A Competitive Algorithm for Minimizing Weighted Flow Time on Unrelated Machines with Speed Augmentation

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1 Cometitive lgorithm for Minimizing Weighted Flow Time on Unrelated Machines with Seed ugmentation Jivitej S. Chadha and Naveen Garg and mit Kumar and V. N. Muralidhara Comuter Science and Engineering Indian Institute of Technology Delhi New Delhi , India BSTRCT We consider the online roblem of scheduling jobs on unrelated machines so as to minimize the total weighted flow time. This roblem has an unbounded cometitive ratio even for very restricted settings. In this aer we show that if we allow the machines of the online algorithm to have ɛ more seed than those of the offline algorithm then we can get an ((1 + ɛ 1 2 -cometitive algorithm. ur algorithm schedules jobs reemtively but without migration. However, we comare our solution to an offline algorithm which allows migration. ur analysis uses a otential function argument which can also be extended to give a simler and better roof of the randomized immediate disatch algorithm of Chekuri-Goel-Khanna-Kumar for minimizing average flow time on arallel machines. Categories and Subject Descritors F.2.2 [NLYSIS F LGRITHMS ND PRB- LEM CMPLEXITY]: Nonnumerical lgorithms and Problems Sequencing and scheduling General Terms lgorithms, Performance Keywords Scheduling, flow-time, cometitive ratio, aroximation algorithms 1. INTRDUCTIN The roblem of online scheduling of jobs on multile machines has been well studied both in the online and schedul- Work artially suorted by the Max-Planck artner grou on roximation lgorithms Work artially suorted by the Max-Planck artner grou on roximation lgorithms Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, requires rior secific ermission and/or a fee. STC 09, May 31 June 2, 2009, Bethesda, Maryland, US. Coyright 2009 CM /09/05...$5.00. ing communities. natural measure to consider in this setting is the weighted sum of flow times of the jobs, where the flow time of a job is defined as the difference in the comletion time and the arrival time (also known as release time of the job. However, most of the research in this area has addressed only the setting of identical multile machines. Garg and Kumar [11] were the first to consider the roblem for related machines. This is the setting when machine i has seed s i so that it finishes one unit of rocessing in time 1/s i. Garg and Kumar [10] gave an (log 2 P -cometitive algorithm for this roblem which has recently been imroved to (log P [1]. For the more general setting of unrelated machines when the time required to rocess a job j on machine i equals ij (and this could be an arbitrary quantity, [12] showed that no cometitive algorithm is ossible. Their examle which showed that no online algorithm can be cometitive used only 3 machines and jobs that had unit rocessing time on 2 of the 3 machines. In light of this strong negative result we ask ourselves if it is ossible to obtain a bounded cometitive ratio by roviding the online algorithm with slightly more resources than those of the offline algorithm. In articular, we assume that each machine of the online algorithm has ɛ more seed which imlies that the work that the offline algorithm can do in 1 unit of time, will require (1+ɛ 1 time for the online algorithm. In this setting we are able to show a simle online algorithm that is ((1 + ɛ 1 2 -cometitive. ur aer also highlights the immense additional ower that a small augmentation of seed can rovide to the online algorithm. In articular, we get the same cometitive ratio for weighted flow time (every job has a weight which may deend on the machine on which the job is scheduled. The schedule that our algorithm rovides is non-migratory a job once it is scheduled on a machine cannot migrate to another machine. However, the offline adversary can be stronger, it can migrate jobs from one machine to another. This shows that seed can also offset the benefits that might accrue with migration. ne another nice roerty of our algorithm is that it disatches a job to the aroriate machine as soon as the job arrives. We would like to oint out that in the absence of seed augmentation we do not even know of any aroximation algorithm (with non-trivial guarantees for the roblem of minimizing flow time on unrelated machines [13]. The key technical contribution of this aer is a simle otential function argument that we believe can be alied to other roblems as well. In articular, we rovide a simle

