HIGH-LEVEL TRANSFORMATIONS DATA-FLOW MODEL OF COMPUTATION TOKEN FLOW IN A DFG DATA FLOW

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1 1 2 Topis: * Dt-flow grphs * (Non)overlpped sheduling * Miniml itertion period Further reding: * Trnsformtions for speed-up * Trnsformtions for low power Prhi, K.K., High-Level Algorithm nd Arhiteture Trnsformtions for DSP Snthesis, Journl of VLSI Signl Proessing, Vol. 9, pp , (1995). Gere, S.H., S.M. Heemstr de Groot, E.R. Bonsm nd M.J.M. Heijligers, Overlpped Sheduling Tehniques for High-Level Snthesis nd Multiproessor Relitions of DSP Algorithms, In: J.C. Lope, R. Hermid nd W. Geisselhrdt (Eds.), Advned Tehniques for Emedded Sstem Design nd Test, Kluwer Ademi Pulishers, Boston, pp , (1998). DATA-FLOW MODEL OF COMPUTATION Dt-flow grphs (DFGs) epliitl represent prllelism in omputtions. A DFG m or m not ontin informtion on ontrol flow. A dt-flow grph is uilt from: * nodes (verties): representing omputtion, nd * edges: representing preedene reltions. 3 4 Emple: := * ; := d; := ; DATA FLOW TOKEN FLOW IN A DFG * A node in DFG fires when tokens re present t its inputs. * The input tokens re onsumed nd n output token is produed.

2 5 6 IMPLICIT ITERATIVE DATA FLOW * Itertion implied strem of input tokens rriving t regulr instnts in time. The omputtion of the DFG is repeted ever time units. * Initil tokens t s uffers. [0] [1] [0] [1] [1] [1] IMPLICIT ITERATIVE DATA FLOW (Ctd.) * Del elements insted of initil tokens. Two nottions re enountered: * epliit del elements * del elements s n edge propert 7 8 ITERATIVE DFG EXAMPLE i m1 d 1 m 2 m 3 d m o 1 7 IDFG(V,E) with: * V: the verte set: V C D I O * C: set of omputtionl nodes * D: set of del nodes * I: set of of input nodes * O: set of output nodes IDFG NOTATION * E: the edge set * (), C gives the durtion of omputtion (tomi, nonpreemptive, restrited lirr) * (d), d D gives the multipliit of del node A seond-order filter setion.

3 9 10 SYNCHRONOUS DATA-FLOW The itertive dt-flow grph is speil se of snhronous dt-flow grph (introdued Edwrd Lee). Chrteristis: * no onditionl nodes D (3) (1) (2) (3) * eh edge hs integer ttriutes for numers of tokens produed t one side nd onsumed t the other: multirte sstem * eh edge hs del ttriute. OPTIMIZATION CRITERIA Most ommonl used: * Time-onstrined snthesis: given the itertion period, use s few proessors s possile or s little hrdwre s possile (tpil for DSP). * Resoure-onstrined snthesis: given multiproessor onfigurtion or set of hrdwre resoures on hip, minimie. Another importnt issue: * Minimition of power. Lee, E.A. nd D.G. Messershmitt, Snhronous Dt Flow, Proeedings of the IEEE, Vol. 75(9), pp , (Septemer 1987) Sutsks: TERMINOLOGY * Sheduling: determine for eh opertion the time t whih it should e performed suh tht no preedene onstrint is violted. * Allotion: speif the hrdwre resoures tht will e neessr. * Assignment: provide mpping from eh opertion to speifi funtionl unit nd from eh vrile to register. Remrks: * The suprolems re strongl interrelted; the re, however, often solved seprtel. * Sheduling (eept for few versions) is NP-omplete heuristis hve to e used. SCHEDULING TERMINOLOGY * Stti sheduling mens: mpping to time nd proessor (funtionl unit, register, et.) is identil in ll itertions. * A stti shedule is either overlpped (epoiting interitertion prllelism) or nonoverlpped. hrdwre units i i 1 nonoverlpped time hrdwre units i ÍÍ ÍÍÍ i 1 ÍÍÍ ÍÍ overlpped time Prhi, K.K. nd D.G. Messershmitt, Stti Rte-Optiml Sheduling of Itertive Dt-Flow Progrms vi Optimum Unfolding, IEEE Trnstions on Computers, Vol. 40(2), pp , (Ferur 1991).

4 13 14 SCHEDULING TERMINOLOGY (Ctd.) * Overlpped sheduling is lso lled: loop folding, softwre pipelining. * The del etween onsumption of input nd prodution of output is lled the lten. In generl. * An ovelpped shedule m llow shorter itertion period or hrdwre utilition, ut: * the serh spe is lrger nd finding optiml solutions hrder. Not overed in this presenttion: * lostti shedules * dnmi shedules (requires run-time sheduler) ÁÁ P ÁÁÁÁÁÁÁ P 3 ÁÁÁÁÁÁÁ ÁÁÁÁÁÁÁ ÁÁÁÁÁÁÁÁ 5 2 P P ÁÁÁÁÁ A lostti shedule ÁÁÁÁÁ THE MINIMAL ITERATION PERIOD (min ) There re four ses: * Ali DFG, nonoverlpped shedule; * Ali DFG, overlpped shedule; * Cli DFG, nonoverlpped shedule; Reple ll del nodes pirs of input nd output nodes. * Cli DFG, overlpped shedule. For the nonoverlpped ses: * Compute the ritil pth, longest pth from n input to n output, in the li grph. min = length of ritil pth i 1 EXAMPLE 4 o 2 (d 1 ) 8 m1 m 2 i 2 (d 1 ) o 1 7 m 3 o 3 (d 2 ) m 4 i 3 (d 2 ) Critil pth in li DFG for nonoverlpped shedule. 6 OVERLAPPED SCHEDULING IN A CYCLIC DFG * Miniml itertion period is given ritil loop. Emple: ( ) ( ) ( ) * When the DFG is li, ritrril smll itertion periods re possile (just duplite the hrdwre s often s neessr; eh op n strt n time s there re no feedk loops in the DFG; see lter on).

