On Energy-Optimal Voltage Scheduling for Fixed-Priority Hard Real-Time Systems

Transcription

1 On Energy-Optmal Voltage Schedulng for Fxed-Prorty Hard Real-Tme Systems HAN-SAEM YUN and JIHONG KIM Seoul Natonal Unversty We address the problem of energy-optmal voltage schedulng for fxed-prorty hard real-tme systems, on whch we present a complete treatment both theoretcally and practcally. Although most practcal real-tme systems are based on fxed-prorty schedulng, there have been few research results nown on the energy-optmal fxed-prorty schedulng problem. Frst, we prove that the problem s NP-hard. Then, we present a fully polynomal tme approxmaton scheme (FPTAS) for the problem. For any ε>0, the proposed approxmaton scheme computes a voltage schedule whose energy consumpton s at most (1 + ε) tmes that of the optmal voltage schedule. Furthermore, the runnng tme of the proposed approxmaton scheme s bounded by a polynomal functon of the number of nput jobs and 1/ε. Gven the NP-hardness of the problem, the proposed approxmaton scheme s practcally the best soluton because t can compute a near-optmal voltage schedule (.e., provably arbtrarly close to the optmal schedule) n polynomal tme. Expermental results show that the approxmaton scheme fnds more effcent (almost optmal) voltage schedules faster than the best exstng heurstc. Categores and Subject Descrptors: C.3 [Specal-Purpose and Applcaton-Based Systems]: Real-Tme and Embedded Systems; F.2.2 [Analyss of Algorthms and Problem Complexty]: Nonnumercal Algorthms and Problems sequencng and schedulng General Terms: Algorthms Addtonal Key Words and Phrases: Fxed-prorty schedulng, real-tme systems, approxmaton algorthms, fully polynomal tme approxmaton scheme, varable voltage processor, dynamc voltage scalng 1. INTRODUCTION Energy consumpton s one of the most mportant desgn constrants n desgnng battery-operated embedded systems such as personal dgtal assstants, dgtal cellular phones, and moble vdeophones. For such systems, the energy consumpton s a crtcal desgn factor because the battery operaton tme s a prmary performance measure. The dynamc energy consumpton E, whch domnates the total energy consumpton of CMOS crcuts, s gven by Ths wor was supported by grant R from the Korea Scence and Engneerng Foundaton. RIACT at Seoul Natonal Unversty provdes research facltes for the study. Authors address: H.-S. Yun and J. Km (correspondng author), School of Computer Scence and Engneerng, Seoul Natonal Unversty, Shlm-dong, Kwana-u, Seoul, , Korea; emal: {hsyun, jhong}@davnc.snu.ac.r. Permsson to mae dgtal/hard copy of all or part of ths materal wthout fee for personal or classroom use provded that the copes are not made or dstrbuted for proft or commercal advantage, the ACM copyrght/server notce, the ttle of the publcaton, and ts date appear, and notce s gven that copyng s by permsson of the ACM, Inc. To copy otherwse, to republsh, to post on servers, or to redstrbute to lsts requres pror specfc permsson and/or a fee. C 2003 ACM /03/ $5.00 ACM Transactons on Embedded Computng Systems, Vol. 2, No. 3, August 2003, Pages

2 394 H.-S. Yun and J. Km E C L N cycle V 2 DD, where C L s the load capactance, N cycle s the number of executed cycles, and V DD s the supply voltage. Because the dynamc energy consumpton E s quadratcally dependent on the supply voltage V DD, lowerng V DD s an effectve technque n reducng the energy consumpton. However, lowerng the supply voltage also decreases the cloc speed, because the crcut delay T D of CMOS crcuts s gven by T D V DD /(V DD V T ) α [Saura and Newton 1990], where V T s the threshold voltage and α s a technologydependent constant. When a gven job does not requre the maxmum performance of a VLSI system, the cloc speed (and ts correspondng supply voltage) can be dynamcally adjusted to the lowest possble level that stll satsfes the job s requred performance. Ths s the ey prncple of the dynamc voltage scalng (DVS) technque. Wth a recent explosve growth of the portable embedded system maret, several commercal varable-voltage processors were developed (e.g., Intel s Xscale, AMD s K6 2+, and Transmeta s Crusoe processors). Targetng these processors, varous DVS algorthms [Aydn et al. 2001; Gruan 2001; Hong et al. 1998; Km et al. 2002; Plla and Shn 2001; Quan and Hu 2001, 2002; Shn and Cho 1999; Shn et al. 2000; Yao et al. 1995] have been proposed, especally for embedded hard real-tme systems. For hard real-tme systems, the goal of voltage schedulng algorthms s to fnd an energy-effcent voltage schedule wth all the strngent tmng constrants satsfed. A voltage schedule s a functon that assocates each tme unt wth a voltage level (.e., a cloc frequency). 1 In ths paper, we consder fxed-prorty real-tme jobs runnng on varable voltage processors. 1.1 Prevous Wor Prevous nvestgatons on the voltage schedulng problem have focused manly on real-tme jobs runnng under dynamc-prorty schedulng algorthms such as the EDF (earlest-deadlne-frst) algorthm [Aydn et al. 2001; Hong et al. 1998; Km et al. 2002; Plla and Shn 2001]. For example, the problem of energy-optmal EDF schedulng has been well understood. For EDF job sets, the algorthm by Yao et al. [1995] computes the energy-optmal voltage schedules n polynomal tme. Although the EDF schedulng polcy maes the voltage schedulng problem easer to solve, fxed-prorty schedulng algorthms such as the RM (rate monotonc) algorthm are more commonly used n practcal real-tme systems due to ther low overhead and predctablty [Lu 2000]. Although there exst several voltage schedulng technques proposed for fxed-prorty real-tme tass (e.g., onlne schedulng algorthms [Gruan 2001; Plla and Shn 2001; Shn and Cho 1999] and offlne schedulng algorthms [Gruan 2001; Quan and Hu 2001, 2002; Shn et al. 2000]), there have been few research results on the optmal voltage schedulng problem for fxed-prorty hard real-tme systems; nether a polynomal-tme optmal voltage schedulng algorthm nor the computatonal complexty of the problem s nown. 1 Throughout the remander of the paper, we use the term voltage schedulng nstead of DVS.

