T. We re going to show tht the fult-tolernt mtrix multiplition n e effiiently implemented on these rrys s well, with shorter exeution time thn tht of the one otined y the projetion vetor µ [ T. In this proposl, without the loss of ny generlity, we re going to synthesize D HSA otined y the projetion diretion vetor µ [ T, whih implements fult-tolernt mtrix multiplition lgorithm. The other two rrys n e synthesized y the sme methodology. Aprt from eing suitle for fulttolernt omputtion, the synthesized rry should hve optiml (i.e. miniml) numer of proessing elements (PEs) for the given mtrix dimensions. Also, the exeution time of the rry should e s smll s possile for the given numer of PEs. This requires sustntil modifition of the systoli rry synthesis proedure desried so fr in the literture [ Mthemtil Bkground Results otined in this setion serve s kground for the synthesis proedure whih will e desried in the next one. We strt our nlysis from the direted grph G (P int P in, D) whih orresponds to the Algorithm. Depending on the reltion etween N nd N, we mp verties of grph G, defined y set P int, either into set P () int or P () int. This is performed y the isomorphisms ϕ or ϕ, defined y the following equlities ϕ ( p [ i j k T ) [ 3i j k T nd (5) ϕ ( p [ i j k T ) [ i 3j k T,

4 E. I. Milovnović et l. / Filomt 9:9 (5), for eh p P int. In this wy we hve otined new direted grphs G () (P () int, D) nd G () ( P () int, D). Now, for the projetion vetor µ [ T we define two mppings H (M, F ) nd H (M, F ) defined y P () int M [ µ e e 3, F 3N 3 nd (6) M [ e µ e 3, F. 3N 3 We use mppings H nd H to otin sets P () int { p [u v w T } nd ˆP () int { p [u v w T } from P (), respetively. These sets re otined ording to the following equlities p u v w M p () + F 3i 3i + j 3 3i + k + 3N 3i j k +, 3N 3 int nd nd (7) u i p v M p () + F 3j + w k 3N 3 i + 3j 3 3j 3j + k + 3N, for i,,..., N, j,,..., N, nd k,,..., N 3. Thus we hve otined two direted grphs G () ( P (), D) nd G() ( ˆP (), D) defined y the following int int index sets { P () int p [ 3i 3i + j 3 3i + k + 3N } T nd ˆP () int { p [ i + 3j 3 3j 3j + k + 3N T }. Some importnt fetures of mppings H nd H nd orresponding direted grphs G ( ( P (), D) nd int ( ˆP (), D), re given in the susequent theorems. G () int Theorem 3.. Mtries M i, i, re nonsingulr nd mppings H i, i,, re injetive. Proof The first prt of the ssertion of Theorem 3. diretly follows from the equlity det M i µ i, i,. The seond prt of the ssertion will e proved y the ontrdition on the exmple of mpping H. Suppose, ontrry to the ssertion of Theorem 3., tht there re t lest two different nodes of grph G () tht re mpped y H into the sme node of grph Ḡ (). Let p () [3i 3i + j 3 3i +k +3N T nd p () [3i 3i + j 3 3i + k + 3N T, p () p (), e position vetors of these nodes. In tht se, ording to (), the following would e vlid M p () + F M p () + F. This implies tht 3i 3i, (8) 3i + j 3 3i + j 3, (9) 3i + k + 3N 3i + k + 3N. ()

5 E. I. Milovnović et l. / Filomt 9:9 (5), From (8) it follows i i. Aordingly, from (9) nd () it follows j j nd k k, so we hve p () p (), whih is in ontrdition with the ssumption tht p () p (). On the sis of Theorem 3. we onlude tht grphs G () nd G () (i.e. G () nd G () ) hve equl numer of nodes whih re differently rrnged in the oordinted spe. Sine M D M D (i.e M D M D), mppings H nd H do not preserve djeny of verties from G (). Theorem 3.. Let l e n ritrry line prllel to µ [ T. If line l ontins t lest one point (vertex) of grph G () (i.e. G () ), then it ontins