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1 Outlie SCHEDULING The schedulig problem. cgiovi De Micheli Schedulig without costrits. Stford Uiversity Schedulig uder timig costrits. { Reltive schedulig. Schedulig uder resource costrits. { The ILP model. { Heuristic methods. Schedulig Circuit model: { Sequecig grph. { Cycle-time is give. { Opertio delys expressed i cycles. Schedulig: { Determie the strt times for the opertios. { Stisfyig ll the sequecig (timig d resource) costrit. TIME Gol: TIME { Determie re/ltecy trde-o. TIME TIME

2 Txoomy Ucostried schedulig. Schedulig with timig costrits: { Ltecy. { Detiled timig costrits. Schedulig with resource costrits. Relted problems: { Chiig. Simplest model All opertios hve bouded delys. All delys re i cycles. { Cycle-time is give. No costrits - o bouds o re. Gol: { Miimize ltecy. { Sychroiztio. { Pipelie schedulig. Miimum-ltecy ucostried schedulig problem Give set of ops V with iteger delys D d prtil order o the opertios E: Fid iteger lbelig of the opertios ' : V! Z, such tht: { t i = '(v i ), ASAP schedulig lgorithm ASAP ( G s (V E)) f Schedule v by settig t S = repet f Select vertex v i whose pred. re ll scheduled Schedule v i by settig t S i = mx t S j d j j:(vj vi)e g g util (v is scheduled) retur (t S ) { t i t j d j i j s:t: (v j v i ) E { d t is miimum.

3 ALAP schedulig lgorithm TIME TIME TIME ALAP( G s (V E) ) f Schedule v by settig t L = repet f Select vertex v i whose succ. re ll scheduled Schedule v i by settig t L i = mi t L j ; d i j:(vi vj)e g g util (v is scheduled) retur (t L ) TIME Remrks ALAP solves ltecy-costried problem. TIME TIME Ltecy boud c be set to ltecy computed by ASAP lgorithm. TIME Mobility: TIME { Deed for ech opertio. { Di. betwee ALAP d ASAP schedule. Slck o the strt time.

4 Opertios with zero mobility: { fv v v v v g. { Criticl pth. Opertios with mobility oe: { fv v g. Opertios with mobility two: { fv v v v g. Schedulig uder detiled timig costrits Motivtio: { Iterfce desig. { Cotrol over opertio strt time. Costrits: { Upper/lower bouds o strt-time dierece of y opertio pir. Fesibility of solutio. Costrit grph model Strt from sequecig grph. Model delys s weights o edges. MAX TIME MIN TIME Add forwrd edges for miimum costrits. { Edge (v i v j ) with weight l ij ) t j t i l ij. Add bckwrd edges for mximum costrits. { Edge (v j v i ) with weight: -u ij ) t j t i u ij { becuse t j t i u ij ) t i t j ; u ij. Vertex Strt time v v v v v v

5 Methods for schedulig uder detiled timig costrits Assumptio: { All delys re xed d kow. Set of lier iequlities. Logest pth problem. Algorithms: { Bellm-Ford, Lio-Wog. Method for schedulig with ubouded-dely opertios Ubouded delys: { Sychroiztio. { Ubouded-dely opertios (e.g. loops). Achors. { Ubouded-dely opertios. Reltive schedulig: { Schedule ops w.r. to the chors. { Combie schedules. For ech vertex: Reltive schedulig method SYN { Determie relevt chor set R(). { Achors ectig strt time. t = mxft d t d g { Determie time oset from chors. Strt-time: { Expressed by: t i = mx R(v i ) ft d t i g { Computed oly t ru-time becuse delys of chors re ukow.

6 Reltive schedulig uder timig costrits Problem deitio: { Detiled timig costrits. { Ubouded dely opertios. Solutio: { My or my ot exist. { Problem my be ill-specied. Reltive schedulig uder timig costrits Fesible problem: { A solutio exists whe ukow delys re zero. Well-posed problem: { A solutio exists for y vlue of the ukow delys. Theorem: { A costrit grph c be mde well-posed i there re o cycles with ubouded weights. Reltive schedulig pproch Alyze grph: u ij v i d d d v v i j d d d v v i j { Detect chors. { Well-posedess test. { Determie depedecies from chors. v j u ij u ij () (b) (c) Schedule ops with respect to relevt chors: { Bellm-Ford, Lio-Wog, Ku lgorithms. Combie schedules to determie strt times: { t i = mx R(v i ) ft d t i g i

7 of cotrol-uit strt Completio of () SYNCH d couter N Vertex Relevt Achor Set Osets v i R(v i ) t t fv g - v fv g - v fv g - v fv g sych Schedulig uder resource costrits Clssicl schedulig problem. { Fix re boud - miimize ltecy. The mout of vilble resources ects the chievble ltecy. Dul problem: { Fix ltecy boud - miimize resources. Assumptio: { All delys bouded d kow. Miimum ltecy resource-costried schedulig problem Give set of ops V with iteger delys D prtil order o the opertios E, d upper bouds f k k = ::: res g: Fid iteger lbelig of the opertios ' : V! Z such tht : { t i = '(v i ), { t i t j d j i j s:t: (v j v i ) E, { jfv i jt (v i ) = k d t i l t i d i gj k types k = ::: res d steps l { d t is miimum.

8 Schedulig uder resource costrits Itrctble problem. ILP formultio: Biry decisio vribles: Algorithms: { Exct: Iteger lier progrm. Hu (restrictive ssumptios). { Approximte: List schedulig. Force-directed schedulig. { X = fx il i = ::: l = ::: g. { x il, is TRUE oly whe opertio v i strts i step l of the schedule (i.e. l = t i ). { is upper boud o ltecy. Strt time of opertio v i : { X l l x il ILP formultio costrits Opertios strt oly oce. { X l x il = i = ::: ILP Formultio Sequecig reltios must be stised. { t X i t j d j X (v j v i ) E { l x il ; l x jl ; d j (v j v i ) E l l X l l x il ; X l X X j mi jjtjj such tht x ij = i = ::: l x jl ; d j i j = ::: (v j v i ) E lx i:t (vi)=k m=l;di x im k k = ::: res l = ::: t Resource bouds must be stised. { Simple cse (uit dely) { X i:t (v i )=k x il k k = ::: res l

9 Opertios strt oly oce. { x = { x x = {... Sequecig reltios must be stised. { x x ; x ; x Resource costrits: { ALUs Multipliers. { x x x ; x N {... Resource bouds must be stised. { = =. { x x x x { x x x x Sigle-cycle opertio. {... { d i = i. Dul ILP formultio Miimize resource usge uder ltecy costrits. Additiol costrit: TIME { Ltecy boud must be stised. TIME { X l l x l TIME TIME Resource usge is ukow i the costrits. Resource usge is the objective to miimize.

10 ILP Solutio TIME Use stdrd ILP pckges. TIME TIME Trsform ito LP problem [Gebotys]. TIME Advtges: { Exct method. Multiplier re =. ALU re =. { Others costrits c be icorported. Objective fuctio:. Disdvtges: { Works well up to few thousd vribles.

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NOP 1 2 * * TIME 1 11 < TIME 2 TIME 3 TIME 4 NOP RESOURCE SHARING cgiovi De Micheli Otlie Resorce-domited circits. { Flt d hierrchicl grphs. Stford Uiversit { Fctiol d memor resorces. Extesios. { No resorce-domited circits. { Cocrret schedlig d bidig.

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