3 Energy-Optmal Voltage Schedulng 395 Up to now, the only sgnfcant research result on the optmalty ssue of fxed-prorty voltage schedulng s the one presented by Quan and Hu [2002], where energy-optmal voltage schedules for fxed-prorty jobs are found by an exhaustve algorthm. However, Quan and Hu dd not justfy ther exhaustve approach. If they had presented the computatonal complexty of the voltage schedulng problem, ther result would have been much more sgnfcant. Snce the worst-case complexty of Quan s algorthm s of hgher order than O(N!), where N s the number of jobs, the algorthm s practcally unusable for most real-tme applcatons. Quan and Hu [2001] also proposed a polynomal-tme voltage schedulng algorthm for fxed-prorty hard real-tme systems, whch s the best nown polynomal-tme heurstc for the problem. Although effcent, beng a heurstc, ths algorthm cannot guarantee the qualty of the voltage schedule computed. 1.2 Contrbutons In ths paper, we gve a complete treatment on the optmal voltage schedulng problem for fxed-prorty hard real-tme systems. As wth the wor of Quan and Hu [2001, 2002], we assume that the tmng parameters of each job s nown a pror. Our problem s dentcal to the one solved by Yao et al. [1995], except that the prorty assgnment s changed from the dynamc EDF assgnment to the fxed assgnment. As llustrated by Quan and Hu [2001], the voltage schedulng problem for fxed-prorty tass s more dffcult to solve because the preempton relatonshp among the tass s much more complex to analyze. Frst, we prove that the optmal voltage schedulng problem s NP-hard, whch mples that no optmal polynomal-tme algorthm s lely to exst. Second, we present a fully polynomal tme approxmaton scheme for the problem. A fully polynomal tme approxmaton scheme (FPTAS) s an approxmaton algorthm that taes any ε (>0) as an addtonal nput and returns a soluton whose cost s at most a factor of (1 + ε) away from the cost of the optmal soluton, wth the runnng tme bounded by a polynomal both n the sze of the nput nstance and n 1/ε [Woegnger 1999]. Gven the NP-hardness of the problem, the proposed approxmaton scheme s practcally the best soluton. The proposed approxmaton scheme computes a near-optmal voltage schedule n polynomal tme. By changng ε, the approxmaton scheme can fnd a voltage schedule that s provably arbtrarly close to the optmal soluton. The rest of the paper s organzed as follows. In Secton 2, we formulate the problem and characterze feasble voltage schedules. We descrbe mportant propertes of an energy-optmal voltage schedule n Secton 3, whch provde a bass of later proofs. In Secton 4, we present the ntractablty result of the problem ncludng ts NP-hardness. The FPTAS for the problem s presented n Secton 5. Expermental results are gven n Secton 6, and we conclude wth a summary and drectons for future wor n Secton PROBLEM FORMULATION We consder a set J ={J 1,J 2,..., J J } of prorty-ordered jobs wth J 1 beng the job wth the hghest prorty. A job J J s assocated wth the followng

4 396 H.-S. Yun and J. Km tmng parameters, whch are assumed to be nown offlne: r J : the release tme of J. d J : the deadlne of J. c J : the number of executon cycles requred for J. We use p J to denote the prorty of the job J. We assume that J has a hgher prorty than J f p J < p J. In the rest of the paper, we use nstead of J as a subscrpt of tmng parameters when no confuson arses (e.g., r, d, and c stand for, r J, d J, and c J respectvely). Note that our job model can be drectly applcable to a perodc real-tme system by consderng all the tas nstances wthn a hyperperod of perodc tass. Snce there s a one-to-one correspondence between the processor speed and the supply voltage, we use S(t), the processor speed, to denote the voltage schedule n the rest of the paper. Gven a voltage schedule, the job executed at tme t can be unquely determned and s denoted by job(j, S, t). A voltage schedule S(t) s sad to be feasble f S(t) gves each job the requred number of cycles between ts release tme and deadlne. (An exact characterzaton of a feasble voltage schedule s gven n Secton 2.1.) As wth other related wor [Quan and Hu 2001, 2002; Yao et al. 1995], we assume that the processor speed can be vared contnuously wth a neglgble overhead both n tme and power. Furthermore, we model that the power P, energy consumed per unt tme, s a convex functon of the processor speed; gven a voltage schedule S(t), the power can be wrtten as a functon of tme by P(S(t)). For smplcty, we assume that all the jobs have the same swtchng actvty and that P s dependent only on the processor speed. The goal of the voltage schedulng problem s, therefore, to fnd a feasble schedule S(t) that mnmzes tf E(S) = P(S(t)) dt (1) t s where t s and t f are the lower and upper lmts of release tmes and deadlnes of the jobs n J, respectvely. For the rest of ths paper, the energy-optmal voltage schedule of a job set J s denoted by S J opt. 2.1 Feasblty Analyss In ths secton, we derve a necessary and suffcent condton for a voltage schedule to be feasble, whch wll provde a bass for the proofs n Secton 3. We frst ntroduce some useful notatons and defntons. W (S,[t 1,t 2 ]) s used to denote the number of cycles executed under a voltage schedule S(t) from t 1 to t 2, that s, W (S,[t 1,t 2 ]) = t 2 t 1 S(t) dt. Among W (S,[t 1,t 2 ]) cycles, W (S,[t 1,t 2 ]) denotes the number of cycles between t 1 and t 2 used for executng a set of jobs J 1, J 2,..., J whose prortes are hgher than or equal to p J. R J and D J represent the sets of release tmes and deadlnes of the jobs n J, respectvely, that s, R J ={r J J J}and D J ={d J J J}.T J denotes the unon of R J and D J, that s, T J = R J D J. Gven a job set J J, C(J ) represents the total worload of jobs n J, that s, C(J ) = J J c J. Furthermore, I J

5 Energy-Optmal Voltage Schedulng 397 represents the mnmum nterval that ncludes the executon ntervals of jobs n J, that s, I J = [mn R J, max D J ]. T J represents the Cartesan product of [r J, d J ], for 1 J, that s, T J = [r J1, d J1 ] [r J2, d J2 ] [r J J,d J J ]. Gven voltage schedules S 1, S 2,...,S n such that S (t) = 0 for all t / [α, β ] for all 1 n and β α +1 for all 1 < n, the concatenaton of S 1, S 2,...,S n s n =1 S = S 1 S 2 S n def = n S (t). Snce jobs should be released before they can be processed, we assume that a voltage schedule S always satsfes the constrant that for any t > 0, W (S, [0, t]) C({J r J < t}). The condton for a voltage schedule S(t) to be feasble can be expressed as follows: Condton I (Feasblty Condton). There exsts a J -tuple ( f J1, f J2,..., f J J ) T J such that 1 J r {t t R J t< f J } W(S,[r, f J ]) C({J p J p J r J [r, f J )}). (2) For a J -tuple ( f J1, f J2,..., f J J ) T J, f J can be consdered as a modfed deadlne of J, whch s equal to or precedes the orgnal deadlne d J. (The meanng of the J -tuple s further clarfed n Secton 3.) If S(t) satsfes Condton I for a gven J -tuple ( f J1, f J2,..., f J J ) T J, J completes ts executon by f J for all 1 J. Such J -tuples are sad to be vald wth respect to J, S(t). Theorem 2.1 gves a proof for the feasblty condton. THEOREM 2.1. be feasble. Condton I s a necessary and suffcent condton for S(t) to PROOF. For the necessary part, suppose that S(t) s feasble, that s, J completes ts executon at f J (r J, d J ] for all 1 J. Then, for any r R J such that r < f J, all the hgher prorty jobs whose release tmes are wthn [r, f J ) complete ther executons by f J. So the total amount of wor that should be done wthn [r, f J ] must be greater than or equal to the sum of worload of the jobs. Thus, we have for all 1 J : =1 W(S,[r, f J ]) C({J p J p J r J [r, f J )}). For the suffcent part, assume that Condton I s satsfed for a J -tuple ( f J1, f J2,..., f J J ). By nducton on, we prove that J s gven ts requred executon cycles c J wthn [r J, f J ] for all 1 J. The base case holds trvally. For the nducton step, assume that the proposton holds for all = 1, 2,..., 1. Let r < r J be the earlest tme pont n R J such that no lower prorty jobs (.e., J for > ) are executed wthn [r, r J ], that s, W (S,[r,r J ]) = W 1 (S,[r,r J ]). If such r does not exst, r s set to r J. Then, a