extly N, (i.e. N ), verties of this grph. Proof Without loss of generlity we will ondut the proof for the grph G (). Suppose p () [ [3i 3i + j 3 3i + k + 3N T nd p () [ 3i + 3i + j 3i + k + 3N 4 T re position vetors of two verties from grph G (). Sine p () p () [ T 3 µ, we onlude tht these verties re pled on the line l prllel to the diretion vetor µ [. By tking, one fter the other, i,,..., N, we onlude tht there re extly N verties of G () on this line. Corollry 3.3. Let L {l} e set of ll lines tht stisfy the onditions of Theorem 3.. Then the intersetion of these lines with some plin not prllel to l, hs extly N N 3 (i.e. N N 3 ) points. Corollry 3.4. Let L {l} e set of ll lines tht stisfy the onditions of Theorem 3.. Depending on the reltion etween N nd N nd y the pproprite hoie of mpping H i, i,, it n e hieved tht intersetion of these lines with some plne not prllel to l, hs extly N 3 min{n, N } points. Note tht in prtie plne tht is orthogonl to line l is usully tken. Aording to Theorem 3. nd Corollries 3.3 nd 3.4 one n get geometril ide of mpping G () (i.e. G () ) into two-dimensionl direted grph Γ (V, ) (i.e. Γ (V, ) whih orresponds to the respetive D SA. However we need expliit nlytil proedure for otining grph Γ (i.e. Γ ). First we define epimorphism S whih is joined to the projetion vetor µ [ T. It mps threedimensionl direted grph G () (i.e. G () ) into two-dimensionl direted grph Γ (V, ) (i.e. Γ (V, ). Epimorphism S is 3 mtrix of the form S [ S T S T [ s s s 3 s s s 3, () where S T nd S T re vetors defining plne orthogonl to the projetion vetor µ [ T. These vetors hve to onform the following onstrints [5:. Sine µ [ T is orthogonl to S T nd S T it follows tht S T µ S T µ, i.e. s + s s 3 nd s + s s 3. (). In order to void mpping 3D grph G () (i.e. G () ) into D grph, the rnk of mtrix S must e, i.e. rnks. 3. To preserve the djeny of nodes tht were present in grph G (), elements of mtrix S n tke vlues only from the set {,, }. Conditions () to (3) do not neessrily provide one of the ruil requirements, whih is tht grph Γ (i.e. Γ ) hs N N 3 (i.e. N N 3 ) verties. The next theorem gives neessry nd suffiient onditions for epimorphism S so tht this requirement is lwys fulfilled.

6 E. I. Milovnović et l. / Filomt 9:9 (5), Theorem 3.5. Neessry nd suffiient onditions tht direted grph Γ (V, ) hs V N N 3 verties re, prt from ()-(3), the following ones or s s 3 nd s s 3, s s 3, nd s s 3. Proof Let p () [3i j k T e position vetor of ritrry point from P (). Then we hve tht int [ (S M ) p () s j + s 3 k. s j + s 3 k (3) In order to hve extly N N 3 ordered pirs of the form (s j + s 3 k, s j + s 3 k), neessry nd suffiient onditions re the ones given y (3). Similrly, the following theorem n e proved. Theorem 3.6. Neessry nd suffiient onditions tht direted grph Γ (V, ) hs V N N 3 verties re, prt from ()-(3), the following ones or s s 3 nd s s 3, s s 3, nd s s 3. Note tht epimorphism S n not e uniquely determined. For exmple, if S [ S T S T T is n epimorphism, so is S [ S T S T T. Therefore, without loss of generlity, ording to Theorems 3.5 nd 3.6, nd onditions ()-(3), we n tke the following epimorphisms [ [ S, S (4) respetively. 