Resource Augmentation for Weighted Flow-time explained by Dual Fitting

Transcription

1 Resource Augmentation for Weighted Flow-time explained by Dual Fitting S. Anand Naveen Garg Amit Kumar Abstract We propose a general dual-fitting technique for analyzing online scheduling algorithms in the unrelated machines setting where the obective function involves weighted flow-time, and we allow the machines of the on-line algorithm to have + )-extra speed than the offline optimum the so-called speed augmentation model). Typically, such algorithms are analyzed using non-trivial potential functions which yield little insight into the proof technique. We propose that one can often analyze such algorithms by looing at the dual or Lagrangian dual) of the linear or convex) program for the corresponding scheduling problem, and finding a feasible dual solution as the on-line algorithm proceeds. As representative cases, we get the following results : For the problem of minimizing weighted flow-time, we show that the greedy algorithm of Chadha-Garg- Kumar-Muralidhara is O ) -competitive. This is an improvement by a factor of on the competitive ratio of this algorithm as analyzed by them. For the problem of minimizing weighted l norm of flow-time, ) we show that a greedy algorithm gives an O -competitive ratio. This marginally im- + proves the result of Im and Moseley. For the problem of minimizing weighted flow-time plus energy, and when the energy function fs) is equal to s γ, γ >, we show that a natural greedy algorithm is Oγ )-competitive. Prior to our wor, such a result was nown for the related machines setting only Gupta- Krishnaswamy-Pruhs). Introduction The problem of online scheduling of obs on multiple heterogeneous machines has been well-studied in scheduling communities. A natural measure of performance is to consider the average weighted flow-time of obs. The flow-time of a ob is the total amount of time it spends in the system, i.e., the difference between its completion time and release date. For practical reasons, it is not desirable to migrate obs across machines. However, pre-emption of obs is a much needed assumption. Indeed there are strong lower bounds on the competitive ratio of any on-line algorithm if we do not allow pre-emption [4]. In the setting of unrelated machines, where the time required to process a ob on machine i equals Department of Computer Science and Engineering, IIT Delhi, {anand,naveen,amit}@cse.iitd.ac.in and this could be an arbitrary quantity), Garg and Kumar [0] showed that no on-line algorithm with bounded competitive ratio is possible. In fact, their example had only 3 machines, and obs that had unit processing time on of the 3 machines. In light of such strong lower bounds, Chadha et al. [6] considered this problem in the speed augmentation model. In this model, we allow each machine in the on-line algorithm -fraction more speed than the corresponding machine in the offline algorithm. In other words, the wor that a machine in the offline setting can do in unit of time will require only + time on the corresponding machine in the on-line setting. They gave an O )-competitive algorithm for this problem. Their algorithm was greedy in nature : when a ob arrives, it is dispatched immediately to the machine for which the increase in the weighted flow-time is smallest. The obective function of minimizing the weighted sum of flow-time has the disadvantage that it can be + very unfair to some obs. One way of improving the fairness criteria is to loo at weighted l norm of flowtime of the obs ) for some. Im and Moseley [] gave an O -competitive algorithm for the problem minimizing weighted l norm of flow-time in the unrelated machines model. However, their algorithm was based on a non-intuitive potential function. Another recent line of research in this direction has been to allow the machines to run at variable speeds. A machine running at speed s incurs an energy cost of fs), where f is assumed to be a well-behaved function. Initial wor in this area assumed f to be of the form s γ, γ >, and more recent wor has allowed f to be any increasing function with some mild constraints). This framewor models modern processors, and also the fact that multi-core architectures have the flexibility of changing the number of processors which are running at any time see e.g., [] and the references therein). A natural obective in this setting is to minimize the sum of weighted flow-time and energy consumed. In this setting, when fs) = s γ, Bansal et al. [3] gave an O γ log γ )-competitive algorithm for this problem. Gupta et al. [] extended this to the case of related machines