6 398 H.-S. Yun and J. Km hgher prorty job J (.e., J l for l < ) released before r (.e., r J < r) must complete ts executon before r; otherwse, snce any lower prorty jobs cannot be executed wthn [r J, r], we have W (S,[r J,r J ]) = W (S,[r J,r]) + W (S,[r,r J ]) = W 1 (S,[r J,r]) + W 1 (S,[r,r J ]) = W 1 (S,[r J,r J ]), whch contradcts the defnton of r. Snce only hgher prorty jobs (.e., J l for l < ) are executed wthn [r, r J ], the amount of remanng worload of the hgher prorty jobs (whch are released wthn [r, r J )) at tme r J s C({J 1 < r J [r, r J )}) W (S,[r,r J ]). So we have W 1 (S,[r J, f J ]) C( {J 1 < r J [r, r J )} ) W (S,[r,r J ]) + C( {J 1 < r J [r J, f J )} ) = C( {J 1 < r J [r, f J )} ) W (S,[r,r J ]). (3) To complete the nducton, we only need to show that W (S,[r J, f J ]) W 1 (S,[r J, f J ]) s not smaller than c J. (Note that J preempts any lower prorty jobs.) From (3) and the assumpton that Condton I s satsfed, we have W (S,[r J, f J ]) W 1 (S,[r J, f J ]) W (S,[r, f J ]) C( {J 1 < r J [r, f J )} ) (from (3).) C({J 1 r J [r, f J )}) C( {J 1 < r J [r, f J )} ) (from (2).) = C({J }) = c J. A job set J s sad to be an EDF job set f for any J, J J (where p J < p J ), d J d J,ord J r J. When the prorty assgnment follows the EDF polcy, we can prove that Condton I s smplfed as follows: Condton II (EDF Feasblty Condton). For any r R J and d D J (where r < d), W (S,[r,d]) C({J [r J, d J ] [r, d]}). LEMMA 2.2. Gven an EDF job set J, a voltage schedule S(t) of J s feasble f and only f Condton II s satsfed. PROOF. Consder a new job set J ={J 1, J 2,..., J J }, where r J = W (S, [0, r J ]), d J = W (S, [0, d J ]), c J = c J, and p J = p J for all 1 J. Because W (S, [0, t]) s a monotoncally ncreasng functon of t, J s also an EDF job set (.e., for any J, J J, where <, d J d J,ord J r J ). Let S (t) = 1 ( t>0) be the voltage schedule of J. Then, we can easly verfy that the ndex of the job job(j,s,t) s the same as that of job(j,s,w(s, [0, t])). Therefore, J J fnshes ts executon by ts deadlne d J under S(t) f and only f ts correspondng job J J fnshes ts executon by d J (= W (S, [0, d J ])) under S. It s well nown that all the jobs n an EDF job set meet ther deadlnes under a constant speed f and only f the utlzaton rato for any tme nterval

7 Energy-Optmal Voltage Schedulng 399 s less than or equal to 1 [Lu 2000]. That s, S s a feasble voltage schedule of J f and only f the followng s satsfed: For any r R J and d D J (where r < d ), C({J J J [r J, d J ] [r, d ]}) d r. (4) Snce (4) s equvalent to Condton II, Condton II s a necessary and suffcent condton for S(t) to be a feasble voltage schedule of J. As shown n Condtons I and II, the complexty of fxed-prorty voltage schedulng manly comes from the nherent exhaustveness n fndng a vald J -tuple. In the EDF schedulng algorthm, t s suffcent for a sngle J -tuple of the orgnal deadlnes to be checed f t satsfes Condton II. 3. SOME PROPERTIES OF OPTIMAL SCHEDULES In ths secton, we explan several propertes for a feasble voltage schedule to be an energy-optmal schedule. These propertes provde a ey nsght n devsng a fast approxmaton algorthm descrbed n Secton 5. The frst property, whch was proven by Quan and Hu [2001], s that an energy-optmal voltage schedule should be a pecewse-constant functon. The exstng optmal voltage schedulng algorthm by Quan and Hu s based on an observaton that f a gven job set satsfes the requrement of an EDF job set, the optmal voltage schedule can be easly computed by Yao s peapower-greedy algorthm [Yao et al. 1995]. Smply applyng Yao s algorthm to a fxed-prorty job set may cause some jobs to mss ther deadlnes. However, f the deadlnes of the jobs are approprately modfed before schedulng, Yao s algorthm can yeld a feasble optmal schedule as shown n Quan and Hu [2002]. The effcency of an optmal voltage schedulng algorthm s, therefore, dependent on how effcently the job set s modfed to be an EDF job set. To gve a better nsght nto our approach for solvng the voltage schedulng problem, we derve an equvalent result to Quan and Hu [2002] usng Condtons I and II. 3.1 Propertes on J -Tuples Gven a J -tuple f = ( f J1, f J2,..., f J J ) T J, J f represents the job set {J 1, J 2,..., J J }, where p J = p J, c J = c J, r J = r J, and d J = f J for all 1 J. We say that a J -tuple f s EDF ordered f J f follows the EDF prorty. Furthermore, J f s sad to be EDF-equvalent to J. We frst establsh a ln between Condtons I and II. LEMMA 3.1. If Condton I s satsfed for a job set J by a voltage schedule S and an EDF-ordered J -tuple f = ( f J1, f J2,..., f J J ), Condton II s satsfed for a job set J f by S. PROOF. For any r R J f and d D J f (r < d), we have r {t t R J (=R J f) t<d} and d = f J for f J D J f ( ={f J1, f J2,..., f J J }).

8 400 H.-S. Yun and J. Km Furthermore, snce f s EDF-ordered, we have Thus, we have for all J J f : J J f s.t. r J (= r J ) [r, d(= f J )), d J = f J f J = d f p J p J (= p J ) d J = f J > f J = d otherwse. p J p J r J [r, d) [r J, d J ] [r, d]. (5) Fnally, by substtutng d for f J n (2), we have W (S,[r,d]) C({J J p J p J r J [r, d)}) = C({J J f p J p J (= p J ) r J (= r J ) [r, d)}) = C({J J f [r J, d J ] [r, d]}). (from (5).) LEMMA 3.2. If Condton II s satsfed for a job set J f by a voltage schedule S where f = ( f J1, f J2,..., f J J )s an EDF-ordered J -tuple, Condton I s satsfed for a job set J by S. PROOF. Let r {t t R J t< f J }. Then, we have r R J f (= R J ), f J D J f (= {f J1, f J2,..., f J J }) and r < f J and substtutng f J for d n Condton II gves W (S,[r, f J ]) C({J J f [r J, d J ] [r, f J ]}). Snce f s EDF-ordered, we have for all J J f (refer to the proof of Lemma 3.1.): p J p J r J [r, f J ) [r J, d J ] [r, f J ]. (6) Therefore, we have W (S,[r, f J ]) C({J J f [r J, d J ] [r, f J ]}) = C({J J f p J p J (= p J ) r J (= r J ) [r, f J )}) = C({J J p J p J r J [r, f J )}). From Lemmas 3.1 and 3.2, we can derve the followng useful theorem that states how a feasble voltage schedule of a job set can be obtaned from ts EDF-equvalent job sets. THEOREM 3.3. Gven a job set J, let F J be the set of all feasble voltage schedules for J. Then, F J = f TEDF F J f, where T EDF s the set of all EDFordered J -tuples for J. PROOF. To show that S F J S f TEDF F J f, assume that J completes ts executon at f J ( d J ) for all 1 J under S F J. Let f = ( f J1, f J2,..., f J J ). Then, J f s an EDF job set. If not, we have for some J, J l J f (where p J < p J l ) r J < d J l (= f Jl ) < d J (= f J ),