4. Systoli Arry Synthesis Bsed on the results otined in the previous setion, we re going to synthesize D HSA tht relizes fult-tolernt mtrix multiplition. Sine fult tolerne is hieved y triplited omputtion of the sme prolem instne followed y mjority voting, we first hve to derive three mtrix multiplition lgorithms equivlent to Algorithm. To do tht, we perform the shifting of verties in the direted grph G () (P () int, D) (i. e. G () ( P () int, D) ) long the oordinte xes to otin two dditionl grphs G (r) (P (r) int, D), (i.e. G (r) ( P (r), D), r, with int the following set of verties P (r) int { p [3i r j + r kt } ( i.e. P (r) int { p [i + r 3j r kt } ) Note tht this shifting is not unique. The only thing we should tke re of is tht sets P (r) int, (i.e. P (r) int ), r,, re mutully disjoint. If we pply mpping H (i.e. H ), defined y (7), on the set of verties of G (r) (i.e. G (r) ), r,, we otin direted grphs defined y the following set of verties (i.e. P (r) int { p [3i r 3i + j 3 3i + k + r + 3N T } ˆP (r) int { p [3j + i 3 3j r 3j + k + r + 3N T })

7 E. I. Milovnović et l. / Filomt 9:9 (5), for r,. Now we re le to define systoli lgorithms tht orrespond to grphs G (r) Algorithm for r : to do for k : to N 3 do for j : to N do for i : to N do u : 3i r; v : 3i + j 3; w : 3i + k + r + 3N ; (u, v, w) : (u, v, w); (u, v, w) : (u, v, w) (u, v, w) (u, v, w ) + (u, v, w) (u, v, w); i.e. Algorithm 3 for r : to do for k : to N 3 do for j : to N do for i : to N do u : i + 3j 3; v : 3j r; w : 3j + k + r + 3N ; (u, v, w) : (u, v, w); (u, v, w) : (u, v, w) (u, v, w) (u, v, w ) + (u, v, w) (u, v, w); (i.e. G(r) ), r,, Denote with Γ (V (r), ) (i.e. ˆΓ (V (r), ) imge of G (r) (i.e. G(r) ), r,,, otined y epimorphism S (i.e. S ). This imge is loted in the D rtesin spe nd orresponds to the D HSA whih implements the fult-tolernt mtrix multiplition lgorithm - Algorithm (i.e. Algorithm 3). Denote the orresponding rry y SA (i.e. SA). Then (x, y) lotions of the proessing elements in the SA rry (i.e verties of Γ ) re determined from the following equlities [ [ 3i r [ x k + PE S y p 3i + j 3 3N 3, (5) j r 3i + k + r + 3N for r,,, j,,..., N nd k,,..., N 3. Direted edges in Γ, whih orrespond to ommunition hnnels in SA, re determined y olumn vetors of mtrix [ [ e e e S D. (6) Similrly, for the rry SA we otin tht PE lotions re determined y [ [ 3j + i 3 [ x k + PE S y p 3j r 3N 3, (7) i r 3j + k + r + 3N for r,,, i,,..., N nd k,,..., N 3, nd ommunition hnnels y the olumn vetors of mtrix [ [ e e e S D. (8)

8 5. Otining Initil Dt Arrngement E. I. Milovnović et l. / Filomt 9:9 (5), So fr we hve otined lotions of the PEs in the x-y plne nd diretions of ommunition hnnels etween them. Now we hve to determine initil dt rrngement tht provides orret omputtion of the fult-tolernt mtrix multiplition lgorithm on the systoli rry. For the ske of revity we will do this only for the rry SA. From Algorithm, i.e. orresponding direted grphs G (r), initil dt rrngement of elements of mtries A, B nd C () in three-dimensionl spe is defined y the set P (r) () P(r) () P(r) (), where P (r) in () { p [ 3i r 3i + k + r + 3N T } P (r) in () { p [ 3i + j 3 3i + k + r + 3N T } P (r) in () { p [ 3i r 3i + j 3 T } in P(r) in for r,,, i,,..., N, j,,..., N nd k,,..., N 3. However this dt rrngement does not provide orret exeution of Algorithm on the rry SA. This is illustrted in the Figure. To simplify, onsider direted grph with four verties, P to P4, whih elong to the internl omputtion spe, P int, where MAC (multiply with umultion) opertion is performed. With,,, nd d input dt elonging to the initil omputtion spe, P in re denoted. For these inputs the following produts, d, nd d should e otined. However, we will otin only two produts, nmely nd d, sine dt re pipelined through the rry (Fig. )). Therefore we hve to rerrnge the spe of initil dt. In essene it is neessry to skew