2 and gave an Oγ )-competitive algorithm. Their results hold for more general class of energy functions, and they need the notion of speed augmentation. By giving extra speed to the processors, one can hope to come within a constant factor of the optimal flow-time, and increasing the speed slightly leads to only a modest increase in energy consumption. Analyzing these algorithms turns out to be quite tricy. When one does not allow speed augmentation, one needs a very strong guarantee from any on-line algorithm: at every point of time t, the number or total weight) of obs waiting at any time in the on-line algorithm must be close to that in the off-line optimum. Indeed, if at time t, the gap becomes large, then the adversary can mae the on-line algorithm pay a lot by maintaining this gap subsequently. However, with speed augmentation, any temporary build-up in the queue as compared to off-line optimum) can be depleted with time. The most powerful approach for analyzing such algorithms has been to design clever potential functions, and show that these algorithms behave well in an amortized sense. Designing such potential functions is quite non-trivial, and often, yields little insight about how to design such functions for related problems. In this paper, we give a more direct approach for analyzing such algorithms. All of these problems have a linear programming or convex programming) relaxation. In this paper, we argue that often, it is much easier and intuitive) to analyze these algorithms by dual fitting, i.e., show that there is a feasible solution to the corresponding dual LP or Lagrangian relaxation) whose obective value is close to that of the on-line algorithm. In fact, we strongly believe that this method is as powerful as the potential function approach, and gives a much more principled approach towards analyzing such algorithms. As examples of our claim, we prove the following results : For the problem of minimizing weighted flow-time in the unrelated setting, we show that the greedy algorithm of Chadha et. al [6] is O ) -competitive. This is an improvement by a factor of on the competitive ratio of this algorithm as analyzed by [6]. For the problem of minimizing weighted l norm of flow-time, we show that a greedy ) algorithm along the lines of [6] gives an O -competitive ratio. + This marginally improves the result of Im and Moseley []. For the problem of minimizing weighted flow-time plus energy in the unrelated setting and when the energy function fs) is equal to s γ, γ, we show that a natural greedy algorithm is Oγ )- competitive. Prior to our wor, such a result was nown for the related machines setting only []. Our analysis also wors for a more general class of energy functions, but this is not the emphasis of our result. We emphasize that although our techniques either improve or match the best nown results for these problems, the main contribution of this paper is the technique of analyzing such algorithms by dual fitting. To illustrate the ideas involved in some more detail, consider the first problem above : minimizing weighted flowtime in the unrelated setting. The LP relaxation for this problem is given in Figure. The constraint 4.3) requires that the ob should be processed completely. Corresponding to this constraint we have a dual variable α. One interpretation of dual variables is the following : suppose we violate the corresponding constraint in the primal LP by a tiny amount, say δ. Then the change in the obective function is roughly α δ. Motivated by this, we thin of the change in the obective function if we change the right hand side of constraint 4.3) from to 0. Assuming no ob is released after, we should define α as the increase in the flow-time of all the obs because of the arrival of see Section 4. for more details). Similarly, the constraint 4.4) says that in any unit time slot, a machine can process unit volume. If we change the right hand side from to 0, and assume that no ob arrives after time t, then the change in the obective function is exactly equal to the total weight of obs waiting at time t so, we define the corresponding dual variable β it as equal to this quantity scaled by some factor). The only remaining thing is to chec that these dual variables are feasible. In the other settings considered in this paper, one can similarly define dual variables in a very natural manner. The bul of the wor lies in checing feasibility of the dual variables defined in this manner but the approach remains very intuitive and principled. In Section 4, we consider the problem of minimizing total weighted flow-time. We extend the result to the problem of minimizing weighted l norm of the flowtime in Section 5. The problem of minimizing the sum of weighted flow-time and energy consumed is discussed in Section 6. Related Wor The on-line problem of minimizing weighted flow-time has no bounded competitive ratio. In fact, this lower bound result holds even for the parallel machines setting [7]. Better results are nown for the setting of

3 unweighted flow-time see e.g., [9] and the references therein). However, Garg and Kumar [0] showed that strong lower bounds on the competitive ratio hold even if all machines are identical and a ob can go on a subset of machines only. Speed augmentation in the online setting was first considered by Kalyanasundaram and Pruhs [3] who used it to get O )-competitive algorithm for minimizing flow-time on a single machine in the nonclairvoyant setting. In the clairvoyant setting, Bansal and Pruhs [] showed that SRPT Shortest Remaining Processing Time first) is O ) competitive for all l -norms of flow time,. They extended this to weighted l norms and showed that HDF Highest Density First) is O )-competitive [4]. Cheuri et al. [8] showed that the immediate dispatch algorithm of Avrahamai and Azar [] is O )-competitive for all l -norms,. Garg and Kumar [6] gave a natural greedy algorithm for the setting of unrelated machines. They showed that this algorithm is O )-competitive for weighted flow-time. Im and Moseley [] extended this to an O + ) )-competitive algorithm for the case of weighted l norm for all. Their algorithm, however, is based on a non-trivial potential function. For the obective of weighted flow-time plus energy ) on a single machine, Bansal et al. [5] gave O γ log γ - competitive algorithm for fractional weighted ) flow plus energy, which then implied O γ -competitive algorithm for integral weighted flow plus energy. log γ Here, the energy function was of the form s γ. The scheduling algorithm was as follows : at any point of time, the speed s is chosen such that the energy consumed is equal to the total fractional weight of obs waiting in the queue. Further, the machine followed the HDF scheduling policy. Recently, Bansal et.al. [3] considered a very general class of energy functions, and gave a constant factor competitive algorithm for the sum of fractional weighted flow-time and energy. This implies an O γ log γ ) -competitive algorithm for the sum of integral weighted flow-time and energy when the energy function is s γ. Gupta et al. [] extended this to the setting of related machines the competitive ratio was Oγ )). 3 Preliminaries We are given a set of m machines, and a set of obs arrive over time. A ob is released at time r and requires units of processing if scheduled on machine i. Further, ob has a weight w if it is scheduled on machine i. Let d = w / denote the density of ob on machine i. A machine can process obs preemptively, but we do not allow obs to be migrated 3 across machines. The flow-time of a ob is the difference between its completion time and release date. If a ob gets processed on machine i, its weighted flow-time is w times its flow-time. 4 Weighted flow-time on Unrelated Machines We consider the problem of minimizing the weighted sum of flow-time of obs. In Section 4., we describe the greedy algorithm of Chadha et al. [6]. In Section 4., we analyze this algorithm. We first describe the LP relaxation and its dual. Then we show that the dual variables can be set as the obs arrive. These dual variables need to satisfy two conditions : i) they should be feasible to the dual LP, and ii) the dual obective value should be close to the weighted flow-time of the greedy algorithm. 4. The greedy algorithm When a ob arrives, the algorithm dispatches it immediately to one of the machines. Let A i t) the set of obs which get dispatched to machine i and are unfinished at time t. Consider a ob A i t). Let p t) denote the remaining processing time of ob. Define the residual density of at time t, d t), as w p t). Each machine follows the highest residual density first HRDF) algorithm : at any time t, machine i processes the ob in A i t) with highest residual density. Observe that if all weights are, then the HRDF policy becomes SRPT Shortest Remaining Processing Time first) policy. Also, it is worth noting, that arrival of new obs does not change the residual density of a ob, and hence, the relative ordering of obs does not change with time. It remains to specify the dispatch policy. When a ob arrives at time t, we compute, for each machine i, the increase in weighted flow-time if we dispatch to machine i. More precisely, let Q denote the quantity 4.) + w A it):d t) d p t) + w + A it):d t)<d w. The dispatch algorithm is greedy : it assigns to the machine i for which Q is minimum. We shall use A to denote this algorithm. We shall denote the weighted flow-time of A by F A. 4. Analysis We first write an LP relaxation for the problem of minimizing weighted flow-time. We shall use the letters

