Efficient Synthesis of Networks On Chip

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1 Efficiet Sythesis of Networks O Chip Alessadro Pito Luca P. Carloi Alberto L. Sagiovai-Vicetelli EECS Departmet, Uiversity of Califoria at Berkeley, Berkeley, CA Abstract We propose a efficiet heuristic for the costrait-drive commuicatio sythesis (CDCS) of o-chip commuicatio etworks. The complexity of the sythesis problems comes from the umber of costraits that have to be cosidered. I this paper we propose to cluster costraits to reduce the umber that eeds to be cosidered by the optimizatio algorithm. The a quadratic programmig approach is used to solve the commuicatio sythesis problem with the clustered costraits. We provide a aalytical model that justifies our choice of the clusterig cost fuctio ad we discuss a set of experimets showig the effectiveess of the overall approach with respect to the exact algorithm.. Itroductio The cotiued growth of the umber of processors ad IP cores that are itegrated o a sigle die [] together with the shift from computatio-boud desig to commuicatio-boud desig [6] lead may researchers to advocate ew desig methodologies to orgaize systematically commuicatio architectures for System-o-Chip (SOC). I [9], we proposed Costrait-Drive Commuicatio Sythesis (CDCS) as a formal method for derivig automatically the implemetatio of a commuicatio etwork of a system from a high-level specificatio: the resultig etwork is a compositio of basic elemets that are istaces take from a library of pre-defied Itellectual Property (IP) commuicatio compoets, such as wires, repeaters, ad switches. I CDCS, the essetial commuicatio requiremets that gover all the poit-to-poit commuicatios amog the system modules are captured as a set of arc costraits i a graph called commuicatio costrait graph. Similarly, the commuicatio features offered by each of the compoets available i the IP commuicatio library are captured as a set of feature resources together with their cost figures. The, every commuicatio architecture that ca be built from the available compoets while satisfyig all costraits is implicitly ecoded as a implemetatio graph matchig the costrait graph. Fially, the optimum etwork is foud by solvig a costraied optimizatio problem with a exact algorithm. Ufortuately, eve though some theoretical results eable the reductio of the size of the search space [9], the computatioal complexity of the exact algorithm udermies its scalability, ad, ultimately, the applicability of the approach to the sythesis of fairly large o-chip etworks. This research was supported i part by the GSRC ad the SRC. I this paper we preset two efficiet heuristics for CDCS that are based o a decompositio of the optimizatio problem ito two steps: () we use quadratic programmig to compute quickly the cost of satisfyig a set of arc costraits with a shared commuicatio medium, ad () we propose two clusterig algorithms to sigle out sets of costraits that should be cosidered together. After providig some backgroud material o CDCS, we discuss how the exact algorithm of [9] suffers from scalability issues ad we summarize the mai ideas of our heuristic approach (Sectio ). I Sectio 3, we start from the assumptio that the cost fuctio of a arc implemetatio is a cocave fuctio ad we show how this eables the efficiet computatio of the cost of implemetig arc costraits with a shared commuicatio medium. The, we preset two distict heuristics for clusterig of costraits: divisive clusterig ad agglomerative clusterig (Sectio 4). Fially, we report o a set of experimets showig the effectiveess of our approach (Sectio 5). Related Work. I [], Beii ad De Micheli preset etwork o chip (NOC) as a ew paradigm for SOC desig based o a approach similar to the micro-etwork stack model []. They discuss the desig problems ad possible solutios for each level of the stack from the applicatio level to the physical level through the topology ad protocol levels. The stadard solutio of the topology selectio problem is the use of a sigle bus, but this