input dt in time s illustrted in Fig. ). in in INPUT: expeted OUTPT:,,, d, d,, d efore skewing P d P step PE P P P3 P4 d d ) P3 P4 d fter skewing ) P P3 P P4 step PE 3 P P P3 P4 d d Figure : Illustrtion of omputtion in the rry ) efore skewing input dt ) fter skewing input dt. Skewing of P (r) in is performed y the timing funtion, t( p) Π T p + α (see [6). This funtion defines time instne when the omputtion in the point p P (r) int is performed. So we hve to determine vetor Π nd onstnt α. Vetor Π [t t t 3 T my e ny vetor orthogonl on the plne tht ontins ll verties of grph G (r)), for some onstnt r, where omputtions from Algorithm re performed t the sme time instne. To nrrow down hoies, the following onstrints for elements of vetor Π re introdued:

9 E. I. Milovnović et l. / Filomt 9:9 (5), t + t + t 3 > ; The norm of Π, Π t + t + t 3, t ij Z, should e s smll s possile, nd Π T e 3 >, Π T e 3 >, Π T e 3 >, or Π T e 3 <, Π T e 3 <, Π T e 3 < Suppose tht omputtion in Algorithm is performed t the vertex p P (r), of grph G(r), for some onstnt int r, with the position vetor p [ 3i r 3i + j 3 3i + k + r + 3N T. Immeditely prior to the omputtion t p, the omputtions in three points of P (r) int given y the following position vetors, p p e 3, p p e 3 nd p p e 3, were performed. These points determine the plin, i.e. they re not olliner. Vetor orthogonl to this plne is ny vetor prllel to vetor Π T ( p p ) T ( p 3 p ) T ( e 3 e 3 ) T ( e 3 e 3 ) T i j k [. Hving in mind the ove mentioned onstrints for the elements of vetor Π, we n tke Π [ T. Constnt α is determined y the pproprite hoie of vertex in grph G (r) where the first omputtion, defined y Algorithm, should our. Let it e vertex defined y position vetor p [u v w T, where i j k. From the onditions t( p ) Π T p + α nd t( p) >, for eh p P (r) int, p p, we otin α 3N +, so we hve tht timing funtion is defined s Rerrngement (i.e. skewing) of P (r) in t( p) u + v + w 3N +. is performed y the following mpping p γ p γ (t( p γ ) + ) e 3 γ, γ {,, }, for eh p γ from P (r) in nd vetors e γ, 3 γ {,, } defined y (4). The rerrnged spe of initil omputtions P (r) in P (r) in () P (r) in () P (r) () is defined y in P (r) in () { p [ 3i r k 3i + k + r + 3N T } P (r) in () { p [ j k r 3i + j 3 3i + k + r + 3N T } P (r) in () { p [ 3i r 3i + j r + 3N 6i j T }. Finlly, y mpping the set P (r) using epimorphism S defined in (4), initil dt rrngement of input in elements A, B nd C () in the (x, y)-plne t the eginning of the omputtion in the rry SA is otined. The rrngement of input dt in the (x, y)-plne is given y [ [ x 3i + k r 3 (i,, i + k + r + N 3) y k + 3N 3 [ [ x 5 3i j k r (, i + j, i + k + r + N 3) (9) (i, i + j, ) y [ x y 3i j + 3N [ r j, 3i j + 3N for r,,, i,,..., N, j,,..., N nd k,,..., N 3. We ssume tht the following identities in (9) re vlid: (i,, t + N 3 ) (i,, t), (, m + N, t + N 3 ) (, m, t), (i, m + N, ) (i, m, ) for eh m,,..., N nd t,,..., N 3. An exmple of the SA rry tht implements fult-tolernt mtrix

10 E. I. Milovnović et l. / Filomt 9:9 (5), lgorithm for the se N 4, N 3 nd N 3 is presented in Fig.. Dshed squres represent dditionl proessing elements whih represent hrdwre overhed due to fult-tolerne Figure : Fult-tolernt mtrix multiplition on the rry SA for the se N 4, N 3 nd N 3. By the similr proedure one n otin initil dt rrngement of input elements A, B nd C () in the (x, y)-plne t the eginning of the omputtion in the rry SA. It is given y (i + j,, j + k + r + N 3) (, j, j + k + r + N 3) (i + j, j, ) [ [ x 5 i 3j k r y i 3j k + 3N [ [ x 3j + k r 3 y k + 3N 3 [ [ x i r y i 3j + 3N for r,,, i,,..., N, j,,..., N nd k,,..., N 3. An exmple of the rry SA tht implements fult-tolernt mtrix lgorithm for the se N 4, N 3 nd N 3 is presented in Fig.. As n e seen from Figs. nd 3 the rry SA hs fewer PEs thn SA sine N > N. ()