5 17 18 CRITICAL LOOP For generl DFG, min is given : min m l ll loops l dl () (d) Reiter, R., Sheduling Prllel Computtions, Journl of the ACM, Vol. 15(4), pp , (Otoer 1968). Renfors, M. nd Y. Neuvo, The Mimum Smpling Rte of Digitl Filters Under Hrdwre Speed Constrints, IEEE Trnstions on Ciruits nd Sstems, Vol. CAS 28(3), pp , (Mrh 1981). i 1 EXAMPLE 4 8 m1 d 1 m 2 m 3 d m o 1 7 min 3, when () 1 nd (*) ON COMPUTING min Remrks: * Diret use of epression for min not effiient (numer of loops in grph m grow eponentill with respet to numer of nodes). * Mn polnomil-time lgorithms hve een pulished; surve in: Dsdn, A., S.S. Irni nd R.K. Gupt, Effiient Algorithms for Optimum Cle Men nd Optmimum Cost to Time Rtio Prolems, 36th Design Automtion Conferene, (1999). * An es to understnd ut not ver effiient method is sed on mtri multiplition. Gere, S.H., S.M. Heemstr de Groot nd O.E. Herrmnn, A Polnomil-Time Algorithm for the Computtion of the Itertion-Period Bound in Reursive Dt-Flow Grphs, IEEE Trnstions on Ciruits nd Sstems I: Fundmentl Theor nd Applitions, Vol. 39(1), pp 49 52, (Jnur 1992). SPEED-UP TECHNIQUES: PIPELINING Insert del elements on ll edges tht re ut ut line through n edge of the ritil pth in the DFG. * Works for li DFGs. * Shedule eomes overlpped. hrdwre units i 1 i time ut line Emple 4 4

6 21 22 SPEED-UP TECHNIQUES: RETIMING * Useful for otining the miniml for nonoverlpped shedule redution of ritil-pth length, oth for li nd li IDFGs. 4 4 SOME REMARKS ON min * Retiming does not ffet min for overlpped sheduling of IDFG s. * The for nonoverlpped sheduling otined fter optiml retiming m still e lrger thn min. * min hs een defined s n integer; frtionl min mkes sense when unfolding is pplied (unfolding retes new DFG of multiple opies of the originl one; see lter). Cho, L.F. nd E.H.M. Sh, Rte-Optiml Stti Sheduling for DSP Dt-Flow Progrms, 3rd Gret Lkes Smposium on VLSI Design, Automtion of High-Performne VLSI Sstems, pp 80 84, (Mrh 1993). Emple SPEED-UP TECHNIQUES: PARALLEL PROCESSING * Works for li IDFGs. * Duplite the IDFG s often s desired speed-up ftor. * Allows n ritrr speed-up, ut is proportionll epensive. i 1 [k] i 1 [k 1] o 1 [k] i 1 [2k] i 1 [2k 1] o 1 [2k 1] i 1 [2k 1] o 1 [2k] proess 2 inputs t time * A tehnique for the duplition of li IDFGs in omintion with proessing multiple inputs t time. * Consider the following IDFG: i[k] s[k] UNFOLDING (1) o[k] * If () 1 nd (*) 2, min * Using unfolding 2, one n reh the vlue min 3 2. * The grph omputes the following differene equtions, ssuming tht one multiplies ftor : s[k] i[k] o[k 1] o[k] s[k 1]

7 25 26 * The preise unfolding lgorithm will not e given here; it mounts to dupliting ll verties in the IDFG suh tht n opies of eh verte is reted (n is the unfolding ftor) nd then to onneting these verties with edges hving n pproprite numer of del elements. The unfolded grph n lso e reonstruted from the equtions. UNFOLDING (2) * The method will e illustrted using the emple IDFG nd unfolding ftor of two, mening tht two inputs will e ville per itertion nd two outputs will e produed. The equtions: s[2k] i[2k] o[2k 1] s[2k 1] i[2k 1] o[2k] o[2k] s[2k 1] o[2k 1] s[2k] * The emple IDFG fter unfolding: i[2k] i[2k 1] s[2k] o[2k 1] s[2k 1] UNFOLDING (3) s[2k 1] o[2k 1] o[2k] * Note tht the unfolded IDFG hs two loops with one del element eh nd omputtionl durtion of 3. Beuse del element retes n offset of two indies (2 inputs re proessed in eh itertion), the effetive itertion period ound is equl to min LOOK-AHEAD TRANSFORMATION (1) * Consider the following omputtion: [n] [n 1] u[n] u[n] [n] [n 1] * It hs one multiplition nd one ddition in the ritil loop with one del element. If () 1 nd (*) 2, min LOOK-AHEAD TRANSFORMATION (2) * Appl look-hed trnsformtion (think of the priniple of lookhed ddition): [n] ([n 2] u[n 1]) u[n] [n] 2 [n 2] u[n 1] u[n] * The new eqution hs one multiplition nd one ddition in u[n 1] the ritil loop with two dels leding to min 3 2 [n] 2. u[n] * The trnsformtion n ffet the originl omputtion (finl wordlength effets). 2 [n 2]