9 Energy-Optmal Voltage Schedulng 401 Fg. 1. An example of EDF-equvalent job sets. whch contradcts a fact that once a hgher prorty job (.e., J ) s released durng the executon of a lower prorty job (.e., J l ), the hgher prorty job completes earler than the lower prorty job (.e., f J < f Jl ). Furthermore, from Lemma 3.1, S(t) s a feasble schedule for the EDF job set J f. Thus, we have S f TEDF F J f. Conversely, gven an EDF-ordered J -tuple f = ( f J1, f J2,..., f J J ), let S F J f be a feasble schedule for the EDF-equvalent job set J f. Then, from Lemma 3.2, S satsfes Condton I for J. Thus, we have S F J. COROLLARY 3.4. Gven a job set J,E(S J opt ) E(SJ f opt ) for any EDF-equvalent job set J f. Furthermore, there exsts an EDF-equvalent job set J f such that S J opt SJ f opt. From Theorem 3.3, there s a one-to-one correspondence between feasble schedules of a fxed-prorty job set J and feasble schedules of J s EDF-equvalent job sets. Snce the energy-optmal schedule S J f opt for an EDFequvalent job set J f can be drectly computed (n polynomal tme) by Yao s algorthm [Yao et al. 1995], the problem of fndng an energy-optmal (feasble) voltage schedule of J s reduced to the problem of fndng an EDF-equvalent job set J f (or to selectng an EDF-ordered J -tuple f) that mnmzes E(S J f opt ). Fgure 1 shows an example of EDF-equvalent job sets and EDF-ordered J - tuples. Fgure 1(a) shows the orgnal job set J ={J 1, J 2 }. In ths example, J 2 has a lower prorty but earler deadlne than J 1,soJ s not an EDF job set. (So Yao s algorthm cannot be drectly appled to J.) In Fgures 1(b) and (c), two job sets are shown, whch are EDF-equvalent to J. The job sets {J 1, J 2 } and {J 1, J 2 } are obtaned by choosng (r J 1, d J1 ) and (d J2, d J2 ) as EDF-ordered J - tuples, respectvely. Both job sets follow the EDF prorty assgnment, 2 and the optmal voltage schedule for each job set can be computed by Yao s algorthm. (As wll be explaned below, the energy-optmal voltage schedule of J s equal or S {J opt dependng on the worload of J 1 and J 2.) Now, we are to restrct the search space of EDF-ordered J -tuples (equvalently, EDF-equvalent job sets). Frst, an EDF-ordered J -tuple f = ( f 1, f 2,..., f J ) does not need to be consdered f for another EDF-ordered J - to S {J 1, J 2 } opt 1, J 2 } tuple f = ( f 1, f 2,..., f J ) ( f), f f for all 1 J. Ths s because for any voltage schedule S(t) feasble under f, S(t) s also feasble under f.wedefne that an EDF-ordered J -tuple f (or J f )sessental f such f does not exst. (The term essental s equvalent to the term NAP n Quan and Hu [2002].) 2 In Fgure 1(c), J need not have an earler deadlne than J for the job set to be an EDF job set; 1 2 d J = d J s suffcent for the job set to be optmally scheduled by Yao s algorthm [Yao et al. 1995]. 1 2

10 402 H.-S. Yun and J. Km Quan s optmal algorthm [Quan and Hu 2002] fnds an optmal voltage schedule by exhaustvely enumeratng all the essental (or NAP) job sets and then applyng Yao s algorthm for each essental job set. Our fast algorthm avods the exhaustveness by carefully enumeratng the essental job sets. 3.2 J -Permutatons It s easy to chec f a J -tuple s EDF-ordered (or essental). On the contrary, t s not obvous how such J -tuples can be enumerated. In ths secton, we descrbe how to construct EDF-ordered J -tuples effcently usng a permutatonbased analyss. Gven a J -tuple f = ( f 1, f 2,..., f J ), let σ f : {1, 2,..., J } {1, 2,..., J } be a permutaton that maps a new tuple ndex when the tuple elements are sorted n a nondecreasng order, that s, f σ 1 (1) f σ 1 (2) f σ 1 ( J ). f f Tes are broen by the prorty, that s, f f = f j where < j, σ f () <σ f (j). (From now on, we call such σ a J -permutaton.) For example, let f = ( f 1, f 2, f 3, f 4 ) = (4, 10, 2, 10). Then, snce f 3 f 1 f 2 = f 4, we have σ(3) = 1, σ (1) = 2, and (from the te-breang rule) (σ (2), σ (4)) = (3, 4). (Equvalently, we have (σ 1 (1), σ 1 (2), σ 1 (3), σ 1 (4)) = (3, 1, 2, 4).) Note that σ 1 () denotes the ndex of the th smallest element n f, that s, f σ 1 () s the th smallest element n f. The followng lemma states that there cannot exst more than one essental J -tuples whose J -permutatons are the same, that s, each essental J - tuple can be unquely addressed by ts correspondng J -permutaton (and, obvously, vce versa). LEMMA 3.5. For any two essental J -tuple f = ( f 1, f 2,..., f J ) and f = ( f 1, f 2,..., f J ) (f f ), σ f σ f. PROOF. Suppose σ f σ f and let (1 J ) be the largest nteger such that f σ 1 () f, that s, f σ 1 f () ( ) f σ 1 () = f = f f σ 1 f () σ 1 () f for all < < J. (7) Wthout loss of generalty, we can assume f σ 1 () < f. Let us consder a new f σ 1 f () J -tuple f = ( f 1, 2,..., f J ), where f = { f f = σ 1 f (), otherwse. From the defnton of f, t can be easly seen that σ f σ f σ f. (We omt the subscrpts n the rest of the proof.) We are now to prove that f s EDF-ordered, that s, for any 1 j < J, f j f or f r J j. (8) Snce f s EDF-ordered, (8) holds for all 1 j < J except for j = σ 1 () or =σ 1 (). So t remans to show that (8) holds for all 1 j <σ 1 () J and 1 σ 1 () < J.

11 Energy-Optmal Voltage Schedulng 403 Fg. 2. The algorthm to buld a J -tuple from a J -permutaton. Case (a). 1 j <σ 1 () J (when J j has a hgher prorty than J σ ().) 1 If f j f, (8) trvally holds. So we only consder j such that f σ 1 () j > f σ 1 (), that s, f j (= f σ 1 (σ ( j ))) > f ( > f σ 1 () σ ()). From the defnton of σ,wehave 1 σ(j)>. Thus, by substtutng σ ( j ) for n Eq. (7), we have f j (= f j ) = f j. From the assumpton, f s EDF ordered, but we have f j = f j > f σ 1 ().Sot must be the case that f σ 1 () r J j. Therefore, we have f σ 1 () = f σ 1 () r J j. Case (b). 1 σ 1 ()< J (when J has a lower prorty than J σ ().) 1 Frst, we can exclude the case when f = f σ (). Otherwse, we have 1 σ () >σ(σ 1 ()) =. (Recall the te-breang rule.) But, by the defnton of σ, f σ 1 (σ ()) (= f ) f and we fnally have σ 1 () f f σ 1 () > f σ 1 () = f, whch contradcts Eq. (7). Second, consder such that f < f σ 1 (). f s EDF-ordered, but we have f σ 1 () > f. So t must be the case that f r Jσ 1 (). Therefore, we have f = f r Jσ 1 (). Fnally, for such that f > f σ (), wehave 1 f σ 1 () = f σ 1 () f = f = f. Thus, f s EDF-ordered. However, snce we have f σ 1 () < f σ 1 () = f σ 1 () and f = f for all 1 σ 1 () J, fs not essental, a contradcton. Therefore, σ f σ f. The proof of Lemma 3.5 also mples how to buld a unque essental job set for σ. LEMMA 3.6. Gven a J -permutaton σ, the algorthm n Fgure 2 fnds a unque essental J -tuple for σ f such a J -tuple exsts. Otherwse, t returns FALSE. PROOF. Frst, suppose that the essental J -tuple for σ exsts and denote t by f = ( f 1, f 2,..., f J ). (Note that f σ 1 (1) f σ 1 (2) f.) We are σ 1 ( J ) to prove that f σ 1 () = f σ (), and the algorthm does not abort n lne 4 for 1 all = J, J 1,..., 1 by nducton on. The base case holds trvally, that