may tur out quite iefficiet from a power cosumptio viewpoit. Istead, [] suggests to use packet-switchig architectures. They focus o providig some examples of kow topologies ad do ot discuss the problem of selectig a optimum oe. A methodology cetered o the simulatio of traces is proposed i [7]: the resultig commuicatio architecture is a itercoectio of well-characterized commuicatio structures similar to buses. Fially, i [4] the itercoectio structure betwee computatio blocks is fixed (a grid) ad predictable. Iformatio are routed i the commuicatio etwork by meas of dedicated switches. Costrait-drive commuicatio sythesis (CDCS) [9] follows a approach that is iheretly differet from the previous oes because it aims to derive a commuicatio architecture as the uio of heterogeeous subetworks that together satisfy the origial commuicatio costraits give by the desiger.. A Heuristic Approach for CDCS A commuicatio costrait graph CG(V,A, p,b) is a directed bipartite coected graph where each vertex v V

2 is associated to a port of a computatioal module of the system ad each costrait arc a A represets a poit-to-poit commuicatio chael betwee two modules. The set of vertices V is partitioed i a set of source ports V s ad a set of target ports V t, each vertex v V has uit degree, a pair p(v) = (p x (v), p y (v)) is associated to each vertex v V to deote its positio o the plae, ad a weight b(a) is associated to each arc a A to deote the required chael badwidth. Give a arc a = (u,v), the distace betwee its ports is deoted as d(a) = p(u) p(v). A commuicatio library L = L N is a collectio of commuicatio liks ad commuicatio odes. Each ode N has a cost c(). Each lik l L has a set of lik properties: () the lik legth (or, distace) d(l) correspods to the legth of the logest commuicatio chael that ca be realized by this lik, () the lik badwidth b(l) correspods to the badwidth of the fastest commuicatio chael that ca be realized by this lik, ad (3) the lik cost c(l) is defied with respect to the other library liks based o a optimality criterio that varies with the type of applicatio. Give a costrait graph C G(V, A, p, b) ad a commuicatio library L = L N, a implemetatio graph I G(CG,L) = G(V N,A ) is a directed graph where each vertex correspods to either a vertex of V or a commuicatio ode istace from L, each arc is associated to a lik istace from L, ad for each arc of CG there is a correspodig path i I G. The cost of a implemetatio graph I G is defied as: C(I G) = N c( )+ a A c(a ). Geerally, for a give library there are may possible implemetatio graphs that satisfy the requiremets expressed by the costrait graph while havig differet costs. I particular, oe implemetatio graph, the optimum poit-to-poit implemetatio graph, is guarateed to exist ad it is derived by implemetig a sigle arc costrait idepedetly from all the others preset i the costrait graph. O the other had, by aalyzig the defiitio of implemetatio graph it is clear that some of its arc implemetatios may share paths (i.e. liks ad/or commuicatio vertices). Figure illustrated the case where 3 arc costraits share a path. This structure is called a 3-way mergig. I geeral, we may have a k-way mergig, with k A. A k-way mergig is characterized by the presece of two commuicatio odes: a ode s at the begiig of the shared path ad a ode t ad the ed. This structure models a shared commuicatio medium such as a bus. Usually, the cost of a implemetatio graph is smaller tha the sum of the costs of its poit-to-poit arc implemetatios. Hece, it is atural to defie the followig costraied optimizatio problem that ca be see as a special case of - iteger liear programmig (ILP). Problem. Give a costrait graph CG ad a commuicatio library L = L N, miimize the cost C(I G) over all implemetatio graphs I (CG,L). u v u u u3 v3 v Figure. Example of 3-Way Mergig. The exact algorithm preseted i [9] to fid the solutio of Problem. is divided i two steps: first a set of cadidate k-way mergigs are geerated ad added to the set of miimum-cost poit-to-poit implemetatios for all the