Competitively Scheduling Tasks with Intermediate Parallelizability

Transcription

1 Copetitively Scheduling Tasks with Inteediate Paallelizability Sungjin I Electical Engineeing and Copute Science Univesity of Califonia Meced, CA si3@uceced.edu Benjain Moseley Toyota Technological Institute Chicago, IL oseley@ttic.edu Eic Tong Dept. of Copute Science and Engineeing Michigan State Univesity East Lansing, MI tong@su.edu Kik Puhs Dept. of Copute Science Univesity of Pittsbugh Pittsbugh, PA 5260 kik@cs.pitt.edu ABSTRACT We intoduce a scheduling algoith Inteediate-SRPT, and show that it is O(log P )-copetitive with espect to aveage waiting tie when scheduling jobs whose paallelizability is inteediate between being fully paallelizable and sequential. Hee the paaete P denotes the atio between the axiu job size to the iniu. We also show a geneal atching lowe bound on the copetitive atio. Ou analysis builds on an inteesting cobination of potential function and local copetitiveness aguents. Categoies and Subject Desciptos F.2.2 [Nonnueical Algoiths and Poble]: Sequencing and scheduling Geneal Tes Algoiths, Theoy Keywods Scheduling, Paallelization, Speedup cuves.. INTRODUCTION Due to the effects of Mooe s law, aound a decade ago chip akes such as Intel hit a theal wall, whee the cost of cooling becae pohibitive if all switches wee devoted to a single high speed pocesso. In esponse the chip akes abuptly switched to pedoinantly poducing ultipocesso chips []. The advantage of ultipocesso chips is Peission to ake digital o had copies of all o pat of this wok fo pesonal o classoo use is ganted without fee povided that copies ae not ade o distibuted fo pofit o coecial advantage and that copies bea this notice and the full citation on the fist page. Copyights fo coponents of this wok owned by othes than ACM ust be honoed. Abstacting with cedit is peitted. To copy othewise, o epublish, to post on seves o to edistibute to lists, equies pio specific peission and/o a fee. Request peissions fo peissions@ac.og. SPAA 4, June 23 25, 204, Pague, Czech Republic. Copyight 204 ACM /4/06...$ that k pocessos with speed s/k would use only about /k 2 faction of the dynaic powe of a single speed s pocesso (assuing the standad cube-oot ule elationship between dynaic powe and speed), but potentially would have the sae pocessing capability; of couse, fully utilizing the pocessing capability of a ultipocesso is a gand challenge. Ou focus hee is on one of these challenges, naely the scheduling of tasks. One (not univesally accepted) vision of the futue is aticulated by Anant Agawal, CEO of Tilea [2].: I would like to call it a coollay of Mooe s Law that the nube of coes will double evey 8 onths. Tilea cuently poduces poducts with ode of 0 2 pocessos [], and poducts with ode of 0 3 pocessos ae in eseach and developent [2]. In such settings, thee will likely often be oe pocessos than tasks, and thus a schedule would have to patition the pocessos aong the tasks. To achieve optial pefoance, the schedule ust conside the paellizability of tasks when patitioning and scheduling. Fo exaple, wheeas soe highly paallel tasks ight be sped up alost linealy when assigned additional pocessos and thus benefit geatly fo being given oe pocessos, othe highly sequential tasks ight not be sped up at all when assigned additional pocessos. In between these two extees lie pehaps the ajoity of tasks which have inteediate levels of paallelizability. The initial otivating questions fo the eseach that we epot on hee ae: What is the best algoith to schedule jobs with inteediate paallelizability, and what wost-case elative eo guaantee does this algoith give? Unde standad assuptions (which we will elaboate on oentaily) it is clea how to optially schedule n fully paallelizable tasks on pocessos: all pocessos ae allocated to the task with the least aount of unpocessed wok. Let us call this algoith Paallel-SRPT. Futhe it is known how to schedule n fully sequential tasks on pocessos in an optially copetitive way: the up to

2 tasks with the least unpocessed wok ae each allocated one pocesso. Let us call this algoith Sequential-SRPT. Fo sequential jobs with sizes ae between and P, Sequential- SRPT is O(log P )-copetitive, with espect to the objective of aveage waiting tie [0]. Futhe, this copetitive atio is best possible fo online algoiths [0]. It was peviously not known how to schedule jobs of inteediate paallelizability in an optially copetitive way, and it was not clea a pioi what the best scheduling policy would be. Pesuably the ight algoith should agee with Paallel-SRPT when jobs ae fully paallelizable, and agee with Sequential-SRPT when the jobs ae sequential. Afte a oent s eflection, the ost obvious popety that both Paallel-SRPT and Sequential-SRPT shae is that they schedule jobs in such a way as to axiize the ate of eduction of the factional nube of unfinished jobs, unde the assuption that the oiginal size of each job was its cuent size. So pehaps the ost natual candidate fo the best