4 4.) 4.3) 4.4) Primal LP min t r x t + i,,tw ) x t = for all i,t x t for all i, t 4.5) 4.6) 4.7) Dual LP max α β it i,t α β it d t r ) + w α, β it 0 for all i,, t for all i,, t x t 0 for all, i, t r Figure : The LP relaxation and its dual i,, t for machines, obs, and integral) time respectively. We assume all release dates r and processing times are integers. We divide the time-line into slots of unit length, i.e., intervals of the form [t, t + ] for all nonnegative integers t. Let x t denote the fraction of the slot [t, t+] used by machine i for processing ob. Note that x t is defined only if t r. The LP relaxation is described in Figure. x t The quantity t refers to the fractional amount of ob processed on machine i. Thus, constraint 4.3) refers to the fact that ob is processed to a fraction of one. Constraint 4.4) refers to the fact that a machine can only perform one unit of processing during [t, t + ]. In the obective function, the first term can be thought of as fractional flow-time xt spends t r ) unit of time before it finishes. fraction of the ob The second term ignoring the factor of /) in the obective function measures the total processing time this term is required because a solution to the LP above could process a ob simultaneously on multiple machines, and so, the flow-time of a ob could be much smaller than its processing time. The following claim, which says that the LP above is indeed a relaxation, was proved in [9]. Claim 4.. [9] For a non-migratory) schedule S, consider the corresponding feasible solution to the LP above. Then the obective value of this solution is at most the weighted flow-time of S. The dual LP of this relaxation is described in Figure. The goal is to prove the following theorem. Theorem 4.. There exists a feasible solution α, β it to the dual LP such that the obective value of this solution, i.e., α i,t β it, is Ω F A ). We now define these dual variables. The variable β it is equal to + times the total weight of obs which 4 are waiting at time t on machine i, i.e., β it = + A it) w. Observe that i,t β it is exactly equal to F A +. Now, we define α. Suppose is released at time t and it gets assigned to machine i. Then α is equal to the increase in weighted flow-time of obs in A i t) and ob ). In other words, α is equal to Q as defined in equation 4.). Lemma 4.. α i,t β it is at least + F A. Proof. The statement follows because α is equal to F A and i,t β it is equal to F A +. It now remains to prove that this is a feasible solution. Lemma 4.. The values α /, β it / form a feasible solution to the dual LP. Proof. Fix a ob and machine i. Let t denote the release date of and t be an arbitrary time after t. We need to show that 4.8) α β it d t t) + w. We can assume that no new ob arrives after indeed, this will not affect the value α, and will only decrease β it. Let A i t) denote the set of obs in A i t) for which d t) d, and A i t) be the obs in A i t) for which d t) < d. We arrange the obs in A i t) in ascending order of residual density; let the sequence of obs be,..., n. Note that machine i will process the obs in A i t) in this sequence. Let the set A i t) consist of obs {,..., r }, and hence) the set A i t) be { r+,..., n }. Suppose at time t, machine i is processing ob. Two cases arise :