give arc costraits, the a istace of the Uate Coverig Problem (UCP) is solved. The algorithm is exact, but it is computatioally expesive ad it scales poorly with the cardiality of the set A of costrait arcs. The mai cotributio of this paper is the idea of solvig Problem. with a heuristic approach that is cetered aroud the efficiet solutio of a clusterig problem [8]. I fact, we focus o fidig a optimal partitio of the set A of costrait arcs to miimize the overall cost of the implemetatio graph that is obtaied by implemetig separately each elemet of the partitio with a optimum sub-etwork of commuicatio library compoets. We propose two distict algorithms for clusterig of costraits: a divisive oe ad a agglomerative oe. The former starts from a sigle cluster cotaiig all costraits arcs ad subsequetly cosider a series of smaller clusters that are derived by splittig the larger oes foud at the previous step. Coversely, the latter starts from cosiderig a set of sigleto clusters, oe for arc costrait, ad subsequetly attempts to merge smaller clusters to derive larger oes. I both cases, oly a subset of all possible clusterig cofiguratio is cosidered, ad the oe with best implemetatio cost is chose. Both algorithms eed a estimatio of the cluster cost which, geerally, is represeted by a fuctio o a metric space (e.g., the maximum distace betwee members of the clusters). Istead, we use the actual implemetatio cost of the cluster. This would be either a poit-to-poit chael if the cluster cotais oly a sigle costrait or a k-way mergig structure if it cotais k arc costraits, with k. For the latter case, this cost fuctio may be quite complex. However, as discussed ext, a reasoable assumptio o the implemetatio cost of oe costrait allows us to derive a closed-form expressio for the cost of a k-way mergig structure. 3. Fidig the Structure of a K-Way Mergig I CDCS, the cost of a -way mergig is defied as the sum of the cost of all arcs plus the cost of all commuicatio odes i the implemetatio graph. This sum depeds o the required badwidths of all the costrait arcs to be merged ad o the positio of the correspodig source ad destiatio ports. We make the followig assumptio o the cost of the implemetatio of a costrait arc: Give a arc a = (u,v) of a costrait graph CG with badwidth b(a) = b the cost of its poit-to-poit impleme- u u3 s t v v3 v

3 tatio is C(a) = f (b) p(u) p(v), where f (b) is a cocave fuctio of b. It is reasoable to assume that the cost of a commuicatio lik depeds o the distace to be covered, while the assumptio of the cocavity of fuctio f (b) is justified by the followig cosideratio: the cost of coverig a distace with a lik supportig badwidth b should be at most twice the cost of coverig the same distace with a lik of badwidth b. With the previous assumptio we ca show that the cost of a -way mergig ca be computed aalytically after derivig the detailed structure of a -way mergig through the solutio of a quadratic programmig problem. For each costrait arc a i i the implemetatio graph I G we pick the lik i the library with the smallest badwidth greater the b(a i ). Let c i be the cost of this lik. Also we pick the lik i the library with the smallest badwidth greater the b(a i) ad let c be the cost of this lik. The, the miimizatio problem ca be writte as: mi s,t c i ( p(u) p(s) + p(v) ) p(t) + c p(s) p(t) where s ad t are the two commuicatio odes i th - way mergig. This problem ca be decomposed ito two idepedet sub-problems alog the x, y coordiates of the plae. For the x coordiate, the miimizatio problem is: mi s,t c i ((p x (u) p x (s)) + (p x (v) p x (t)) ) + c (p x (s) p x (t)) This ca be re-writte as mi x x T Px + x T q + r, where: x = (p x (u), p x (v)) T, P = i= c i c c i= c, i q = c i p x (u) c, r = i p x (v) c i (p x (u) + p x (v) ) Note that P is resposible for all the quadratic terms while q for all the liear terms. The matrix P is positive semidefiite, the objective fuctio is covex ad the problem has oly oe solutio [3]. The solutio to the ucostraied miimizatio problem ca be computed i