algoith to schedule tasks with inteediate paallelizability would again be to assue that the eaining unfinished wok of each job was its oiginal wok, and then geedily axiize the ate that the factional nube of unfinished jobs is being educed by. Quite supisingly (at least to us) we show in Section 3 that the copetitive atio of this natual hybid algoith is lage. Ou ain esult is a less obvious algoith (though still siple and natual), which we call Inteediate-SRPT, that we show is optially copetitive fo all inteediate levels of paallelizability. We now descibe the Inteediate-SRPT algoith, intoduce ou natual odel fo inteediate paallelizability, state ou uppe bound on the copetitive atio of the Inteediate-SRPT algoith, and finally state ou geneal atching lowe bound on the copetitive atio of any algoith. Inteediate-SRPT Algoith Desciption: If thee ae at least tasks, the tasks with the least unpocessed wok ae each allocated one pocesso (this is like Sequential- SRPT). If thee ae stictly fewe than tasks, the pocessos ae evenly patitioned aong the tasks (this is essentially the Round Robin o Pocesso Shaing Algoith). Modeling Inteediate Paallelizability: We assue that fo each job j thee exists an α j (0, ) such that the speedup cuve fo job j is Γ j(x) = x fo x, and Γ j(x) = x α j fo x. The speedup cuve gives the ate that wok is pocessed if the job is allocated x pocessos. Note that α j = 0 coesponds to a sequential job and α j = coesponds to a fully paallelizable job. We believe speed-up cuves of the fo x α j give a natual way of intepolating inteediate degees of paallelizability without being gounded in any specific achine odel. Theoe. Fo jobs of inteediate paallelizability, the algoith Inteediate-SRPT has copetitive atio O() 4 /( α) log P with espect to aveage waiting tie, whee α = ax j α j. In paticula, this holds fo the special case that each α j = α. Theoe 2. Fo all α [0, ), the copetitive atio of evey algoith with espect to aveage waiting tie, esticted to instances with tasks with speedup cuves of the fo Γ j(x) = x fo x, and Γ j(x) = x α fo x, is Ω(log P ). Taken togethe, these esults show (again soewhat supisingly to us) that scheduling jobs that ae even slightly less than fully paallelizable is oe like scheduling sequential jobs than like scheduling fully paallelizable jobs. The lowe bound fo the natual hybid algoith shows that its eo is that it will soeties allocate too any pocessos to one job. This is the ight stategy if the jobs ae fully paallelizable, but can lead to a lage elative eo if the jobs ae even a bit less than fully paallelizable. The algoith Inteediate-SRPT coects this eo by functioning as Sequential-SRPT when the syste is undeloaded, and shaing the pocessos equally when the syste is oveloaded. Theoe and Theoe 2 togethe establish that the optial copetitive atio jups fo to Θ(log P ) the instant α <. Theoe is poved in Section 2. Theoe 2, is poved in Section 4. But fist, we eview standad odeling assuptions and notation, and eview the ost closely elated papes in the liteatue.. Standad Modeling Assuptions and Notation Thee ae identical unit-speed pocessos. Each task/job j has thee chaacteistics: a elease tie j when it aives in the syste, a size o wok aount p j [, P ] specifying the aount of pocessing that has to be pefoed on job j to finish it, and a speed-up cuve Γ j(x) specifying the ate at which wok on job j is pocessed if assigned x pocessos. A job is fully paallelizable if Γ j(x) = x, and is sequential if Γ j(x) = x fo x, and Γ j(x) = fo x. A job j has inteediate paallelizability if thee exists an α (0, ) such that the speed-up cuve fo j is Γ j(x) = x fo x, and Γ j(x) = x α fo x. If p j(t) is the aount of unpocessed wok on job j at tie t, then the factional nube of jobs at tie t is p j (t) j p j. The flow/waiting/esponse tie fo job j in a schedule S is F S j = C S j j which is the length of tie between when the job is eleased and when the job is copleted in schedule S, and the aveage flow/waiting/esponse tie of the schedule is j F S j /n. Within the context of this pape, the copetitive (appoxiation/wost-case) atio of an online scheduling algoith A is the axiu ove all inputs I with job sizes in the ange [, P ], of the atio between total flow tie fo the schedule poduces by A on I and the optial flow tie fo instance I (this is essentially just a easue of wost-case elative eo)..2 Related Liteatue The speedup cuves odels was intoduced into the liteatue in [5], who showed that equally patitioning the pocessos aong the jobs is 2-copetitive fo total waiting tie if jobs have abitay speedup cuves and all jobs ae eleased at the sae tie. The othe standad way to easue the quality of an online scheduling algoith, beside copetitive atio, is esouce/speed augentation analysis [9, 3]. An online algoith A is s-speed c-copetitive if fo all inputs I, the cost fo A on I with s-speed pocessos is at ost c ties the optial cost fo I on speed pocessos. An algoith is scalable if it is ( + ɛ)-speed O()-copetitive fo all fixed constant ɛ > 0.