5 A i t) : Here, t t) + ) l= p l t). Further, all obs among,..., n belong to A i t ). Recall that α = min i Q i Q. So, + )α + )Q d p t) + w + w A i t) A i t) ) r ) = d p l t) + d p l t) + w l= l= + A i t) w d t t) + ) + r w l + w + l= d t t) + ) + w + + )β it A i t) w where the second last inequality follows from the fact that if A i t), then d t) d, and so p t)d w. A i t) : Again, t t) + ) l= p l t), and all obs among,..., n belong to A i t ). As above, + )α + )Q d p t) + w + A i t) = d r l= d r l= p l t) p l t) ) ) + + )β it + w + l=r+ A i t) w w l + + w + d l=r+ d t t) + ) + w + + )β it satisfy the inequal- Thus, we have shown that α ity 4.8) the in both cases., βit n l= p l t) w l ) Proof of Theorem 4.. The desired dual feasible solution is α /, β it / Lemma 4.). Lemma 4. shows that the dual obective value is at least +) F A = Ω) F A. 5 Extension to l norm We now extend our result to weighted l norm of the flow-times of obs. We state the obective function more 5 precisely. For a schedule S, let C be the completion time of ob. Define the weighted l -norm of the flowtime vector as w i) C r ) where i) is the machine on which gets scheduled. It will be easier to deal with the th power of this quantity. Let F S, denote w i)c r ). 5. Scheduling Algorithm We define the scheduling algorithm in two stages the first algorithm A is given a speed-up of + ), and then the schedule B is given a speedup of + ) over A. Our algorithm A again behaves in a manner similar to the greedy algorithm described in Section 4.. When a ob arrives, it gets immediately dispatched to one of the machines, and hence, we can define the quantities A i t), p t), d t) as before. Again, each machine follows the highest residual density first HRDF) algorithm. It remains to describe the dispatch policy. For a ob A i t), let R t) be the remaining time before finishes. i.e., C t, where C is the completion time of assuming no obs arrive after time t. Suppose a ob arrives at time t. Given a machine i, define Q as ) w A it):d t) d p t) + 5.9) + + w R t) + p ) ) R t). + A it):d t)<d Now, assign to the machine i for which Q is smallest. Note that this is different from the increase in F A, if we assign to machine i it does not loo at the age of a ob. The quantity is ust measuring the increase in flowtime if we assume all obs in A i t) are released at time t. Clearly, this cannot yield a competitive algorithm because if the age of a ob is very high, then we should process it soon. However, we will show that after giving one more round of extra speed to the machines, the resulting schedule becomes constant-competitive. Before we describe the schedule B, we note one property of the HRDF policy. Fix a machine i, and consider the obs which get dispatched to i in A the HRDF policy gives a total ordering on these obs. Indeed, consider two such obs and which are dispatched to i. Let t denote maxr, r ). Then HRDF prefers if d t) d t), otherwise it prefers. It is,

6 easy to show that this gives a total ordering on the obs dispatched to i, and the HRDF policy processes the obs according to this ordering. In the schedule B, the set of obs dispatched to any machine i will be the same as in A. Fix a machine i. In B, the obs will be prioritized by the ordering given by HRDF policy in A as described above). Further the machine i will process at + )-times the speed of i in A. 5. Analysis We first establish some useful inequalities. Lemma 5.. Suppose a,..., a m are non-negative integers. Let b i denote a + + a i. Then, m m 5.0) a i+ b i b m i= Taing a = = a m =, we get 5.) m i m i= i= a i b i. m i. i= Proof. Consider the area under the curve y = x where x varies from 0 to b m. Then, it is easy to chec that this area is equal to b m. Now, one way of upper bounding this area is the following : for each i =,..., m, construct a rectangle of height b i and base between the points b i and b i on the x-axis. It is easy to chec that the sum of the areas of these rectangles is at least the area under the curve mentioned above. The lower bound is proved similarly. Proof. Consider a ob, and suppose its completion time is C. Then its contribution to F S, is w C r ) here i is the machine on which gets scheduled. Now, t w x t t r ) dt w C r ) x t dt t = w C r ). Thus the first term in the obective function is at most F A,. Now, if a ob is scheduled on machine i by S, then t x tdt =, and so, t w p x t dt = w p w C r ). Therefore, the second term in the obective function is also at most F A,. Rest of the plan of the proof is to come up with a feasible dual solution whose obective value is close to F A,. The dual LP is shown in Figure. We assume that the machines in A have + )-extra speed. For a machine i and time t, define β it as A it) w R t). For a ob, define α as the quantity Q defined in in 5.9) where i is the machine to which gets dispatched. The following lemma shows that the dual variables as defined above have obective value close to the cost of schedule B. Lemma 5.. Let a, b be non-negative integers. for any, 0 < <, ) 3 5.) a + b) + )a + b. Then, Proof. If b a, then a+b) + ) a +)a. Otherwise, a + b) + ) b. We now write an LP relaxation for this problem. Here, it will be convenient to thin of time t as a continuous variable. The LP relaxation is described in Figure. The variable x t should now be thought of as the rate at which machine i is processing ob at time t. The constraints are same as those in the LP for total weighted flow-time case Figure ). We now argue that it is indeed a relaxation. Lemma 5.3. Consider a schedule S where machines run at their original speed), and let x t be the corresponding LP solution. Then, the obective value of this LP solution is at most F S,. 6 Lemma 5.4. The obective value for the solution α, β it defined above, i.e., α i t β itdt is Ω + F B,). Proof. Note that 5.9) i t β it = i = t A it) w R t) dt w i) t r R t) dt, where i) is the machine to which gets dispatched. For a fixed, let us try to understand the right hand side above. Let t 0 = r < t <... < t r be the times at which R t) is discontinuous this happens precisely when a ob of higher density than gets dispatched to machine i) at time t. If t is not one of such times, then R t) decreases linearly at the rate of +. So