closed-form by settig the gradiet of the differetiable fuctio equal to zero: (x T Px + x T q + r) = Px + q = x = P q. The iverse of the matrix always exists uless all costs are zeroes: P = i= c i c i= c i c i= c i The cost of the -way mergig is, the, p = qt x + r. Note that P is a two-by-two matrix because the structure of a -way mergig has oly switches s,t. However, oe could evisio other structures with more tha two commuicatio odes. While the sizes of P ad q would chage accordigly, the ature of the optimizatio problem would remai the same as log as the routig path of the data from the source to the target ports is decided a priori. If this is ot the case, the optimizatio problem becomes a istace of geometric programmig that ca be still solved easily [3]. 4. Hierarchical Clusterig of Costraits We use the cost of a -way mergig to estimate the cost of a cluster of costraits ad we focus o clusterig as the c first step towards fidig a optimal implemetatio graph. As clusterig is kow to be NP-complete, we propose two heuristic algorithms: divisive clusterig ad agglomerative clusterig. Both algorithms explore efficietly a subset of all possible clusterig by geeratig them usig a hierarchical techique. : DivisiveClusterig(CG ) : SG derivesimilaritygraph(cg ) 3: SF computespaigforest(sg ) 4: i 5: while SF > do 6: (e, sol[i], cost[i]) selectsplittigedge(sf ) 7: remove e from SF 8: i i + 9: ed while : retur sol[argmi i cost[i]] : selectsplittigedge(sf ) : for all edges e SF do 3: remove e from SF 4: sol[e] Implemet(SF ) cost(t) 5: cost[e] t SF t 6: add e to SF 7: ed for 8: pick e that miimizes cost[e] 9: retur (e, sol[e], cost[e]) Divisive Clusterig. This algorithm is cetered aroud the otio of similarity fuctios betwee pairs of costraits, which captures the advatage of implemetig two costraits with a shared commuicatio medium as opposed to realized them with a dedicated coectio. The similarity fuctio σ(a,a ) betwee two costraits a ad a i CG is defied as σ(a,a ) = c(a,a ) c(a ) c(a ), where c(a,a ) deotes the cost of implemetig a,a as a two-way mergig, while c(a i ) is the cost of implemetig a i as a poit-to-poit coectio. From the costrait graph CG(V,A, p,b), we derive a directed complete similarity graph SG(W, E, ω) as follows: () a vertex w(a) W is associated to each costrait arc a A, () a edge e = (w,w ) E is draw betwee ay pair of vertices w,w W, ad (3) a weight ω(e) is attached to each edge e = (w,w ) such that ω(e) = σ(a,a ), where w i = w(a i ). The divisive clusterig algorithm is divided i three steps: First, the similarity graph SG is derived from the costrait graph C G. This step requires ( )/ similarity fuctio computatios. The, the miimum spaig forest SF is foud for the similarity graph SG. This step takes time O(E logv) ad, sice SG is a complete graph, returs a miimum spaig tree. This spaig tree captures the similarities amog the vertices of SG (i.e. the costrait arcs of CG), as more similar vertices ed up beig adjacet i the tree. The algorithm proceeds by removig oe edge at the time from SF, which becomes a spaig forest with two trees after the first removal, ad, sees the geeratio of a ew tree after each subsequet removal. The edge to be removed is foud ivokig the routie selectsplittigedge, which returs also the cluster-

4 d d y x 4 3 j+ / j+/ b c Figure. Experimet settigs (l), σ vs. positio ad badwidth (c), σ vs. positio ad cost (r) ig ecoded by the ew forest together with its cost effectiveess (i.e. the cost of the correspodig implemetatio divided by the umber of trees i the forest). After ivocatios of routie selectsplittigedge, SF becomes a forest without edges, while distict implemetatios have bee cosidered ad stored, together with their costs, i the array variables sol ad cost. Fially, the clusterig with miimum cost effectiveess is retured. Overall, assumig that the solutio of the quadratic programmig problem is obtaied with a costat umber of operatios, the algorithm complexity ca be derived as follows: the mai while loop is executed times ad at the i-th iteratio, the