3 [4] showed that patitioning the pocessos equally aongst the jobs is (2+ɛ)-speed O()-copetitive with espect to aveage waiting tie fo jobs with abitay speedup cuves. [6] showed that the algoith that patitions the pocessos equally aongst the latest aiving jobs is scalable. [3] gives essentially optially copetitive algoiths fo scheduling jobs with abitay speedup cuves in a setting of identical speed scalable pocessos whee the objective is total waiting tie plus enegy (in this setting one essentially gets speed augentation fo fee). As [3] focused on non-claivoyant scheduling algoiths, the copetitive atios wee supe-constant. [7] shows a scalable algoith is achievable when the schedule has access to a job s paallelizability. [4, ] give essentially optially copetitive algoiths fo scheduling jobs with abitay speedup cuves fo the objective of axiu waiting tie. [7] consides scheduling jobs with abitay speedup cuves and with pecedence constaints. [0] also shows that the copetitive atio with espect to aveage waiting tie of Sequential-SRPT is O(log n ), and give a geneal atching lowe bound fo all online algoiths. Thee is a lage liteatue on online scheduling. One good suvey fo poviding backgound on elated esults is [5]. 2. ANALYSIS OF INTERMEDIATE-SRPT Ou goal in this section is to pove Theoe, which uppe bounds the copetitive atio fo the Inteediate- SRPT algoith. 2. Analysis Oveview Ou analysis will be based on a soewhat novel cobination of potential function and local copetitiveness aguents. Let A(t) and OP T (t) be the unfinished jobs at tie t in the algoith s and optial solution s schedules, espectively. In Subsection 2.3, we will define a potential function Φ(t) that satisfies the following standad popeties: Bounday Condition: Φ(0) = Φ( ) = 0. Discontinuous Changes Condition: the potential function can only decease when a job aives, o is copleted by ou algoith o the optial solution. Continuous Changes Condition: at any tie t when no job aives o copletes, A(t) + d Φ(t) c OP T (t). dt By integating ove tie, one can see that the existence of such a potential function suffices to show that the algoith is c-copetitive fo the total flow tie objective. We efe the eade to [8] fo details. The novelty in ou analysis lies in poving the Continuous Changes Condition. Most analyses based on potential functions ely on esouce (speed) augentation to pove this condition. We will patition tie into oveloaded and undeloaded ties. Let O denote the set of oveloaded ties t when A(t), and U denote the set of undeloaded ties when A(t) <. Theoe then follows easily fo the following thee leas. Intuitively Lea shows that duing the oveloaded ties, the unfinished jobs fo the algoith can be chaged to the unfinished jobs fo optial at that tie (a local copetitiveness aguent). Intuitively Lea 2 shows that duing the oveloaded ties, the incease in the potential function can be chaged to the unfinished jobs fo optial at that tie. Togethe Lea and Lea 2 show that the Continuous Changes Condition holds at oveloaded ties. Lea 3 then shows that the Continuous Change Condition holds at undeloaded ties. Lea. At all ties t O, A(t) (3 + log P ) + 2 OP T (t) Lea 2. At all ties t O, d dt Φ(t) O()4/( α) log P OP T (t) Lea 3. At all ties t U, A(t) + d dt Φ(t) O()2/( α) OP T (t) In Subsection 2.2, we pove Lea. In Subsection 2.3, we define the potential function Φ and pove the Bounday Condition, and the Discontinuous Changes Condition. In subsection 2.4 we pove Lea 2. In subsection 2.5 we pove Lea Local Copetitiveness Duing Oveloaded Ties This section is devoted to poving Lea and is an adaptation of a siila esult fo [0]. We will need to define additional notation. At any tie, we classify jobs based on eaining length. A job whose eaining length is in [2 k, 2 k+ ) is in class k fo intege 0 k k ax = log P. Note that the nube of initial job classes is log P. We define one special class to denote jobs whose eaining length is stictly less than. Fo scheduling