2 roof of the randomized immediate disatch algorithm of Chekuri et.al. [9] for minimizing average weighted flow time on arallel machines which also imroves the cometitive ratio from (ɛ 3 to (ɛ 2. Related Work Seed augmentation in online scheduling roblems was first considered by Kalyanasundaram and Pruhs [14] who used it to get imroved on-line algorithms for minimizing average (unweighted flow-time on a single machine in the non-clairvoyant setting. They obtained an (ɛ 1 - cometitive algorithm for this roblem. Without resource augmentation no online algorithm for this roblem can be Ω(log n-cometitive [17]. Bechetti and Leonardi [7] showed that the randomized multi-level-feedback algorithm is (log n cometitive for single machines and (log n log P -cometitive for arallel machines in the non-clairvoyant setting. Bansal and Pruhs [5] showed that in the clairvoyant setting some well-known algorithms like SRPT and Shortest Job First are ɛ 1 -cometitive for the L norm of flow-time for all values of 1. There has been considerable work on minimizing average (unweighted flow-time on arallel machines [16, 3, 4, 2] these results established (min(log P, log n/m-cometitive algorithms for this roblem. Leonardi and Raz [16] also showed that no randomized on-line algorithm can do better. With ε-resource augmentation, Chekuri et.al. [9] showed that the immediate disatch non-migratory algorithm of zar et.al. [2] is (ɛ 1 -cometitive for all L norms. They also showed that the following simle stateless algorithm has cometitive ratio ( log 1 ε for the average (unweighted flowtime : when a job arrives, 3 assign it to a machine icked uni- ε formly at random. s mentioned above, Garg and Kumar showed that there cannot be a bounded cometitive ratio if we make the model slightly more by adding the restriction that each job can go only on a subset of machines. For the roblem of minimizing weighted flow time on arallel machines, Chekuri, Khanna and Zhu [8] showed that no online algorithm can have a bounded cometitive ratio. urs is the first non-trivial cometitive ratio for minimizing weighted flow-time when there are more than one machine. More recently, Bansal et. al. [6] show that seed augmentation can hel us overcome the hardness imosed by non-reemtion [15]; they show a olynomial time (1- aroximation algorithm for minimizing flow time on a single machine with (1 additional seed when no reemtion is allowed. 2. PRELIMINRIES ND THE SCHEDUL- ING LGRITHM We consider the on-line roblem of minimizing total weighted flow-time on unrelated machines. Job j is released at time r j and requires ij units of rocessing if scheduled on machine i. Further, job j has a weight w ij if it is scheduled on machine i. Let d ij = w ij/ ij denote the density of job j on machine i. We shall use to denote the schedule roduced by our algorithm, and the otimal schedule PT. We assume that a machine in our algorithm has (1 + ε more seed than the corresonding machine in PT. Let ε be such that (1 + ε 2 = 1 + ε. 2.1 Fractional Flow time In the following sections, we shall comare the fractional flow-time of our algorithm to that of the otimal schedule. Let x i,j,t be a variable between 0 and 1 which tells us the extent to which job j is scheduled on machine i in the eriod [t, t + 1. x i,j,t = 1 denotes that all of the t th time-slot on machine i is occuied by job j. The (weighted fractional flow-time of job j is then defined as F j = i,t w ij x i,j,t ( t rj ij The fractional flow time of a schedule is denoted by F and equals the sum of the fractional flow times of the jobs. The following three claims follow as an easy consequence of the definition of fractional flow time. Claim 2.1. In any schedule the fractional flow time of a job would be at most its (integral flow time. Claim 2.2. To minimize fractional flow time, each machine should rocess the jobs assigned to it in decreasing order of density. In our schedule,, job j would be assigned entirely to one machine, say i. In this scenario F j, as defined above, is equivalent to w ij ij 2 + ij(t dt (1 r j where ij(t is the remaining rocessing time of job j on machine i at time t in schedule. In fact, in the above exression the first term is at most the second term [10]. Let P = j wijij denote the total weighted rocessing time of schedule. Hence, F P. 2.2 Relation between fractional and integral flow time Let B be a non-migratory schedule. Consider a job j scheduled on machine i in this schedule. Let β j be the time at which a (1 + ε 1 fraction of job j is comleted. Then the fractional flow time of j is at least εw ij(β j r j/(1 + ε. Suose we increase the seed of each machine by a factor (1 + ε. We modify the schedule B to get a schedule B for these faster machines as follows. Job j is scheduled in the first ij/(1 + ε slots of the ij slots in which this job is scheduled in B. Hence the comletion time of j in B is β j and the flow time is w ij(β j r j. This imlies that for each job, its flow time in B is less than (1 + ε 1 times its fractional flow time in B. Henceforth, we would only be interested in the fractional flow time of. 2.3 The lgorithm Each machine maintains a queue of jobs assigned to it. t time t let i(t be the set of unfinished jobs scheduled on machine i in our algorithm. We can think of PT also as maintaining a queue of jobs for each machine. When a job arrives at time t, it goes to the queue of the machine on which it eventually gets rocessed. Let i(t be the set of unfinished jobs scheduled on machine i in PT. Let j (t be the residual rocessing time of job j at time t in our algorithm and j (t be the corresonding quantity in PT.

3 Suose a job k arrives at time t. For each machine i, we comute the quantity Q(i, k = w ik j (t + ik j (t d ij >d ik Job k is assigned to the machine i for which Q(i, k is minimum. This secifies the rule for assigning a job to a machine. Since each machine rocesses jobs assigned to it in decreasing order of density, the quantity Q(i, k measures the increase in the flow-time of jobs which are in the queue of machine i (at time t if we assign k to machine i. Note that there is no increase in the flow-time of jobs of density more than d ik, but the flow time of jobs whose density is less than or equal to d ik increases by ik times their weight. 3. NLYSIS ur analysis relies on a otential function argument. Define = Φ i(t k i (t k i (t k (t k (t and let Φ = i Φi. Φ is zero both at the start and the end of the algorithm. Φ(t is continuous at all times other than at the arrival times of jobs. Φ(t is also differentiable almosteverywhere (the sloe changes only when a job finishes. We show that the total decrease in Φ is at least ε(f P /2 F PT (Lemma 3.1. We also show that the total increase in Φ is at most F PT (Lemma 3.2. Since total decrease in Φ equals the total increase in φ, we have, ε(f F PT P /2 Decrease in Φ Since F P, this imlies = Increase in Φ F PT F 2(1 + ε 1 F PT We now rove the two lemmas. Let t 1 < t 2 < < t l be the times at which jobs are released. Lemma 3.1. The sum of the decreases in Φ in the oen intervals (t l, t l+1, l is at least ε(f F PT. Proof. From time t to t + δ we calculate the decrease in Φ i. Let our algorithm schedule job a on machine i and PT schedule job b. Note that a is the highest density job in i(t while b would be the highest density job in i(t. We consider 2 cases deending on which of a, b has the higher density. We first consider the case when a has higher density. The residual rocessing time of a decreases by δ(1 + ε and the decrease in Φ i due to this equals δ(1 + ɛ + δ(1 + εd ia ( a (t δ(1 + ε j (t We assume δ small enough so that δ(1+ε a (t and hence the increase in Φ i is at least the first term. Similarly, the residual rocessing time of b decreases by δ and the decrease in Φ i due to this equals δ d ij d ib which is greater than δ d ib d ij >d ib + δd ib ( b (t δ j (t assuming δ b (t. There is an additional decrease in Φ i due to the fact that the residual rocessing times of a and b decrease simultaneously. This additional decrease is in the term a (t b (t d ib and equals (1 + εδ 2 d ib. Therefore the total decrease in Φ i is at least δε j (t δ(1+ɛ We next consider the case when b has higher density. The decrease in Φ i due to the decrease in the residual rocess time of a is now, assuming δ(1 + ε a (t, at least which is greater than δ(1 + ɛ d ij d ia d ia d ij >d ia j (t Similarly, the residual rocessing time of b decreases by δ and the decrease in Φ i due to this, again assuming δ b (t, is at least δ j (t. Hence the total decrease in Φ i is, once again, at least δε j (t. This imlies that the total decrease in Φ during (t l, t l+1