7 5.3) 5.4) 5.5) Primal LP min ) xt w t r ) + p x t dt i, t x t dt = i t r x t i, t 5.6) 5.7) 5.8) Dual LP max α β itdt i t α β it d t r ) + w p α, β it 0 i,, t i,, t x t 0, i, t r Figure : The LP relaxation and its dual define t r+ as C ), 5.0) t r R t) dt = = = = r l=0 t l+ t + l + ) + ) r l=0 t l+ R t) dt t + l R t + l ) + )t t l) ) dt r R t + l ) R t l+ )) l=0 r R t + l ) R t l )) l=0 for B. It is easy to chec the invariant that for any time t and ob, R Bt) RA t). Now, at time r, let V A be the total remaining processing time of obs which have precedence over we have already mentioned that HRDF induces a total ordering on obs). Define V B similarly for the schedule B. Our invariant implies that V B V A. Suppose, for the sae of contradiction, that does not finish at time t in B. Then B processes obs of priority higher than during [r, t], and the total processing done by B during this interval is + ) t r ). If V denotes the total processing time of obs of priority higher than which get released during [r, t], it follows that where we define R t 0 ) as 0. Let ) denote the quantity r R t + l ) R t l )). l=0 Now, observe that Q as defined in 5.9) is exactly measuring the increase in w i ) ) if gets dispatched to machine i and no obs arrive after. Thus, α is exactly equal to w i). Combining this with 5.9) and 5.0), we see that α β it =. i t + It remains to prove that F B,, where F B, denotes the th power of the flow-time of in the schedule B. We begin with a useful claim first. Claim 5.. For a ob and time t r, if R t) t r ), then ob finishes by time t in B. Proof. For the sae of this proof, let R A t) denote the remaining processing time of at time t in A, i.e., R t), and R B t) be the corresponding quantity 7 V + V B > + ) t r ). But then, the total remaining processing time of such obs in A at time t will be at least V +V A +)t r ) V +V B +)t r ) > t r ). This implies that R t) > t r ), a contradiction. Fix a ob, and let t 0 = r < t <... < t r < t r+ = C be as defined above with respect to schedule A). Let F t) be the flow-time of assuming no obs arrive after time t in schedule A). So, F t) = R t) + t r ). Let t be the first time in A such that R t) t r ). Since R t), as a function of t, is continuous except with finite discrete positive umps, it must be the case that at time t, R t) = t r ). Claim 5. shows that will finish by time t in B. Therefore, F B, t r ) R t) Suppose t lies between t m and t m+. Then R t)

8 R t + m), and observe that R t + m) R t + m) + = m l=0 R t + l ) R t l+ )) m R t + l ) R t l )). l=0 Now, we prove that α, β it are nearly feasible to the dual LP. Lemma 5.5. The values α /γ, β it /γ form a feasible solution to the dual LP, where γ is Ω )). Proof. Fix a ob and machine i. Let t denote the release date of r, and fix a time t t. We will show that α 5.) β it Ω )) d t t) + ). This will imply the lemma because d t t) + ) d t t) + w p. We will mae one simplifying assumption no new ob arrivals happen during [t, t ] indeed such ob arrivals will not change α, but would only increase β it. Now, arrange the obs in A i t) in descending order of d t) values let this order be,..., m. We first observe that at time t, d t ) d t )... d m t ). Indeed, machine i processes obs in this order. So at time t, if it is processing ob l, then, d t ) = = d l t ) =, d l t ) d l t) and d l+ t ) = d l+ t),..., d m t ) = d m t). We develop some more notation. Let A i t) denote those obs in A i t) for which d t) d, and A i t) be the remaining set of obs. Assume that A i t) = {,..., r }, and so, A i t) = { r+,..., m }. Recall that α = min i Q. So, 5.) α Q + = d A i t) p t) + p ) + A i t) w R t) + p + ) R t) ) Now, we simplify each of the terms above. Claim 5.. ) d A i t) p t) + + w R t ) A i t) + 3 ) d t t) + ). Proof. During the period [t, t ], t t) + ) amount of processing is done. So, A t)p t) p t )) = i A i t) p t) A i t ) p t ) + )t t). Also, observe that R l t ) = l + s= p s t ). Therefore, ) A i t) p t) + + p t ) t t) 5.) A i t ) + ) 5.0) A i t ) p t ) + ) t t)) p t )R t ) + 3 A i t ) ) + t t)). Now, multiplying both sides by d, and using the fact that A i t), we get p t )d p t)d = d w d t) w. This implies the lemma. Bounding the second term in equation 5.) is more non-trivial. We begin with the following claim. Claim 5.3. w A i t) R t) + p ) ) R t) + + w R t) + 3 A i t) ) 8 d p 8