algoith S, let δ S (t) denote the nube of jobs that ae alive at tie t in schedule S and V S (t) denote the total volue of this schedule, whee the volue is defined to be the su of eaining lengths of jobs that ae still alive. Note that δ A (t) = A(t) and δ OP T (t) = OP T (t). We define the volue diffeence V (t) = V A (t) V OP T (t). Fo function f {V, V, δ}, we define f h,k (t) to be the function f esticted to jobs in class at least h and at ost k. We siilaly define f =k (t). In Lea 4 we bound the volue by which ou algoith can be behind optial, and then use this Lea in the poof of Lea 5, which bounds the nube of jobs by which ou algoith can be behind optial. It is easy to see that Lea iediately follows fo Lea 5 and the obsevation that the nube of jobs in class is at ost. Lea 4. Fo any tie t O, V k (t) 2 k+ Poof. Fist, fo tie t, we define tie t to be the ealiest tie such that [t, t) O. Next, we define t k to be the latest tie in [t, t) pio to tie t in which a job of class stictly highe than k was pocessed by soe achine. If thee is no such tie t k, then we set t k = t. We fist obseve that V k (t k ) 2 k+. By the definition of t k, it follows that fo any tie t k ɛ fo any ɛ > 0, δ A k(t k ɛ). It ay be the case that soe job entes class k at tie t k by the algoith s pocessing, but this only eans that δ A k(t k ) when esticted to jobs that aived stictly pio to tie t k. The volue of such jobs is esticted to at ost 2 k+ because each such job has a axiu eaining length of 2 k+. Finally, jobs that aive

4 at tie t k do not affect V k (t k ) since such jobs incease both Vk OP T (t) and Vk(t). A We next obseve that V k (t) V k (t k ). This follows because by the definition of O, each achine is pocessing one job duing [t k, t] and by the definition of t k, each job pocessed cannot be in a class lage than k. Thus, ou algoith copletes as uch wok on jobs in classes at ost k duing this tie peiod as OPT and the esult follows. k ax Lea 5. Fo any tie t O, δ A 0,k ax (t) (k ax + 2) + 2δ OP T k ax (t) Poof. We foulate δ A 0,k ax (t) as follows: δ A =k(t) = = k ax k ax k ax V A k (t) 2 k V =k (t) + V OP T =k (t) 2 k V k (t) V k (t) + V OP T =k (t) 2 k 2 k V k ax k ax (t) V k (t) + 2 kax 2 k+ V (t) + 2δ OP T 2 0 0,k ax (t) k ax δ OP T (t) + 2δ 0,k OP T ax (t) (k ax + 2) + 2δ OP T k ax (t). The fist inequality follows since 2 k is the iniu eaining length of any job in class k. The fouth inequality follows by assuing the jobs in δk OP T have eaining length 2 k+. The fifth inequality follows fo the pevious lea, obseving that we can eliinate the negative te and add a positive te δ OP T (t). 2.3 Potential Function Analysis In this section, we define the potential function Φ, and then we pove the Bounday Condition and the Discontinuous Changes Condition. Definition of the Potential Function: Let p A i (t) and p OP i T (t) denote the eaining pocessing tie of job i in the algoith s and optial solution s schedules at tie t, espectively. Let z i(t) = ax{p A i (t) p OP i T (t), 0}. Recall that A(t) and OP T (t) denote the unfinished jobs in the algoith s and optial solution s schedules, espectively. Let ank(i, t) = in{, j A(t), j i } whee without loss of geneality we assue that each job aives at a unique tie. Note that ank(i, t) fo all i and t. We define the potential function as follows: Φ(t) = i A(t) z i(t) Γ i(/ank(i, t)) Thoughout the analysis, the following siple lea will be useful. Poposition. Fo any B and C whee B C and any job j, it is the case that Γ j (B) Γ j (B) B C. Poof. The poposition follows iediately by the assuption that Γ j is a concave function and Γ j(0) = 0. Bounday Condition: It is easy to see that Φ(0) = Φ( ) = 0 fo the definition of the potential function Φ. Discontinuous Changes Condition: Fist conside when a job aives at tie t. In this case thee is no change in the potential function. This is because the ank fo evey job eains the sae fo all jobs that aive befoe tie t. Futhe, fo the job i that aives at this