12 404 H.-S. Yun and J. Km s, f σ 1 ( J ) = d J σ 1 ( J ) = f σ 1 ( J ). For the nducton step, assume that the proposton holds for all = J, J 1,...,+1. Let J H ={J σ () < J 1 σ 1 ()<σ 1 ()} (as n lne 3 of the algorthm). Note that any job n J H has the hgher prorty than J σ 1 () and that f σ 1 () d J σ 1 () and f σ 1 () f σ 1 ( + 1). Case (a). J H =. Suppose that f σ 1 () < d J σ 1 () and f σ 1 () < f, that s, σ 1 (+1) f σ 1 (1) f σ 1 () < mn { d Jσ 1 (), f } σ 1 (+1) f σ 1 (+1) f σ 1 ( J ). Let f = ( f 1,..., f σ 1 () 1, mn{d J σ 1 (), f σ 1 (+1) }, f,..., σ 1 f ()+1 J ). Then, f s EDF ordered, and f s not essental, a contradcton. Therefore, we have f σ 1 () = mn { d Jσ 1 (), f σ 1 (+1)} = mn { d Jσ 1 (), f σ 1 (+1)} = fσ 1 (). Case (b). J H. For all J σ 1 () J H,wehave f σ 1 () <f from the defnton of σ (Recall σ 1 () the te-breang rule.), and f σ 1 () r J σ 1 () snce f s EDF-ordered. Suppose that f σ 1 () < mn{r J J J H }, f σ 1 () < d J σ 1 (), and f σ 1 () < f, that s, σ 1 (+1) f σ 1 (1) f σ 1 () < mn ({ d Jσ 1 (), f } { σ 1 (+1) rj J J H}) f σ 1 (+1) f σ 1 ( J ). Let f = ( f 1,..., f σ 1 () 1, mn({d J σ 1 (), f σ 1 (+1) } {r J J J H }), f σ 1 ()+1,..., f J ). Then, t can be easly shown that f s EDF-ordered. Thus, f s not essental, a contradcton. Therefore, we have f σ 1 () = mn ({ d Jσ 1 (), f } { σ 1 (+1) rj J J H}) = mn ({ { d Jσ 1 (), f σ (+1)} 1 rj J J H}) = f σ (). 1 Furthermore, we have for both cases r Jσ 1 () < f σ 1 () mn ( {r J J J H } { f σ 1 (+1)}) = mn ( {rj J J H } { f σ 1 (+1)}), and the algorthm does not abort n lne 4 at teraton, whch completes the nducton. If the algorthm does not abort, the J -tuple bult by the algorthm s always a correct EDF-ordered J -tuple, mplyng the exstence of such J -tuple for σ. Therefore, f such J -tuple does not exst, the algorthm eventually returns FALSE. If a J -permutaton σ has the correspondng EDF-ordered J -tuple f, t s sad to be vald. Furthermore, f f s essental, σ s sad to be essental. From the above argument, we can establsh one-to-one correspondences between EDF-ordered J -tuples and vald J -permutatons, and between essental J - tuples and essental J -permutatons. Fgure 3(a) shows a job set wth three jobs, and Fgures 3(b) (d) show ts EDF equvalent job sets wth ther J - permutatons. Among 3!(= 6) possble J -permutatons, only three permutatons are vald (and essental).

13 Energy-Optmal Voltage Schedulng 405 Fg. 3. An example of J -permutatons. (a) A job set and ts EDF-equvalent job sets for whch (σ 1 (3), σ 1 (2), σ 1 (1)) = (b) (2, 3, 1), (c) (2, 1, 3), and (d) (3, 2, 1), respectvely. ((σ 1 (3), σ 1 (2), σ 1 (1)) = (1, 2, 3), (1, 3, 2), and (3, 1, 2) are not vald J -permutatons.) Fg. 4. The algorthm to buld a J -tuple from a bt-vector. Based on the algorthm n Fgure 2, we descrbe another way to enumerate J -tuples. In the followng, r J and d J are nterpreted as symbolc values, not as real numbers. Then, R J D J has 2 J dstnct symbolc values. Furthermore, the algorthm n Fgure 2 s assumed to assgn symbolc values to elements of a J -tuple wth the followng te-breang rule n lne 5: (a) r J = r J j ( < j ):r J <r Jj, (b) d J = d J j ( < j ):r J <r Jj, (c) r J = d J j : r J < d J j. Gven a J -tuple f = ( f 1, f 2,..., f J ), let ζ f : R J D J {0, 1} be a bt-vector of length 2 J such that { 1 t = f for some 1 J, ζ f (t) = 0 otherwse. The algorthm n Fgure 4 constructs a J -tuple from an arbtrary bt-vector ζ : R J D J {0, 1}. The correctness of the algorthm can be proved n a smlar manner as the algorthm n Fgure 2.

14 406 H.-S. Yun and J. Km 3.3 An Alternatve Formulaton The problem formulaton gven n Secton 2 s based on the voltage schedule S(t). In ths secton, we descrbe an alternatve formulaton, based on the followng ntutve property, whch states that each job runs at the same constant speed f the voltage schedule s an optmal one. LEMMA 3.7. For an energy-optmal voltage schedule S(t), S(t 1 ) = S(t 2 ) for any t 1 and t 2 such that j ob(j, S, t 1 ) = job(j,s,t 2 ). PROOF. Gven an optmal schedule S(t), suppose that S(t 1 ) S(t 2 ) for some t 1 and t 2 such that job(j, S, t 1 ) = job(j, S, t 2 ). Gven that S(t) s optmal, there exst t 1, t 2, S 1, S 2, and t such that S(t) = S 1 for t 1 t t 1 + t, S(t) = S 2 for t 2 t t 2 + t, and S 1 S 2. Let S(t) be defned by S 1 + S 2 S(t) t 1 = 2 t t 1 + t, t 2 t t 2 + t, S(t) otherwse. Then, t s obvous that S(t) s feasble, and E(S ) < E(S), a contradcton. From Lemma 3.7, t can be shown that the voltage schedulng problem s equvalent to determnng the allowed executon tme a allocated to each J. Gven a feasble voltage schedule S, the correspondng tuple of the allowed executon tmes (a 1, a 2,...,a J ), called a tme-allocaton tuple, can be unquely determned. Conversely, gven a tme-allocaton tuple A = (a 1, a 2,...,a J ), the correspondng voltage schedule S A can be unquely constructed by assgnng the constant executon speed c /a to J. A s sad to be feasble f the correspondng voltage schedule S A s feasble. Let us now consder the exact condton for a tme-allocaton tuple A = (a 1, a 2,...,a J ) to be feasble by rewrtng Condton I n Secton 2 n terms of A. Condton III (Feasblty Condton for Tme-Allocaton Tuples). There exsts a J -tuple ( f J1, f J2,..., f J J ) T J such that 1 J r {t t R J t< f J } a f J r. (9) J /p J p J r J [r,f J ) LEMMA 3.8. feasble. Condton III s a necessary and suffcent condton for A to be PROOF. Gven a job set J ={J 1, J 2,..., J J } and a tme-allocaton tuple A = (a 1, a 2,...,a J ) for J, consder a new job set J ={J 1, J 2,..., J J }, where c J = a, r J = r J, d J = d J, and p J = p J for all 1 J, that s, J s dentcal to J except for the worload. Let S (t) = 1 ( t>0) be the voltage schedule of J. Then, t s obvous that the response tme of J under S A s the same as that of J under S. Thus, A s feasble f and only f S s a feasble voltage schedule for J.