9 Proof. Using binomial expansion, R t) + = p ) R t) + + R t) + l= ) ) l R t) l p. l + If 8 R t), then it is easy to chec that the above expression is at most +/ R t). So, assume that R t) 8 let J denote the set of such obs. For such obs, it is again easy to chec using the binomial expansion) that 5.3) w J + R t) + p ) ) R t) + J w R t) + ) 8 p J w Observe that there is a ob in J the one which is processed last by i) for which + )R t) J p t). So, we get J w J p t)d 8 + ) w, where the first inequality follows from the fact that A i t). Substituting the above in 5.3) implies the lemma. We now simplify the expression in the claim above. Recall that the set of obs in A i t) are ordered,..., r, and those in A i t) are r+,..., m. Also, note that + )R s t) = s s = p s t). Let l be the highest index for which R l t) 6t t). We now split the sum in the right hand side of Claim 5.3 in two parts, and analyze each of them. Claim 5.4. l s=r+ w s R s t) )) Ω d t t). A i t), w s p s t) d. Now, l s=r+ = 5.0) p s t) + R s t) l s=r+ l s=r+ p s t) Rs + t) + p s t) ) p s t) + R s t) + + l R l t) + + s=r+ p s t) l s=r+ ) p s t) Since l s=r+ p s t) = + )R l t), and R l t), we get the result. 6t t) Claim 5.5. m w s R s t) s=l+ + ) m s=l+ w s R s t ). Proof. For such obs s, R s t) 6t t). During [t, t ] machine i can process at most + )t t) t t) volume of obs. So, R s t ) 8 ) Rs t). Combining Claims 5., 5.3, 5.4, 5.5, we get the statement of the lemma. Finally, we have our main theorem. Theorem 5.. Algorithm A is an O + competitive algorithm for minimizing the weighted l -norm of flow-time. Proof. Let O denote the optimal schedule. We have + F B, ) Lemma 5.4 α Ω γ γ i t β itdt γ Lemmas 5.3, 5.5 F O,. Taing th root gives the theorem. 6 Weighted Flow-time + Energy on Unrelated Machines We now consider the problem of minimizing the sum of weighted flow-time and energy. We are also given an energy function which has the form fs) = s γ, where γ is a constant parameter. In other words, if a machine runs at speed s, then the energy consumed per unit time is s γ. We define = γ. We again need a notion of speed augmentation. Let f denote the function ) - Proof. Consider obs s, s = r +,..., l. Since s 9 fs) = f s )).

10 Note that f allows us to incur the same energy cost as f, however with an extra speed-up of We shall. allow our algorithm to use f, whereas the optimal offline algorithm will use f. Note that the notion of f is ust for the purpose of designing and analyzing the algorithm. The algorithm will of course incur the cost as dictated by f. For technical reasons, which will become clear later, it will be easier for us to wor with fractional weighted flow-time of obs. Let p t) be the remaining processing time of ob at time t in a schedule). The fractional remaining weight of the ob at time t is defined as w t) = pt) p w = d p t), where d is the density of ob and i is the machine to which it gets dispatched. The fractional weighted flow-time of ob is defined as w t)dt. t r Our algorithm will be designed for minimizing the sum of total fractional weighted flow-time of obs and the energy consumed. It will follow from standard arguments that one can derive a competitive algorithm with an extra γ-factor in the competitive ratio) for minimizing the sum of total weighted flow-time of obs and the energy consumed. 6. The algorithm In this section, we describe the scheduling algorithm. A. For a machine i and time t, recall that A i t) denotes the set of waiting obs at time t in the queue of machine i. Let W i t) denote the total fractional remaining weight of the obs in A i t). At time t, a machine i will run at the speed s t which satisfies fs t ) = W i t). The machine follows HDF Highest Density First) policy : at any time, it processes the ob in its queue with highest density. When a ob, arrives we compute for each machine i the increase in the fractional flow-time if we dispatch to i call this quantity Q. The ob is dispatched to the machine for which Q is smallest. Note that the total fractional weighted flow-time of our algorithm is equal to the total energy consumed if we use f). This algorithm is essentially the same as that of Bansal et al. [5]. 6. Analysis We first give some intuition behind the analysis. One can write a convex program for this problem. The idea is again to set the variables in the Lagrangian dual. Recall one crucial and easy to see) property we used while analyzing the algorithms in previous section : if after some time t, we do not dispatch any obs to a machine i, then the total weight of remaining obs at a time t t on machine i can only decrease as 0 compared to the case when more obs get dispatched to i after time t. This monotonicity property may not hold here because as more obs get dispatched to a machine, its speed also increases. In fact, it turns out that such a property may not hold if we are woring with the HRDF policy and loo at the total weight of remaining obs. However, if we wor with the HDF scheduling policy and only loo at the total remaining fractional weight of obs, then such a property holds. We first show this result. Then we analyze the scheduling algorithm as done in previous sections. The fact that we are now woring with a convex rather than a linear program maes the calculations more non-trivial, but the definition of dual variables remains the same. In the following discussion, we shall assume, for ease of notation, that there is only one machine. We can then apply the result to any of the machines in our input instance. Given an input instance with only one machine), we shall define several other instances, which we shall denote by I or I. We shall apply our algorithm A on each of these instances. For an instance I, we shall let A I t) to be the set of obs waiting at time t in I when we run A on I). Define w It), pi t) similarly. We shall use W I t) to denote the total fractional weight of obs in A I t), i.e., A I t) wi t). If we do not put a superscript instance I, then we will be refering to the actual input instance. Suppose in an instance I, no ob gets released after a certain time t. For a value W, 0 W W I t), we define the ob I W, t) as follows : run the algorithm A on I, and consider the time t t when W I t ) = W. Then I W, t) is the ob which is being processed at time t. A monotonicity property : Consider two instances I and I which are identical except that there is a ob in I which does not appear in I. Further, assume that no obs are released after time r in either of the instances. Claim 6.. For all W, 0 W W I r ), d I W,r) d I W,r) Proof. The set of obs A I r ) and A I r ) are identical except for the ob which appears in the latter set. Arrange the obs in A I r ) by decreasing density. Let this ordering be,..., r. The corresponding order for A I r ) will be same but with inserted somewhere in the sequence. Since the two instances are identical till r, w I l r ) = w I l r ) for all l. Now, I W, r ) is the ob l, where the index l satisfies : w I l r )+...+w I r r ) W < w I l+ r )+...+w I r r ). Since the corresponding sequence in I will contain as well, the ob I W, r ) would be a ob which comes