tie, z i(t) = 0. Thus, thee is no change in the potential. Next obseve that optial copleting a job has no effect on the potential. Now conside the case whee the algoith copletes soe job i at tie t. In this case, the potential function can only decease. To see that this is the case, conside any job j A(t). If j < i, then thee is no change in job j s te in the potential function. Howeve, if j > i then ank(j, t) ay decease by at ost one. Since Γ j is non-deceasing, Γ j(/ank(j, t)) can only incease fo a job j whee j i. Since this is in the denoinato of the te in the potential function coesponding to job j and z j(t) is non-negative, the potential function can only decease. 2.4 Potential Function Change Duing Oveloaded Ties In this subsection we pove Lea 2. If A(t) 0 log P, then Lea 2 iediately follows fo Leas 6 and 7. If 40 4 /( α) log P OP T (t) A(t), then Lea 2 iediately follows fo Lea 7. If A(t) 0 log P and 40 4 /( α) log P OP T (t) A(t) (which in tun iplies that OP T (t) ), then Lea 2 iediately 4 4 follows fo Leas /( α) 8 and 9. Fist we conside the case whee the algoith has a lage nube of jobs copaed to. Lea 6. If A(t) 0 log P, then OP T (t) A(t) /2 2 log P A(t) /4. Poof. The lea iediately follows fo Lea by noticing that t O. Lea 7. At all ties t, the ate of incease in the potential due to optial pocessing the jobs is at ost ( A(t) + OP T (t) ). Poof. Let qi OP T (t) be the nube of achines assigned to job i by OP T at tie t. The change in the potential due to optial pocessing the jobs can then be bounded as follows: OP T (t) + Γ i(/ank(i, t)) Γ i(/ A(t) ) = OP T (t) + A(t) ( A(t) + OP T (t) ) [Since Γ i is non-deceasing] qi OP T (t) / A(t) qi OP T (t) [By Poposition ] (t) / A(t) The second inequality holds since fo each job i with qop i T, it is the case that. Γ i (/ A(t) )

5 Lea 8. At any tie t whee OP T (t), the ate of incease in the potential due to optial pocessing the jobs is at ost α OP T (t) α. Poof. As befoe, let qi OP T (t) be the nube of achines assigned to job i by OP T at tie t. Let Γ be a function such that Γ(x) = x fo 0 x and Γ(x) = x α fo x. Recall that ank(i, t) fo all i and t fo the definition of ank. The change in the potential due to optial pocessing the jobs can then be bounded as follows: = Γ i(/ank(i, t)) Γ i(/) [Since Γ i is non-deceasing] Γ i(q OP T i (t)) [Since Γ i() = ] Γ(qi OP T (t) (/ OP T (t) ) α [Due to the concavity of Γ] = α OP T (t) α Lea 9. At any tie t whee A(t) 0 log P and OP T (t), the ate of incease in the potential due to the algoith /( α) pocessing jobs is at ost Poof. When A(t) the algoith assigns the shotest jobs each on a unique achine. Let A (t) denote these jobs. Notice that z i(t) deceases at a ate of one fo each job in A (t) \ OP T (t). Thus, we have that the change in the potential due to the algoith is at ost: i A (t)\op T (t) i A (t)\op T (t) A (t)\op T (t) i= (3/4) Γ i(/ank(i, t)) i ank(i, t)) The fist inequality easily follows fo Poposition and by obseving that /ank(i, t)) since ank(i, t). The second to last inequality holds since OP T (t) (/4) and A (t) =. 2.5 Undeloaded Ties Ou goal in this subsection is to pove Lea 3. Let t be a tie such that A(t). If OP T (t) A(t), then Lea 3 iediately follows fo Lea 7; the potential can only decease when the algoith pocesses jobs. Hence we assue that OP T (t) A(t). Fist we bound the incease in the potential function due to the pocessing of optial. Again, let qi OP T (t) be the nube of pocessos assigned to job i at tie t by OP T. The incease in the potential is at ost the following. OP T (t) + OP T (t) + Γ i(/ank(i, t)) Γ i(/ A(t) ) [Since Γ i is non-decasing],q i OP T (t) / A(t) (qi α (/ A(t) ) α (qi α i (/ A(t) ) α i The second to last inequality holds fo the following eason. Conside any job i such that qi OP T (t) (/ A(t) ) α i. Then we have (qop i T (t)) α i (/ A(t) ) α i. Hence the total contibution of such jobs is at ost OP T (t). Since ou goal is to bound the total change of the potential plus