4 is at least tl+1 t l ε i j (t dt. dding over all l, and using (1, we get that the total decrease in Φ during these intervals is at least ε(f P /2 F PT. The otential function increases when jobs arrive. Lemma 3.2. The total increase in Φ at times t l, l is at most the fractional flow time of PT. Proof. Let k be a job which arrives at time t and is scheduled by on machine 1 and by PT on machine 2. The increase in Φ 1 equals w 1k w 1k j 1 (t j 1 (t j (t + 1k 1k Similarly increase in Φ 2 is w 2k j 2 (t d 2j >d 2k +w 2k j 2 (t d 2j >d 2k 2k j (t + 2k d 1j j 1 (t d 1j j 1 (t d 2j j 2 (t d 2j d 2k d 2j j 2 (t d 2j d 2k j (t (2 j (t (3 j (t (4 j (t (5 Note that the quantity in exression (2 equals Q(1, k while exression (4 equals Q(2, k. ur choice of machine to schedule job k imlies that (2 + (4 0. Since (3 < 0, the total increase in Φ is at most (5. But (5 is also the increase in the fractional flow time of PT due to the arrival of job k. Hence total increase in Φ is at most the fractional flow time of PT. 4. CMPRISN WITH MIGRTRY P- TIMUM So far we have comared our algorithm with a non-migratory otimum. We now show that in fact our algorithms are also cometitive with resect to a migratory otimum, mpt. Let ij(t denote the residual rocessing time of job j on machine i at time t in PT. We need to re-analyse the increase in Φ when a job arrives. Let k be the job which arrives at time t and is scheduled on machine 1 by, while mpt sends α i ( i αi = 1 fraction of k to machine i. The increase in Φ 1 due to job k being scheduled on machine 1 by is as before and equals w 1k j (t + 1k j (t (6 w 1k j 1 (t 1j(t 1k d 1j j 1 (t d 1j 1j(t (7 The increase in Φ = i Φi due to αi fraction of k being scheduled on machine i in mpt equals i + i α i w ik α i w ik d ij >d ik j (t + ik (8 ij(t + ik ij(t (9 d ij >d ik Note that we are ignoring the change in Φ 1 due to the fact that job k is assigned on machine 1 by our algorithm and to an extent α 1 by PT. ccounting for this would only decrease Φ 1. ur choice of machine to schedule job k imlies that (6+ (8 0. Since (7 < 0, the total increase in Φ is at most (9. But (9 is also the increase in fractional flow time of mpt due to the arrival of job k. Hence total increase in Φ is at most the fractional flow time of mpt. The rest of the analysis continues as before and hence F 2(1 + ε 1 F mpt 5. SIMPLER (ND BETTER NLYSIS F THE RNDMIZED LLCTIN LGRITHM FR PRLLEL MCHINES In this section, we consider the setting of arallel identical machines. Consider the following simle algorithm : when a job arrives assign it randomly to one of the machines. For a articular machine, we follow the highest-density first rule to schedule the jobs assigned to this machine. We use the same definitions as in Section 4. Note that i(t, j (t, Φ it and Φ(t are random variables. We first observe that it is easy to characterize PT if we allow it to migrate jobs and rocess a job simultaneously on multile machines (this can only reduce its flow-time. Lemma 5.1. There is an otimal solution which distributes each job evenly on all the machines. Proof. We can think of the m machines as a single machine with m times the seed. View the otimal solution as a schedule on this single machine. We can get back an otimal solution for the m machine case by slitting each unit time interval in this schedule into m equal arts. Suose job k arrives at time t. We would like to comute the increase in E[Φ]. Fix all the random choices before job k. So Φ(t is now a known quantity. Suose we send k to machine 1 (this haens with robability 1/m, where m is the number of machines. Then the increase in Φ 1 is at most Q(1, k. Hence, the exected increase in Φ due to scheduling of job k by our algorithm is (1/m i Q(i, k. n the other hand, the increase in Φ due to PT scheduling 1/m fraction of job k on each machine equals (8+(9 with α 1 = α 2 = = α m = 1/m. This in turn is equal to (1/m i Q(i, k lus the increase in the fractional flow time of PT due to arrival of job k. Therefore, the exected increase in Φ due to the arrival of job k is at most the increase in the fractional flow time of PT. Since this is true for all random choices made rior to job k, we can remove the conditioning on the random choices made before job k. Hence, we have that the increase in E[Φ] is at most the increase in flow time of PT.