11 after l it could be l as well) in the sequence. This proves the inequality. Corollary 6.. For any time t r, W I t ) W I t ) Proof. Suppose the inequality is not true let T be the set of time t which violate this inequality. Let t be the infimum of T. Since, W I t ), W I t ) are continuous with t, we must have W I t ) = W I t ) call this quantity W. Thus, the speed at time t is the same in both the schedules. But Claim 6. shows that density of I W, t ) is at most the density of I W, t ). Thus, we are processing a ob of higher density at time t in I than that in I. So, for arbitrary small > 0, W I t + ) W I t + ). This contradicts the definition of t. We now prove the monotonicity property. Consider two instances I and I with the following properties. There is a time t such that till t, both instances are identical. In I, no ob gets released after t, whereas I may see some more obs after time t including t). We would lie to prove that W I t ) W I t ) for all time t t. Note that this is not entirely obvious because addition of new obs will speed up the machines, and so obs will get processed faster. Lemma 6.. Let I, I and time t be as above. Then for any time t t, W I t ) W I t ). Proof. The proof is by induction on the number of obs in I which get released during [t, t ]. If there are no such obs, then I and I are identical, and so we are done. Now, let be the last ob which is released in I during t, t ]. Let I be the instance which is same as I except for the ob. By induction hypothesis, W I t ) W I t ). So we ust need to compare W I t ) and W I t ). But this follows from Corollary 6., where I = I, I = I. The Convex Program and its Lagrangian Dual : We first write a convex programming relaxation. For a ob, let s t be the speed with which we are processing it at time t on machine i. The convex program is given in Figure 3. We show that it is indeed a relaxation. Lemma 6.. Consider a non-migratory schedule S for an instance I of this problem. Let s t be the corresponding solution to this convex program. Then the obective value of the solution s t for this convex program is at most a constant times the sum of the weighted fractional flow-time and the energy consumed by S. Proof. The constraint 6.5 is stating that a ob must be processed to full extent. So, s t must satisfy this constraint. Let us loo at the obective value. For a ob, its fractional weighted flow-time of can be written as ) d p t)dt = d s t dt dt i t r t r t t = d s t t r )dt. t r Thus, the first term in the obective function is capturing the weighted fractional flow-time, and the second term represents the total energy consumed. Let us now show that the third term is at most a constant times the cost of S. Suppose S schedules a ob on machine i. Let T be the time by which i processes half of, i.e., does / amount of processing on. Let s be the average speed of i while processing during [r, T ], i.e., s = T T r r s t dt = p T r ). Since the energy function is a convex function, it is easy to chec that the energy consumed while processing is at least T r )s γ = s sγ. Also, the total energy consumed by S is at least the sum over all obs of the energy consumed while processing only. Further, the fractional weighted flowtime of is at least T r ) w that s γ + w ) 4s = wp 4s 4 w.. Now observe w i) p i), So, we see that the cost of S is at least 4 where i) is the machine on which gets dispatched by S. Now the third term in the obective function is w s t dt = w i) s i)t dt,i t r t r This proves the lemma. = w i). The Lagrangian dual where we have Lagrangian variables for the constraints 6.5), is shown in Figure 3 the s t variables are defined only for t r ). Setting the Dual Variables We now show how to set the α variables. Fix a ob. We define α as the increase in weighted fractional flow-time assuming no new ob arrives after, i.e., α = min i Q, where Q is the increase in weighted fractional flow-time if we dispatch to machine i. Our result will follow from the following theorem.

12 min 6.4) 6.5) 6.6) Primal Relaxation ) s td t r )dt + f s t dt t r t + w s tdt,i t r s t dt = i t r s t 0 i,, t r Lagrangian Dual max α [ max s α i t s t α t )] 6.7) d t r ) w ) f s t dt 6.8) s t 0 i,, t Figure 3: The convex programming relaxation and its Lagrangian dual Theorem 6.. Fix a time t and machine i. Then, the maximum value of ) α 6.9) s t d t r ) w p f s t, where the maximum is taen over s t for all obs, s t 0, is at most W i t ) ). Before we prove this theorem, let us see why it implies the desired result. Corollary 6.. The algorithm A is Oγ)-competitive for the obective of minimizing the sum of weighted fractional flow-time and energy. This yields an Oγ )- competitive algorithm for the obective of minimizing the sum of weighted flow-time and energy. Proof. Let F A be the total weighted fractional flowtime of the algorithm A. Then the theorem above shows that the value of the Lagrangian relaxation is at least α )F A. But, α is equal to F A, and hence, the value of the Lagrangian relaxation is at least F A. But, we now that the optimal value of the Lagrangian dual is Oopt), where opt denotes the value of the optimal solution Lemma 6.). Thus, F A is at most Oopt/). Now, we need to compute the energy incurred by A. If we assume that the energy function is f, then the energy consumed is same as F A. This is so because at any point of time t, we choose the speed s such that fs) = W t). However, the energy function γ, is f, and so we pay an extra factor of ) which is only a constant. This proves the first statement. For the second statement, we run the algorithm A, but now we speed up each machine by a factor of +). Call this schedule B. Using standard arguments see e.g., [6]) the total weighted flow-time of B is at most Oγ) times the total weighted fractional flow-time of A. Further, the energy consumed by B is only a constant factor more than that by A. We now prove Theorem 6.. Fix a time t and machine i for the rest of this discussion. Consider the values s t which maximize the quantity 6.9). We first observe that exactly one of these values will be non-zero. Indeed, suppose there are two obs and such that s t, s t > 0. Then either the values s t +δ, s t δ, or s t δ, s t +δ will improve the quantity in 6.9). Thus, we will be done if we prove the following lemma. Lemma 6.3. For any ob, r t, and s 0, ) α s d t r ) w fs) W i t ) ). Proof. Fix a ob which gets released at time t. Rearranging terms, it is enough to show that for all s, α d t t) w s γ + W it ) ). s We would lie to find the value of s which minimizes the right-hand side. By using differentiation, one can easily chec that the right hand side above is at least W i t ). Thus, we will be done if we show that α d t t) w W i t ) Using Lemma 6., we can assume that no obs arrive after time t because this will not change the left hand side above, and will only increase the right hand side). At time t, let the obs in A i t) arranged by descending