15 Energy-Optmal Voltage Schedulng 407 Fg. 5. Soluton spaces for (a) an EDF job set and (b) a fxed-prorty job set. After replacng S and c J Condton III. n Condton I by S and a, respectvely, we have By applyng the same argument to Condton II, we have the followng condton for EDF job sets. Condton IV (EDF Feasblty Condton for Tme-Allocaton Tuples). For any r R J and d D J (where r < d), a d r. J/[r J,d J ] [r,d] Now the voltage schedulng problem can be reformulated as follows: Fnd a tme-allocaton tuple A = (a 1, a 2,...,a J ) such that E(S A ) s mnmzed subject to Condton III (or Condton IV for an EDF job set). The energy consumpton of the voltage schedule S A can be computed drectly: E(S A ) = J =1 a P(c /a ). (10) The set of feasble tme-allocaton tuples represents the soluton space for the voltage schedulng problem stated n terms of tme-allocaton tuples. For an EDF job set, the soluton space s specfed by a conjuncton of lnear nequaltes that can be drectly obtaned from Condton IV. However, ths s not the case for a fxed-prorty job set; the exstental quantfer n Condton III s not always removable. Consequently, the soluton space for an EDF job set s a convex set whle the soluton space for an arbtrary fxed-prorty job set may not be a convex set. Before we present an ntractablty result for the voltage schedulng problem n the next secton, we llustrate the nherent complexty of fxed-prorty voltage schedulng based on the results explaned n ths secton. Fgures 5(a) and (b) show the soluton spaces for an example EDF job set and an example

16 408 H.-S. Yun and J. Km fxed-prorty job set, respectvely. As a fxed-prorty job set, we use the job set {J 1, J 2 } of Fgure 1. As an EDF job set, we use the same job set {J 1, J 2 } n Fgure 1 wth the same tmng parameters, but the prorty assgnment s changed such that t follows the EDF prorty assgnment, that s, p J2 < p J1. For the EDF job set, we have the followng constrant: a 1 d J1 r J1 a 2 d J2 r J2 a 1 + a 2 d J1 r J2. Smlarly, we have the followng constrant for the fxed-prorty job set: a 1 d J1 r J1 a 2 r J1 r J2 (Fgure 1(b)) a 1 d J2 r J1 a 1 + a 2 d J2 r J2 (Fgure 1(c)). In Fgures 5(a) and (b), the soluton spaces for the EDF job set and the fxed-prorty job set are depcted as a convex regon and a concave regon, respectvely. (Each pont n the shaded regons represents a feasble schedule.) In general, the soluton space of any EDF job set wth N jobs s represented by a convex set n R N, whereas the soluton space of a fxed-prorty job set s represented by a concave set. Note that for EDF job sets, the objectve functon, the total energy consumpton, can be effcently mnmzed by an optmzaton technque for a convex set (as n Yao s algorthm). However, optmzaton problems defned on a concave set are generally ntractable. 4. INTRACTABILITY RESULT In ths secton, we present some observatons related to the complexty ssue of the optmal fxed-prorty schedulng problem. We frst show that the decson verson of the problem s NP-hard. THEOREM 4.1. Gven a job set J and a postve number K, the problem of decdng f there s a feasble voltage schedule S(t) for J such that E(S) Ks NP-hard. PROOF. Wthout loss of generalty, we assume that the energy consumpton (per CPU cycle) s quadratcally dependent on the processor speed. That s, the nstantaneous power consumpton (per tme) s cubcally dependent on the processor speed, that s, P(t) = S(t) 3. (The reducton can be easly modfed for other power functons.) We prove the theorem by reducton from the subset-sum problem, whch s NP-complete [Garey and Johnson 1979]: SUBSET-SUM INSTANCE: A fnte set U, a sze s : U Z +, and a postve nteger B. Queston: Is there a subset U U such that u U s(u) = B? Gven an nstance U (= {u 1,...,u U }), s, B of the subset-sum problem, we construct a job set J and a postve number K such that there s a voltage schedule S(t) ofj wth E(S) K f and only f U U, u U s(u) = B. The correspondng job set J conssts of 2 U +1 jobs as follows: J = { } J 1, J 2,..., J 2 U +1

17 Energy-Optmal Voltage Schedulng 409 where and p J = for all 1 2 U +1, r J2 + 1 = s(u +1 ) + 3 s(u j ), r J2 + 2 = d J2 +1 = +1 j =1 j =1 3 s(u j ), j =1 3 s(u j ), d J2 +2 = 2 s(u +1 ) + 3 s(u j ), c J2 +1 = 8 γ s(u +1 ), c J2 +2 = 8 s(u +1 ) for all 0 U 1, r J2 U +1 = 0, d J2 U +1 = B + j =1 3 s(u j ), c J2 U +1 = 3 4 B, j =1 where γ s the unque postve soluton of the followng quadratc equaton: γ 2 + γ = ( 12 ) 3 8 <γ <1. 3 Furthermore, K s set to be K = (8 3 + γ 3 ) 4 U 83 s(u ) + 2 B. From the constructon of J,wehave ( r J2 +2 < r J2 +1 =rj2 +2 +s(u +1 ) ) ( <d J2 +2 =rj2 +1 +s(u +1 ) ) ( < d J2 +1 =dj2 +2 +s(u +1 ) ), [,d ] [ rj2 +2 J 2 +1, d rj2 U +1 J 2 U +1] for all 0 U 1 and Let κ : {0, 1} U where and [, d ] [ rj2 +2 J 2 +1, d ] rj2 +2 J 2 = for all 0 U =1 T J be a functon defned by κ((b 1, b 2,...,b U )) = ( f 1, f 2,..., f J ) f 2 +1 = d J2 +1, f 2 +2 = r J2 +1 f b +1 = 0, f 2 +1 = f 2 +2 = d J2 +2 f b +1 = 1 for all 0 U 1, f 2 U +1 = d J2 U +1. Then, the set of essental job sets of J s gven by { J f f = κ(b), b {0, 1} U }.