A(t) by OP T (t), we will ignoe OP T (t), and poceed with ou sting of inequalities. (qi α (/ A(t) ) α (/ OP T (t) ) α (/ A(t) ) α (/ OP T (t) )α = OP T (t) (/ A(t) ) α OP T (t) α A(t) α ( ) α+2 A(t) + 2 α α OP T (t) 2α+2 The fist inequality is iediate fo the fact that 0 α <. The last inequality can be easily shown by consideing two cases whethe A(t) 2 α+2 α OP T (t) o not. Now we conside the decease in the potential function due to the algoith pocessing jobs. When A(t) < then the algoith gives each job equal shae of evey pocesso. Thus, fo all i A(t) \ OP T (t) it is the case that z i(t) deceases at a ate of Γ i(/ A(t) ). Thus, we have the decease due to the algoith is as follows. i A(t)\OP T (t) Γ i(/ A(t) ) Γ i(/ank(i, t)) i A(t)\OP T (t),ank(i,t) A(t) /2 ( A(t) \ OP T (t) A(t) /2)(/2) α ( 2 ) A(t) (/2)α 6 A(t) (/2) α Γ i(/ A(t) ) Γ i(2/ A(t) So fa we have shown that d ( ) dt Φ(t) α+2 A(t) + 2 α α OP T (t) 6 A(t) (/2) α 2α+2 A(t) + O()2 /( α) OP T (t) This copletes the poof.

6 3. LOWER BOUND FOR GREEDY ALGO- RITHM In this section, we pove that the following natual geedy hybid of Paallel-SRPT and Sequential-SRPT has a supelogaithic lowe bound on its copetitive atio. Desciption of Geedy Algoith: At all ties allocate pocessos to jobs in such a way as to axiize the instantaneous ate at which the factional nube of unfinished jobs would be deceased, if it was the case that the oiginal wok of each job was its eaining unpocessed wok. Using a siple exchange aguent one can pove that if each job j has a speedup cuve of the fo Γ j(x) = x α fo α (0, ), then this policy can be ipleented in the following geedy way: We abitaily nube the pocessos fo to. At each decision point, the achines schedule jobs in ode fo achine to achine. When it is achine i s tun to schedule a job, let p(i, j) be the nube of pocessos fo to i that have been assigned to job j. Pocesso i chooses job j that axiizes Γ(p(i,j)+) Γ(p(i,j)) p j (t). Lea 0. This Geedy algoith has a copetitive atio that is Ω(ax{P, n /3 }). Poof. Let ɛ = α. Conside an input instance whee ɛ jobs of size ae eleased at tie 0. Fo tie 0 to tie, one job of size is eleased evey ɛ tie units. Finally, at tie +, we elease a job of size ɛ evey tie units fo X = 2 tie units (a total of ɛ X ɛ jobs ae eleased in this final phase). This geedy algoith will devote all achines to the job of size and coplete it just as the next size job aives. This follows by consideing the last pocesso. It balances the choice of ɛ ( ) ɛ vesus. Given that ɛ > 0, it will always choose to assign the achine to the size job. At tie, this geedy algoith will still have all ɛ jobs of size eaining. In this next unit of tie, it can only coplete at ost units of wok on these ɛ jobs; in paticula, it cannot finish any of these jobs and cannot educe the pocessing tie to less than. Afte tie +, it will assign all pocessos to the newly aived job until the stea ends. The total flow tie incued will thus be fo the jobs of size eleased pio to tie, X fo the jobs of size eleased afte tie +, and ( ɛ + )(X + + ) fo the jobs of size up to the end of the long stea. We ignoe the flow tie incued to coplete these long jobs afte the end of the stea. The doinant te is ( ɛ )X fo the size jobs duing the long stea. On the othe hand, an altenative algoith (not necessaily optial but siple to conceptualize) will assign ɛ achines to the size jobs fo tie 0 to tie copleting the by tie. On the eaining ɛ achines, it assigns one achine to each job of size as that job aives. Because it opeates efficiently, each such achine will coplete its assigned size job exactly tie unit late. Duing this tie, each job will coplete just as its achine is needed to schedule the next aiving size job because the nube of jobs that aive duing unit of tie is exactly ɛ. By tie +, this algoith will have copleted all of these jobs and will now devote all achines to the stea of size jobs that aive copleting each one just as the next aives. Befoe tie +, this algoith incus a total tie of 2 ɛ fo the size jobs since each of these jobs is scheduled iediately on one pocesso. The lage jobs each coplete within tie units of aival fo a total flow tie of 2 2 ɛ. Finally, duing the stea of length X, ɛ each job incus a flow tie of fo a total flow tie of X. The Ω(P ) bound follows fo the obsevation that P =. The Ω(n /3 ) bound follows fo the obsevation that n = Θ( 3 ɛ ). 