ier procedure has to remove/add i edges. So the umber of costat time operatios is the sum of the first itegers. The complexity is the O( ). From its defiitio, similarity fuctio σ depeds o the costraits specificatio ad o the commuicatio library. However, by reportig o a couple of experimets, we show here that the depedecy from the library is ideed quite weak. I Figure, two boudig boxes are depicted such that their positio i the Euclidea plae depeds o two parameters j ad. Let s assume that the boudig boxes defie two distict areas where each of two arc costraits may respectively reside ad that the positio of the source ad the destiatio port of each costrait are chose radomly ad uiformly withi each boudig box. For a give, we sweep parameter j to icrease the distace betwee the two costraits: for each positio of the boudig boxes, we radomly geerate pairs of costraits ad compute the correspodig value of σ. Fially, we calculate the arithmetic mea. We repeated this procedure for two differet experimetal scearios. First, we set the badwidth of the two costraits to be the same (equal to b) ad we repeat the above procedure for differet values of b. I this experimet we cosider a commuicatio library characterized by the followig pairs of badwidth ad cost per uit legth: {(,),(,.8),(3,.6), (4,3.4),(5,4.),(6,5.)}. It is easy to verify that the library is cocave, meaig that the cost per uit legth as a fuctio of the badwidth is a cocave fuctio. Figure illustrates the depedecy of the mea value of σ o both ad the required badwidth b. I the secod experimet, we keep the required badwidth of the two costraits equal to ad we cosider a library presetig two kid of liks. The first lik ca support a badwidth equal to with a cost per uit legth c. The secod lik ca support a badwidth equal to with a cost per uit legth c = c ( + δ) where δ. Notice that for each possible value of delta the library is still cocave. Figure (right) shows that the similarity fuctio does t deped sesibly o the parameter δ ad, therefore, either o the characteristics of the library. To uderstad this fact, let s recosider the quadratic programmig approach of Sectio 3. The cost of -way mergig ca be writte as c x = x T Px+x T q+r = 4 qt P q+r, where r is idepedet from the cost c. Matrix P is the oly factor depedig o c ad is writte as follows: P = c + c c c + c c = c c + c c (3 + )δ ( + δ ) + δ + δ + δ where the equality holds whe c = c ( + δ). The matrix eigevalues are λ = /c ad λ = /(c (3+δ)). They determie the shape of the paraboloid over which the cost is computed. While λ is idepedet from δ, λ has a very smooth depedecy. Hece, the value of the quadratic fuctio computed at q is almost the same for all the values of δ (ote: q ad r do t deped o δ). I summary, the similarity fuctio ca be cosidered as techology idepedet, i.e. oly depedet o the problem specificatio. : Agglomerative Clusterig(CG ) : K / 3: for all costraits arcs a i R do 4: K K a i 5: ed for 6: l 7: while K l > do 8: (i, j) argmi u,v [, l] σ(k u,k v ) 9: K l+ K l \{k i,k j } : K l+ K l+ {k i,k j } : l l + : ed while 3: retur K opt where opt = argmi t [, ] c(k t ) Agglomerative Clusterig. This is a greedy algorithm that, at each step, cosiders more complex clusters that are derived by composig the simpler clusters foud at the previous step. Before, we defied the similarity fuctio betwee two costraits. Now, we exted the cocept to clus-

5 S4 D3 S D D7 D9 D D3 S3 Exact Algorithm Divisive Agglomerative A cost time cost time cost time , t/o S D6 D5 S7 S7 S6 S9 D7 D9 S S6 D S8 S8 D4 S5 S S9 DD6 S5 D8 D4 D D8 S S D D5 S4 S3 Figure 3. Exact algorithm (left), results table (ceter), agglomerative clusterig (right) ters of costraits. The ituitio is the same: the similarity fuctio measures the advatage of implemetig two clusters with the same commuicatio medium versus usig two dedicated media: The similarity fuctio betwee two costrait clusters k i,k j is σ(k i,k j ) = c(k i k j ) c(k i ) c(k j ). This fuctio is used by the agglomerative clusterig algorithm to derive a optimum clusterig. The algorithm first builds a sigle cluster for each costrait i CG. The set of these clusters is deoted as K = {a,...,a }. The, at each step of the while loop, the two more similar clusters are greedily selected ad merged together. Thus, the algorithm cosiders distict clusterig cofiguratios before returig the optimal oe from a implemetatio cost viewpoit. The complexity of this algorithm is O( 3 ). 