18 410 H.-S. Yun and J. Km To compute the energy consumpton of an essental job set by Yao s algorthm [Yao et al. 1995], we frst compare the ntensty of each nterval. Let I 1 = c J2 +2 r J2 +1 r J2 +2, I 2 = c J2 +1 d J2 +1 r J2 +1, and Then, we have and I 1 = 8 s(u +1) s(u +1 ) I 3 = I 4 = 8 (1 + γ ) s(u +1) 2 s(u +1 ) c J2 +1 d J2 +2 r J2 +1, I 5 = c J 2 U +1 B + δ. = 8 > I 2 = 8 γ s(u +1) 2 s(u +1 ) I 4 = c J c J 2 +2 d J2 +2 r J2 +2 = 4 γ>2> 3 4>I 5 = = γ>i 3 = 8 γ s(u +1) s(u +1 ) 3 4 B B+δ =8 γ>i 5. So the energy consumpton of S J f opt for f = κ((b 1, b 2,...,b U )) can be computed as follows: E ( S J ) U f opt = E + E L =1 where (8 3 + γ 3 ) 4 83 s(u ) (= (8 s(u )) 3 + (8 γ s(u )) 3 ) s(u ) E = 2 (2 s(u )) 2 (1 + γ ) s(u ) (= (8 (1 + γ ) s(u )) 3 ) 4 (2 s(u )) 2 and 4 B 3 ( c 3 E L = (B + J U =1 b = 2 U +1 s(u )) 2 (B + U =1 b (d J2 1 d J2 )) 2 Snce we have (1 + γ ) s(u ) = γ + 3 γ 2 + γ s(u ) 4 4 = (1 + 4/(3 83 )) + γ s(u ) 4 = (8 3 + γ 3 ) 4 83 s(u ) + s(u ), we can rewrte E(S J f opt ) as follows: E ( S J ) f opt = (8 3 + γ 3 ) 4 U 83 s(u ) + x + 4 B3 (B + x) 2 =1 b = 0, b = 1 ).

19 where x = Energy-Optmal Voltage Schedulng 411 U =1 b s(u ). It can be easly shown that E(S J f opt ) has the mnmum (83 + γ ) U =1 s(u ) + 2.B (= K )atx=b. That s, E(S J f opt ) K f and only f whch s equvalent to U (b 1, b 2,...,b U ) {0, 1} U, b s(u ) = B, =1 U U, u U s(u) = B. It s obvous that the transformaton can be done n polynomal tme. Therefore, the problem s NP-hard. From the NP-hardness proof, the problem seems unlely to have polynomal tme algorthms that compute optmal solutons. The NP-hardness of the problem strongly depends on the fact that extremely large nput numbers are allowed, as wth some other NP-hard problems (e.g., the subset-sum problem and the napsac problem [Garey and Johnson 1979]). The NP-hardness n the ordnary (but not strong) sense does not rule out possblty of exstence of a pseudopolynomal tme algorthm or an FPTAS. Snce our problem s an optmzaton problem that handles real numbers, we focus our attenton on the FPTAS n the next secton. 5. A FAST APPROXIMATION SCHEME In ths secton, we present a fully polynomal tme approxmaton scheme (FP- TAS) for the problem. We frst consder a dynamc programmng formulaton that always fnds the optmal soluton, but may run n exponental tme. Then, the dynamc programmng formulaton s transformed nto an FPTAS by usng a standard technque, the roundng-the-nput-data technque [Woegnger 1999]. The technque brngs the runnng tme of the dynamc program down to polynomal by roundng the nput data so that suffcently close nput data are treated by a representatve data [Sahn 1976]. The relatve error of an approxmaton scheme depends on how we defne the closeness; the smaller the threshold value for the closeness, the smaller the relatve error. For a smaller error bound, however, the computaton tme becomes longer. 5.1 Algorthm for Optmal Solutons We frst present an exponental-tme optmal algorthm based on the propertes of optmal voltage schedules descrbed n Secton 3. The exponental-tme algorthm essentally enumerates all the essental job sets. However, unle Quan s exhaustve algorthm [Quan and Hu 2002], t enumerates the essental job sets ntellgently wthout actually enumeratng all of them. Furthermore, t s based

20 412 H.-S. Yun and J. Km Fg. 6. An example llustratng the optmal algorthm. (a) An orgnal job set; and (b) an essental job set defned by a J -tuple f = ( f 1, f 2,..., f N 3, f N 2, f N 1, f N )=(d 2,d 2,...,d N 1, r N 3,r N 3,r N 3 ). Jobs n each subnterval between the thc dashed lnes follow the EDF prorty assgnment and can be optmally scheduled by Yao s algorthm. on dynamc programmng formulaton so that t can be easly transformed nto an FPTAS by the standard technque. In formulatng the problem by dynamc programmng, we frst dentfy approprate overlappng (or reusable) subproblems to whch dynamc programmng can be appled teratvely. We note that the optmal substructure of our problem s naturally reflected by blocng tuples, whch are just sequences of tme ponts n T J n strctly ncreasng order. (We formally defne the blocng tuples later n ths secton.) That s, the optmal soluton of the orgnal problem can be bult by just mergng the optmal schedules of the subntervals defned by a blocng tuple. Fgure 6 shows an example job set and ts correspondng EDF-equvalent job set whose tme nterval s parttoned by a blocng tuple (r N, r N 3, d N 1,...,r 2,d 2 ), whch s depcted by a set of the dashed thc lnes n Fgure 6(b). Note that jobs n each subnterval follow the EDF-prorty assgnment. The orgnal problem s parttoned nto subproblems by parttonng the overall tme nterval nto subntervals such that jobs n each subnterval follow the EDF prorty assgnment. If a job s released wthn a subnterval wth ts deadlne outsde the subnterval, the deadlne can be modfed to the end of the

21 Energy-Optmal Voltage Schedulng 413 subnterval. Each parttoned nterval can be optmally scheduled n polynomal tme by Yao s algorthm [Yao et al. 1995]. The challenge s how to fnd the set of subntervals whose optmal subschedules buld an energy-optmal voltage schedule Basc Idea: The Frst Example. We now explan the basc dea of the optmal algorthm by descrbng the optmal algorthm on a smple but llustratve job set J ={J 1, J 2,..., J N }n Fgure 6(a), where r +1 < r < d +1 < d for 1 < N. (Note that f the prortes of jobs are reversed, the job set follows the EDF prorty.) For ths job set, an essental job set J e (such as one n Fgure 6(b)) s parttoned nto J1 e, J 2 e,...,je e such that each J (1 ) follows the EDF prorty assgnment and the unon I of executon ntervals of jobs n J e (.e., I = J J e [r J,d J ]) does not overlap wth I j (= J Jj e[r J,d J ]) for all 1 j. To be more concrete, for all 1 < j <, J J e, J J j e, d J r J. Therefore, the optmal voltage schedule S J e opt of J e s equal to the concatenaton of the optmal voltage schedules of J e, that s, S J e opt (t) e =1 SJ opt (t). Note that S J e opt can be drectly computed by Yao s algorthm [Yao et al. 1995] snce J e follows the EDF prorty assgnment. Therefore, the energy-optmal fxed-prorty voltage schedulng problem s further reduced to the problem of fndng a partton that gves the energy-optmal voltage schedule for the whole tme nterval. In defnng a partton, we use a blocng tuple. For example, assume that f N s selected as r N 3 as n Fgure 6(b). Then, both f N 1 and f N 2 should be selected as r N 3, so that the job set becomes EDF-equvalent and, furthermore, essental. As shown n Fgure 6(b), these three jobs are separated from the other jobs by a thc vertcal lne at tme r N 3. These jobs consttute the frst parttoned job set J1 e. The remanng job sets J 2 e,...,je can be constructed by applyng the same argument. In ths way, any essental job set can be parttoned and represented by a blocng tuple. Let b = (b 1, b 2,...,b l )(b 1 < b 2 < <b l,b j T J ) be a blocng tuple where 1 j < l, J s.t. b j = r b j +1 d. Then, the correspondng EDF-ordered J -tuple f = ( f 1, f 2,..., f N ) s gven by f = b j s.t. r [b j 1, b j ) for all 1 N. We call such [b j 1, b j ]anatomc nterval. For example, the ntervals [r N, r N 3 ] and [r N, d N ] n Fgure 6(a) are atomc, but the nterval [r N, d N 1 ] s not atomc. (Later, we wll formally defne the term atomc nterval n arbtrary job sets other than ths example.) Let t h be the hth earlest tme pont n T J, and let S h, g represent the energy-optmal voltage schedule defned wthn [t h, t g ] for the job set J h, g defned by J h, g ={J r J [t h,t g )}