4. GENERAL LOWER BOUND Ou goal in this section is to pove Theoe 2, which gives a logaithic lowe bound on the copetitive atio of any algoith. This lowe bound is an adaptation of the lowe bound poof fo [0]. The poof is slightly oe coplex because online algoiths can exploit the fact that the jobs have inteediate paallelizability to catch up on jobs that they should have finished ealie. We constuct a faily of input instances paaeteized by α whee each instance is coposed of two pats. In the fist pat, jobs ae eleased in phases. Each phase has long jobs and shot jobs that foce the online algoith to choose between copleting alost all of the shot jobs befoe the halfway point of a phase o copleting all the jobs in the phase by the end of the phase. The faily of input instances is stuctued such that any deteinistic online algoith on at least one instance in the faily ust face a tie T whee it has at least Ω( log P ) unfinished jobs wheeas the optial algoith at the sae oent in tie will have at ost /2 unfinished jobs. The second pat of the input instance stats at tie T and pesents a stea of jobs of size fo P 2 consecutive stating ties. We foally define the faily of input instances as follows. Fist, we need to define the following tes. Let ɛ = α. We define a length eduction facto = /2( 2 ɛ ); the length of the long jobs will be ultiplied by (equivalently divided by a facto of /) in each phase of the input instance. We choose the nube of achines such that 2 ɛ is an intege. We choose the longest job length P 2 2 ɛ + 2 such that the axiu nube of phases L = /2 log P is an intege and log 2 P < 2 ɛ P / ɛ + The fist pat has at ost L = /2 log P phases nubeed fo 0 to L. Each phase 0 i L has a phase length p i = P i and a stat tie s i = i j=0 pj. Duing phase i, /2 long jobs of length p i ae eleased at tie s i, and shot jobs of length ae eleased at ties s i + j fo 0 j p i/2. The advesay begins by eleasing the jobs in phase 0 stating at tie s 0 = 0. In geneal, suppose the advesay has eleased the jobs in phase i stating at tie s i whee i L. The advesay decides at tie s i + p i/2 whethe o not to (i) begin the second pat of the input instance at tie s i + p i/2 o (ii) to elease the next set of jobs stating at tie s i+ as follows. If the online algoith has at least log P eaining wok fo length jobs eleased in phase i at tie s i + p i/2, then the advesay begins the second pat of the input instance at tie s i + p i/2. Othewise, if i < L, then the advesay eleases the jobs in phase i + stating at tie s i+. If i = L, then the advesay stats the second pat of the input instance at tie s i + p i. This leaves us with two possible cases. In the

7 fist case, the advesay stats the second pat of the input instance at soe tie T = s i + p i/2 whee 0 i L. In the second case, the advesay stats the second pat of the input instance at tie T = s L + p L. We now ague that fo both cases, the optial flow tie is bounded by O(P 2 ). As in [0], we define a notion of a standad schedule fo phase i that has the goal of copleting all jobs eleased in phase i by tie s i + p i. Each of the /2 long jobs ae pocessed non-peeptively by one achine fo the entie phase. Fo the length jobs eleased at tie s i + k whee 0 k p i/2, /2 of the ae copleted by using /2 achines at tie s i + k and the othe /2 ae copleted using /2 achines at tie s i + k + p i/2. The total flow tie of this standad schedule fo phase i is 2p i + /2(p i/2) 2. We now show that the optial flow tie is O(P 2 ) fo the fist case by giving a specific schedule with flow tie O(P 2 ). The standad schedule is used fo all