5 Since, we have that ε(f F PT P /2 Decrease in Φ ε(e[f P /2] F PT Decrease in E[Φ] The rest of the argument roceeds as before and this lets us claim that E[F ] 2(1 + ε 1 F PT 6. CKNWLEDGEMENTS We would like to thank Pushkar Triathi and shish Sangwan for many useful discussions. 7. REFERENCES [1] S. nand, Naveen Garg, and mit Kumar. n (log P cometitive algorithm for minimizing flow time on related machines. Unublished manuscrit, [2] Nir vrahami and Yossi zar. Minimizing total flow time and total comletion time with immediate disatching. lgorithmica, 47(3: , [3] Baruch werbuch, Yossi zar, Stefano Leonardi, and ded Regev. Minimizing the flow time without migration. In CM Symosium on Theory of Comuting, ages , [4] Baruch werbuch, Yossi zar, Stefano Leonardi, and ded Regev. Minimizing the flow time without migration. SIM J. Comut., 31(5: , [5] N. Bansal and K. Pruhs. Server scheduling in the L norm: rising tide lifts all boats. In CM Symosium on Theory of Comuting, ages , [6] Nikhil Bansal, Ho-Leung Chan, Rohit Khandekar, Kirk Pruhs, Clifford Stein, and Baruch Schieber. Non-reemtive min-sum scheduling with resource augmentation. In IEEE Symosium on Foundations of Comuter Science, ages , [7] Luca Becchetti and Stefano Leonardi. Nonclairvoyant scheduling to minimize the total flow time on single and arallel machines. J. CM, 51(4: , [8] C. Chekuri, S. Khanna, and. Zhu. lgorithms for weighted flow time. In CM Symosium on Theory of Comuting, ages CM, [9] Chandra Chekuri, shish Goel, Sanjeev Khanna, and mit Kumar. Multi-rocessor scheduling to minimize flow time with esilon resource augmentation. In CM Symosium on Theory of Comuting, ages , [10] Naveen Garg and mit Kumar. Better algorithms for minimizing average flow-time on related machines. In ICLP (1, ages , [11] Naveen Garg and mit Kumar. Minimizing average flow time on related machines. In CM Symosium on Theory of Comuting, ages , [12] Naveen Garg and mit Kumar. Minimizing average flow-time : Uer and lower bounds. In IEEE Symosium on Foundations of Comuter Science, ages , [13] Naveen Garg, mit Kumar, and Muralidhara V N. Minimizing total flow-time: The unrelated case. In ISC, ages , [14] Bala Kalyanasundaram and Kirk Pruhs. Seed is as owerful as clairvoyance. In IEEE Symosium on Foundations of Comuter Science, ages , [15] Hans Kellerer, Thomas Tautenhahn, and Gerhard J. Woeginger. roximability and nonaroximability results for minimizing total flow time on a single machine. In CM Symosium on Theory of Comuting, ages , [16] Stefano Leonardi and Danny Raz. roximating total flow time on arallel machines. In CM Symosium on Theory of Comuting, ages , [17] R. Motwani, S. Phillis, and E. Torng. Non-clairvoyant scheduling. Theoretical Comuter Science, 130(1:17 47, 1994.