22 414 H.-S. Yun and J. Km where r J = r J, c J = c J, p J = p J, and d J = mn{d J, t g }. Then, we have E ( 1 Sopt) J = E(S1, TJ ) = mn E(S h j,h j +1 ) 1 = h 1 < h 2 < <h = J j =1 and [ ] th j, t h j +1 s atomc for all j = 1,..., 1}. Gven an atomc nterval [t h j, t h j +1 ], S h j,h j +1 can be drectly computed by Yao s algorthm. In ths way, the optmal voltage schedulng problem s reduced to a varant of the subset-sum problem. That s, for such job sets as n Fgure 6, our problem can be formulated as follows: Select a tuple (h 1, h 2,...,h )(1=h 1 < <h = J ) of ntegers such that the sum q h1,h 2 + q h2,h 3 + +q h 1,h s mnmzed subject to [t h, t h+1 ] s atomc for all 1 <, where q h j,h j +1 denotes E(S h j,h j +1 ) (whch can be drectly computed by Yao s algorthm) Basc Idea: The Second Example. The example job set n Fgure 6 s llustratve n showng how our problem can be formulated by dynamc programmng. However, the easly parttonable structure comes from the fact the job set follows the reverse EDF prorty. For example, n Fgure 6, snce f N s set to be r N 3, whch s wthn the executon ntervals of J N 1 and J N 2, f N 1 and f N 2 cannot be larger than f N (or r N 3 ) so that the modfed job set should be EDF-equvalent. Furthermore, f N 1 and f N 2 are set to be the maxmum possble value, f N, for the modfed job set to be essental. If the prorty pattern s not the same as the example job set n Fgure 6, the parttonng becomes dffcult. For example, the essental job sets n Fgures 3(c) and (d) cannot be obtaned by the parttonng procedure just explaned. In Fgure 7(a), J 4 has the lowest prorty and the latest deadlne, whch maes f 4 to be d 4 for all essental job sets (Fgures 7(a) (c)). Therefore, any atomc nterval (e.g., [r 3, r 1 ], [r 1, d 1 ], or [r 3, d 3 ]) contans partal worload of J 4, whch we call a bacground worload. In the followng, we frst explan how to extend the dynamc programmng formulaton to handle the bacground worload. Then, we descrbe how to explore essental job sets of a gven arbtrary job set (as n Fgure 3) by dynamc programmng. From Lemma 3.7, the job J 4 n Fgure 7 runs at the same speed f the voltage schedule s an optmal one. For the tme beng, let us assume that the constant speed s among S C ={s 1,s 2,s 3 }. (For now, S C s set to be the set of all the possble constant speeds n the optmal voltage schedule. In Secton 5.2, we explan how the set S C s selected such that the sze of S C s bounded by a polynomal functon.) For each s S C, we frst compute the amount of bacground worload

23 Energy-Optmal Voltage Schedulng 415 Fg. 7. An example of bacground worload. Fg. 8. An example llustratng the algorthm on a job set wth bacground worload; (a) atomc ntervals (obtaned from the job set n Fgure 7(b)). The optmal schedules for two atomc ntervals where the speeds of bacground worload of J 4 are (b) s 1, (c) s 2 and (d) s 3, respectvely. The voltage schedules for overall tme ntervals where the speeds of J 4 are (e) s 1, (f) s 2, and (g) s 3, respectvely. of J 4 for each atomc nterval, and then fnd the mnmum-energy essental job set (among those n Fgures 7(b) (d)) by usng the smlar procedure to the prevous case n Fgure 6. However, unle the prevous case, we dscard any job set for whch the sum of bacground worloads executed n overall tme nterval s less than the total worload of J 4. Fgure 8(a) shows the atomc ntervals [r 3, r 1 ] and [r 1, d 1 ], whch are obtaned from the essental job set n Fgure 7(b). Fgures 8(b) (d) show the optmal voltage schedules for the atomc ntervals, where J 4 runs at the speed s 1, s 2, and s 3, respectvely. The worloads of jobs J 1, J 2, and J 3 are denoted by c 1, c 2, and c 3, respectvely, and the bacground worloads are denoted by w. The amount of the bacground worload (and the resultant optmal voltage schedule) for each atomc nterval and speed can be easly computed by a slghtly modfed

24 416 H.-S. Yun and J. Km Fg. 9. The algorthm to buld a strongly blocng tuple from a J -permutaton. verson of Yao s algorthm [Yao et al. 1995]. That s, when the crtcal nterval s selected, f the speed to be assgned (by the ntensty of the crtcal nterval) s less than or equal to the speed of the bacground worload, we assgn the speed of the bacground worload to all the unscheduled tme ntervals (ncludng the crtcal nterval). Then, the amount of bacground worload can be drectly computed as n Fgures 8(b) (d). Once the bacground worload and the optmal voltage schedule are computed for each atomc nterval, we apply the same procedure as n the job set n Fgure 6 to fnd the mnmum-energy essental job set and the energy-optmal voltage schedule. In explorng the soluton space, we should dscard any nfeasble schedules. Fgure 8(e) shows an nfeasble schedule, where J 4 runs at s 1 and cannot complete ts executon untl ts deadlne. The voltage schedule n Fgure 8(g) s feasble, but not an optmal one. Thus, only the schedule n Fgure 8(f) s not removed n the prunng procedure and s compared wth another schedules obtaned from the essental job sets n Fgures 8(c) and (d) Puttng It Altogether. We now descrbe the optmal algorthm for arbtrary job sets based on the observatons from the example job sets. Frst, we formally defne the terms strongly atomc nterval and strongly blocng tuple. Gven a vald J -permutaton σ, the algorthm n Fgure 9 bulds the correspondng strongly blocng tuple b σ = (b 1, b 2,...,b ), where b 1 < b 2 < <b and b T J for all 1. The algorthm s dentcal to the algorthm n Fgure 2 except for lnes 2, 8, 9, and 11. In lne 8, f σ 1 () s selected as an element of a strongly blocng tuple f t parttons the executon nterval. Defnton 5.1. Gven a vald J -permutaton σ, the tuple b σ bult by the algorthm n Fgure 9 s called a strongly blocng tuple. An nterval [t, t ]s strongly atomc f there s a strongly blocng tuple b = (b 1, b 2,...,b ) such that [t, t ] = [b, b +1 ] for some 1 <. Furthermore, the job set J [t,t ] s defned by J [t,t ] ={J J J,r J [t,t )}