5. Simulatio ad Result We implemeted the proposed clusterig algorithms i a C++ package called SENC (Sythesis Egie for Networks-o-Chip). To derive the optimum -way mergig topology, SENC uses the NEWMAT matrix library [5]. The table of Figure 3 reports the experimetal results obtaied by ruig the two clusterig algorithms ad the exact optimizatio algorithm from [9]. We used a commuicatio library with oly three types of liks havig respectively badwidth, 5, ad cost per uit legth,.,.4. We radomly geerated four costraits graphs with arc cardiality A equal to 5,,5,3. For each algorithm, we report the cost of the implemetatio graph ad the CPU time i millisecods (o a 75MHZ, 56Mbyte Petium III). The expoetial ature of the exact algorithm is clear: for A = 3 it does ot retur the solutio i the allotted time ( miutes). As expected from the complexity aalysis, divisive clusterig is faster tha agglomerative clusterig. However, the latter returs better results that are quite close to the exact solutio (whe this is computed). Note that the cost of the implemetatio graph may be omootoic i the umber of requiremets because addig ew costraits may lead to the geeratio of ew mergigs with costraits that previously were implemeted as poit-to-poit dedicated liks. The two diagrams of Figure 3 show the sythesis results for the exact algorithm ad the agglomerative clusterig oe (case A = ). For each costrait a i, S i is the source port ad D i the target port. Note the similarities betwee the results obtaied with the two approaches. Also otice how outliers are sigled out by the clusterig algorithm (particularly costraits 3, 4, 9, 8). Costraits 3,4, ad 9 are so short ad with low badwidth that are better implemeted as poit-to-poit chaels. Arc costrait 8 is orieted orthogoally to all its eighbors, which are log arc costraits that get merged together. Cocludig Remarks. We divided the optimizatio problem of CDCS [9] ito two steps: clusterig of costraits ad optimal sythesis of the clustered costraits. We cast the latter as a quadratic programmig problem. For the former, we developed two heuristic algorithms, divisive clusterig ad agglomerative clusterig, whose complexity is respectively O( ) ad O( 3 ), where is the umber of costraits. Experimetal results show that the agglomerative algorithm is closer to the exact solutios tha the divisive oe, which, however, rus faster. Eve though we focused o bus-based commuicatio etworks, the applicability of our approach is quite geeral because chagig the cluster implemetatio does ot chage the ature of the optimizatio problem. Refereces [] A. Alla, D. Edefeld, W. J. Jr., A.B.Kahg, M. Rodgers, ad Y. Zoria. Techology Roadmap for Semicoductors. IEEE Computer, 35():4 53, Ja.. [] L. Beii ad G. De-Micheli. Networks o-chips: A ew soc paradigm. IEEE Computer, Jauary. [3] S. Boyd ad L. Vadeberghe. Covex Optimizatio. available at boyd/cvxbook.html,. [4] W. J. Dally ad B. Towles. Route packets, ot wires. I Proc. of the Desig Automatio Cof., pages ,. [5] R. Davies. Newmat C++ Matrix Library. available at itro.htm,. [6] R. Ho, K. Mai, ad M. Horowitz. The Future of Wires. Proc. of the IEEE, 89(4):49 54, Apr.. [7] K. Lahiri, A. Raghuatha, ad S. Dey. Efficiet exploratio of the soc commuicatio architecture desig space. I Proc. Itl. Cof. o Computer-Aided Desig, pages 44 43,. [8] B. Mirki. Mathematical Classificatio ad Clusterig. Kluwer Academic Publishers, 996. [9] A. Pito, L. P. Carloi, ad A. L. Sagiovai-Vicetelli. Costrait-Drive Commuicatio Sythesis. I Proc. of the Desig Automatio Cof., pages IEEE, Jue. [] J. Walrad ad P. Varaija. High Performace Commuicatio Networks. Morga Kaufma,Sa Fracisco,.