phases up to but not including phase i. Fo phase i, the /2 long jobs ae ignoed and each length job is assigned its own achine iediately upon aival. Thus, by tie T = s i +p i/2, only the /2 long jobs of phase i eain. Fo tie T + k whee 0 k P 2, the jobs of length eleased at tie T +k ae each assigned thei own achine and copleted by tie T + k +. Finally, at tie T + P 2, each of the /2 long jobs of size p L ae assigned to 2 achines and copleted by tie T + P 2 + p i/2 α. Clealy, the oveall flow tie fo this feasible schedule is O(P 2 ). We now show that the optial flow tie is O(P 2 ) fo the second case by giving a specific schedule with flow tie O(P 2 ). The standad schedule is used fo all phases. Thus, by tie T = s L + p L /2, no jobs eain. Fo tie T + k whee 0 k P 2, the jobs of length eleased at tie T + k ae each assigned thei own achine and copleted by tie T + k +. Clealy, the oveall flow tie fo this feasible schedule is O(P 2 ). We now show that the online flow tie fo both cases is at least Ω(P 2 log P ). By the definition of the fist case, the online algoith has at least log P eaining wok fo the length jobs eleased in phase i at tie T = s i + p i/2. Thus, the online algoith has at least log P unfinished jobs fo tie T to tie T + P 2. Using only the flow tie fo this tie inteval, we see that the online algoith incus a total flow tie of at least P 2 log P and the theoe follows fo this case. We now conside the second case. In ou analysis, we opt fo siplicity athe than poving the ost accuate bound. The fist key obsevation is that in phase i fo 0 i L, online copletes at least p i/2 log P of the total available wok fo the length jobs by tie s i + p i/2, the halfway point of phase i. This eans that at ost log P wok can be copleted on the long jobs, possibly fo ealie phases, duing the tie inteval [s i, s i + p i/2]. We will pove that at tie T, the aount of unfinished wok fo the /2 long jobs fo phase i fo 0 i L is at least 2 ɛ pi. This iplies that the nube of long 2 2 ɛ + 2 jobs with eaining length at least fo phase i at tie T is at least 2 ɛ. Given that thee ae L = /2 log 2 2 ɛ + 2 P phases, we have that the total nube of jobs with eaining length at least at tie T is at least L 2 ɛ = 2 2 ɛ + 2 (log 2 P ) 2 ɛ which is Ω( log 2 2 ɛ + 2 P ). Thus, the total flow tie incued in inteval [T, T + P 2 ] is Ω(P 2 log P ). Conside the /2 long jobs fo phase i. Fo ou pevious obsevation, we can coplete at ost (L i) log P 2 log 2 ɛ 2 ɛ + 2 ɛ 2 ɛ + 2 pi wok on these log P P / /2 jobs duing the fist half of phases i to L. Duing the second half of phases i to L, the best we can do is devote 2 achines to each job fo the entie second half of these phases; note that we ignoe the pocessing equied by any unfinished length jobs fo phase i in the second half of phase i. Given that the phase lengths fo a geoetic pogession with ultiplicative facto, the total tie available in these second halves of phases is stictly less than p i. Thus, the total aount of wok that can be copleted in the second half of these phases is stictly less than 2 2α p i which is equal to 2 2 pi. Thus, consideing only the second half of these phases, thee is stictly 2 2 ɛ + oe than 2ɛ 2 ɛ pi unfinished wok fo these /2 long jobs + 2 fo phase i at tie T. Taking into account how uch wok can be done in the fist half of these phases, we see that the total unfinished wok on the /2 long jobs fo phase i at 2 tie T is at least 2 the second case. Acknowledgent 2 ɛ 2 ɛ + 2 pi, and the theoe follows fo Sungjin I s wok was suppoted in pat by NSF Awad CCF , and was patially done while the autho was at Duke. Kik Puhs wok was suppoted in pat by NSF gants CCF-5575, CNS-25328, and an IBM Faculty Awad. 5. REFERENCES [] [2] [3] Ho-Leung Chan, Jeff Edonds, and Kik Puhs. Speed scaling of pocesses with abitay speedup cuves on a ultipocesso. Theoy Coput. Syst., 49(4):87 833, 20. [4] Jeff Edonds. Scheduling in the dak. Theo. Coput. 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