11 E. I. Milovnović et l. / Filomt 9:9 (5), Figure 3: Fult-tolernt mtrix multiplition on the rry SA for the se N 4, N 3 nd N 3. A shemti of the voting logi is presented in Fig. 4. It onsists of 3 [ min{n,n }+ 3 three-input multiplexors nd [ min{n,n } 3 mjority voters. Selet signls for ll multiplexors re ommon. The inputs re seleted in irulr mnner. In eh yle the mjority voter tkes three inputs nd genertes the result. More detils onerning the mjority voter n e found in [. voted vlue voted vlue Voter Voter multiplexors MUX MUX MUX MUX MUX MUX selet signls output from PE PE PE PE3 PE PE PE4 PE3 PE PE PE PE n- n- n-3 PEn PEn- PEn- PEn+ PEn PEn- Figure 4: A detil of the voting logi. By the proposed sheme single trnsient nd intermittent fults n e orreted. A numer of multiple fult ptterns n e tolerted lso, if they do not ffet the sme element of the resulting mtrix. Fult detetion nd lotion re not neessry for fult-tolerne, errors re msked onurrently with norml opertion of the systoli rry. 6. Disussion Aording to (5) nd (7) the following result n e proved. Theorem 6.. Hexgonl systoli rry otined y the projetion vetor µ [ T tht implements Algorithm (i.e. Algorithm 3) onsists of Ω N 3 (N + ) ( i.e. Ω N 3 (N + )) () proessing elements.

12 E. I. Milovnović et l. / Filomt 9:9 (5), Aording to () it follows tht optiml numer of PEs in the SA depends on mutul reltion etween N nd N. This lso mens tht this reltion diretly determines whether Algorithm or Algorithm 3 should e used. Corollry 6.. Hexgonl systoli rry synthesized for projetion vetor µ [ T tht implements Algorithm, i.e. Algorithm 3, hs Ω N 3 (min{n, N } + ) () proessing elements. Note tht hexgonl SA synthesized for the diretion µ [ T tht implements fult-tolernt mtrix multiplition otined in [7 hs the sme numer of PEs. Totl exeution time, T tot, of mtrix multiplition lgorithm on systoli rry inludes time to lod dt into the rry, T in, tive exeution time, T exe, nd time to drin the rry, T out, i.e. T tot T in + T exe + T out. If G (P, E) is direted grph tht orresponds to n lgorithm, nd t( p) is timing funtion, then tive exeution time n e determined from the eqution T exe mx t( p) min t( p) +, p P r [. In our se int the following result holds. Theorem 6.3. Totl exeution time of Algorithm (i.e. Algorithm 3) on the systoli rry with optiml numer of PEs otined for the diretion µ [ T is T tot 3 mx{n, N } + min{n, N } + N 3. (3) Proof Timing funtion of the rry SA tht implements Algorithm is t( p) u + v + w 3N +. Consequently, tive exeution time is T exe 3N +N +N 3 4. In [4 hexgonl SA with optiml numer of PEs whih omputes mtrix produt ws synthesized for the diretion µ [ T. It onsists of N 3 N PEs, nd hs totl exeution time T tot T exe, i.e. T in T out. The rry SA synthesized in this pper represents n extension of the one otined in [4 for N 3 PEs. Nmely, two PEs re dded in eh row of PEs. Sine elements of mtrix A propgte in tht diretion, the initiliztion time is inresed for two time units, i.e. T in, while T out remins. Aordingly, we hve tht totl exeution time of Algorithm on the rry SA, synthesized y the equtions (5), (6) nd (9), is T tot T in + T exe + T out 3N + N + N 3. (4) Similrly, it n e onluded tht totl exeution time of Algorithm 3 on the rry SA, synthesized y equtions (7), (8) nd (), is T tot N + 3N + N 3. (5) Now, sttement of the theorem diretly follows from the equtions (4) nd (5). Hexgonl rry synthesized in [7 hs totl exeution time equl to T tot 3 mx{n, N }+ min{n, N }+ N 3 3. This mens tht the rry synthesized in this pper hs shorter exeution time for T tot min{n, N } time units whih is hieved y the sme numer of PEs s in [7. 