Scheduling with Integer Time Budgeting for Low-Power Optimization

Transcription

1 Schedlng wth Integer Tme Bdgetng for Low-Power Optmzaton We Jang, Zhr Zhang, Modrag Potkonjak and Jason Cong Compter Scence Department Unversty of Calforna, Los Angeles Spported by NSF, SRC.

2 Otlne Introdcton to nteger tme bdgetng (ITB) problem Low-power schedlng Expermental reslts Conclsons and ftre work

3 Integer Tme Bdgetng (ITB) Problem Defnton Slack: the amont of extra delay that each component (ether a small gate or a large modle) of a desgn can tolerate wthot volatng the gven tmng constrant Tme bdgetng problem The problem of dstrbtng the slacks to dfferent modles of a desgn to optmze some objectves (sch as area, power) Example: redce area/power by sng slower adders Integer tme bdgetng problem The slacks mst be ntegral vales Applcatons: schedlng, gate szng, nterconnect plannng D 1 = 1ns + + D 2 = 1ns + D 3 = 1ns T = 5ns Slacks: S 1 =S 2 =S 3 =3ns

4 Motvaton Power (mw) 5 3 Maxmzng sm of weghted slack mght be sboptmal for power optmzaton v 1 v 2 v Delay (ns) Power-delay tradeoff-crve v 1 v 2 5ns v 1 v 2 v 3 v 3 Total slack = 6ns Total power = 16mW Total slack = 5ns Total power = 11mW

5 Related Work Network-flow flow-based algorthm A Unfed Theory of Tme Bdget Management, [Ghas[ et al., ICCAD 4] Can solve the ITB problem optmally wth lnear objectve fncton Desgn Closre Drven Delay Relaxaton Based on Convex Cost Network Flow [Ln et al., DATE 7] Can handle convex objectve fncton Mathematcal programmng approach A Mathematcal Formlaton of Integer Tme Bdgetng Problem [Cong et al., TECHCON 7] Handles convex objectve fncton Inttve and easy to be ncorporated nto systems wth applcaton on-specfc desgn constrants

6 A Mathematcal Formlaton of Tme Bdgetng Problem Power (mw) v 1 v 2 v 3 v 4 v 5 T v 6 Drected Acyclc Graph: G = (V, E) s : start tme of node v d : mnmm latency of v b : tme bdget at node v Lnearly constraned separable convex optmzaton problem Mn Sbject s b s s Delay (ns) Each f s a sngle-varable convex fncton + b d T V = 1 s f ( b ) to : j v v v e(, V PIs j) POs E

7 Totally Unmodlar Constrant Matrx v 1 v 2 s 2 + d 2 s 3 s 1 + d 1 s 3 v 3 s 1 s 2 s 3 T s s s J 1 J 1 J 2 J 1 b 1 1 b 2 1 b 3 Theorem 1: The ITB constrant matrx s a TUM

8 Optmzng Separable Convex Objectve Power (mw) 1 Power (mw) Delay (ns) Delay (ns)

9 Otlne Introdcton to nteger tme bdgetng (ITB) problem Low-power schedlng Expermental reslts Conclsons and ftre work

10 Applcaton to Low-Power Schedlng Motvaton Schedlng and tme bdgetng are hghly correlated Problem Consder the schedlng and bdgetng problem together to mnmze the average power nder tme constrant T Man dea Integrate or ITB problem wth the SDC based schedlng [Cong and Zhang, An effcent and Versatle Schedlng Algorthm Based on SDC Formlaton, DAC 6]

11 Low-power Schedlng Problem Formlaton Gven: A data flow graph G A latency constrant T A set of optonal schedlng constrants ncldng cycle tme constrant, relatve tmng constrants and resorce constrants. + * v 1 v 2 * v 3 + v 4 A set of power-delay tradeoff crves for each type of operaton sch as addton, mltplcaton, etc. Objectve v 5 DFG Example Get a vald schedlng whch satsfes all the constrants and mnmze the total power

12 Low-Power Schedlng Each node v V op s assocated wth a node bdgetng varable bv(v ) whch denotes the # of clock cycles that operaton v lasts n the fnal schedle Adjst the followng constrants Data dependence constrant (, v) E d : sv beg beg ()) + bv() sv sv beg (v) Latency constrant T v V op : sv beg ()) + bv(v) T Throghpt constrant wth ntaton nterval II v V op : bv(v) II Optmzng total node power Mn V op = pw 1 op v ) ( ( bv( v )) We can optmally mnmze the total node power n polynomal tme

13 Consderaton of Resorce Bndng Optmzng total FU power Constrant matrx s no longer totally nmodlar wth the reqrement that: all operatons sharng a same fncton nt mst have same slacks The problem s NP-complete (redcton from 3-SAT) 3 Proposed herstc Mn Frst solve the contnos verson and obtan the optmal fractonal bdget fb(v ) for each node v Perform a global rondng by mnmzng the least-sqares sqares error Objectve fncton s separable convex F = f * j 1 j pw op f j ) ( ( bv ( f j )) Mn V op = 1 ( bv( v ) fb( v )) 2

14 Low-power Schedlng Flow User Constrants CDFG Target platform modelng (Power-delay crves) Resorce Sharng Informaton Low Power Schedler Constrant generaton ITB constrants Dependency constrants Objectve generaton Mathematcal Programmng Solver STG (State Transton Graph) Resorce Bndng

15 Otlne Introdcton to nteger tme bdgetng (ITB) problem Low-power schedlng Expermental reslts Conclsons and ftre work

16 C specfcaton C-to- xplot behavoral synthess LLVM compler SSDM (System-Level Synthess Data Model) SSDM/CDFG Behavoral synthess SSDM/FSMD RTL generaton Platform descrpton & constrants FSM wth Datapath n VHDL Mlt-cycle path constrants Magma RTL-to to-slcon v4.2 (+ TSMC 9nm lbrary) GDSII

17 Expermental Reslts Comparson wth Max Weghted Slack LPS: Mnmze total node power WMS: Maxmze maxmm weghted slack Power, area, and cycle tme comparsons between WMS and LPS (Latency constrant = 1.2x the longest path length)

18 Expermental Reslts Consderatons of Resorce Sharng LPS_NRS: Mnmze total node power (wthot consderng resorcee sharng) LPS_RS: Mnmze total fncton nt power (consderng resorce sharng) ILP: ILP-based approach to drectly mnmze total FU power (optmal solton) Actal power consmpton LPS_RS s wthn 6% of the ILP exact approach and otperforms LPS_NRS by 3%

19 Conclsons and Ftre Work Conclsons A mathematcal programmng formlaton of the nteger tme bdgetng problem wth great flexblty and extensblty Applcaton to low-power schedlng problem Ftre works Apply or ITB formlaton to other problems

20 Thanks.

21 Desgn Closre Drven Delay Relaxaton Based on Convex Cost Network Flow [DATE7] Problem formlaton Desgn Closre Drven Delay Relaxaton problem Essentally a ITB problem wth convex objectve fncton Solton Transformaton to a convex cost nteger dal network flow problem Comparson wth mathematcal programmng (MP) approach Network flow based algorthm has a better worst-case complexty MP approach allows the tlzaton of the leadng-edge edge mathematcal programmng solvers MP approach can be easly extended to spport applcaton specfc c constrants