7. Conlusion Systoli rrys re suitle elertor rhitetures for mtrix multiplition. They re omplex CMOS VLSI iruits tht exploit mssive prllelism t dt level. However, shrinking the dimensions of trnsistors mke this iruits prone to vrious kind of fults. Relile opertion of these iruits n e hieved y some kind of redundny, whih in generl involves dditionl hrdwre or time, tht s onsequene

13 E. I. Milovnović et l. / Filomt 9:9 (5), dereses system throughput. Therefore the min design hllenges re oriented towrd deresing hrdwre omplexity nd inresing the throughput. In this pper we desrie systemti proedure for the designing of D hexgonl rrys tht n tolerte single trnsient nd intermittent fults during mtrix multiplition (MM). In this proposl we strt from the hexgonl rry whih implement MM lgorithm with pipeline period λ, nd y involving pproprite modifitions t lgorithm level, otin the rry with λ 3. Then two redundnt omputtions re introdued so tht fult-tolerne is hieved y triplited omputtion followed y mjority voting. This is hieved with miniml hrdwre nd time overhed. The proposed proedure is formlly desried y expliit formuls nd n e used s softwre tool for utomti synthesis of fult-tolernt rrys. Referenes [ C. Constntinesu, Trends nd hllenges in VLSI iruit reliility, IEEE MIro, Vol. 3, 4 (3), pp [ M. Duois, M. Annvrm, P. Stenstrom, Prllel omputer orgniztion nd design, Cmridge University Prss, New York,. [3 J. A. B. Fortes, K. -S. Fu, B. W. Wh, Systemti design pprohes for lgorithmilly speified systoli rrys, In: Computer Arhiteture: Conepts nd Systems, (V. Milutinović, ed.), Elsevier Si. Pul. Co., New York, 988. [4 I. Ž. Milovnović, M. P. Bekkos, I. N. Tselepis, E. I. Milovnović, Forty-three wys of systoli mtrix multiplition, Inter. J. Comput. Mth., Vol. 87, 6 (), pp [5 S. G. Sedukhin, The designing nd nlysis of systoli lgorithms nd strutures, Progrmming, Vol., (99), pp. -4. (In Russin) [6 S. G. Sedukhin, G. Z. Krpetin, Design of optiml systoli systems for mtrix multiplition of different strutures, Report 85, Comput. Center Sierin Division of USSR Ademy of Siene, Novosiirsk, 99. (In Russin). [7 N. M. Stojnović, E. I. Milovnović, I. Stojmenović, I. Ž. Milovnović, T. I. Tokić, Mpping mtrix multiplition lgorithm onto fult-tolernt systoli rry, Comput. Mth. Appl., Vol. 48, - (4), pp [8 J. -C. Tsy, P. -Y. Chng, Some new designs of D rry for mtrix multiplition nd trnsitive losure, IEEE Trns. Prll. Distri. systems, Vol. 6, 4 (995), pp [9 C. R. Wn, D. J. Evns, Nineteen wys of systoli mtrix multiplition, Inter. J. Comput. Mth., Vol. 6, - (998), pp [ J. H. Weston, C. N. Zhng, H. Li, Some spe onsidertions on VLSI systoli rry mppings, IEEE Trns. Ciruits Systems II, Vol. 48, 4 (), pp [ C. N. Zhng, Systemti design of systoli rrys for omputing multiple prolem instne, Miroeletronis J., Vol. 3, 7 (99), pp [ C. N. Zhng, T. M. Bhtir, W. K. Chou, Optiml fult-tolernt design pproh for VLSI rry proessors, IEE Pro. Comput. Digit. Teh., Vol. 44, (997), pp. 5-. [3 C. N. Zhng, J. H. Weston, Y, -F. Yn, Determining ojetive funtions in systoli rry design, IEEE Trns. Very Lrge Sle Integrtion (VLSI) Systems, Vol., 3 (994), pp