13 order of density be,..., m, and suppose the density of lies between those of and +. Assume that does not get dispatched to i the other case is similar. Let J denote the set of obs {,..., } and J be the obs in { +,..., m }. Let Wi t) be the total fractional remaining weight of obs in J at time t, i.e., l= w l t). Define Wi t) for J similarly. Suppose it too x units of time to process ob if were dispatched to i note that would be processed after J and before J ). Further let T 0 be the time at which the obs in J finish processing on machine i. Then we can bound α by 6.30) α w T 0 t) + w x + W i t)x. Indeed, because of the arrival of, the fractional flowtime of obs in J will decrease because the machines will now run at faster speed. So the increase in fractional weighted flow-time can be bounded by the weighted flow-time of plus the increase in that of obs in J. The latter is at most x Wi t) because the delay in the completion of a ob in J is at most x note that the obs in J will finish earlier). We now consider two cases depending on the time t. First assume that t T 0. So, the machine i is processing a ob in J at time t. Claim 6.. T 0 t d Wi t ) W i t) ). Proof. Fix a time u between t and T 0. Let W i u) be the total remaining fractional weight at time u. Then, the speed of machine i at this time is Wiu). So, in a small amount of time du, we process Wiu) du volume of obs. Since the density of these obs is at least d, the decrease in the total fractional remaining weight, dw is at least d W i u) ) du. So we can set up the differential equation Wit ) W i t) )dw d W γ Integrating this gives the claim. Corollary 6.3. x is at most min T0 t du. w, p ) W i t) ). The second inequality follows because while the ob is getting processed, the remaining fractional weight is at least W i t). So the speed will be at least W i t). Now, Corollary 6.3 and inequality 6.30) imply that α d T 0 t) + d w + x Wi t) d T 0 t) + w + Wi t) ) = d T 0 t ) + w + d t t) Claim 6. + W i t) ) d t t) + w + W i t ) This implies the lemma. Now, we consider the case when t > T 0, i.e., the machine i is processing a ob of J at time t. So, W t ) is same as Wi t ). Claim 6.3. t T 0 d W i t) W i t ) ) Proof. Consider a time u between T 0 and t. Let W i u) be the remaining fractional weight at time u. Then, as in the proof of Claim 6., the total processing done during [u, u + du] is Wiu) du. Since the density of obs in J is at most d, the total fractional weight dw processed during this time is at most dwiu) du. Thus we get the differential equation : W i t) W it ) )dw d W Integrating this gives the result. t T 0 du. Again, using Corollary 6.3 and inequality 6.30), we get α d T 0 t) + d w + W i t )x d t t) d t T 0 ) + w + )Wi t ) Claim 6.3 This implies the result. d t t) + w + W i t ) Proof. If were dispatched to machine i, we can show by the same proof as that of Claim 6. that the machine i will process for d W i t) + w ) Wi t) ) amount of time. But the latter is at most w. This proves the first inequality. 3 References [] N. Avrahami and Y. Azar. Minimizing total flow time and total completion time with immediate dispatching. In SPAA, pages 8. ACM, 003.

14 [] N. Bansal and K. Pruhs. Server scheduling in the l p norm: A rising tide lifts all boats. In ACM Symposium on Theory of Computing, pages 4 50, 003. [3] Nihil Bansal, Ho-Leung Chan, and Kir Pruhs. Speed scaling with an arbitrary power function. In SODA, pages , 009. [4] Nihil Bansal and Kir Pruhs. Server scheduling in the weighted lp norm. In LATIN, pages , 004. [5] Nihil Bansal, Kir Pruhs, and Clifford Stein. Speed scaling for weighted flow time. SIAM J. Comput., 394):94 308, 009. [6] Jivite S. Chadha, Naveen Garg, Amit Kumar, and V. N. Muralidhara. A competitive algorithm for minimizing weighted flow time on unrelatedmachines with speed augmentation. In STOC, pages , 009. [7] C. Cheuri, S. Khanna, and A. Zhu. Algorithms for weighted flow time. In STOC, pages ACM, 00. [8] Chandra Cheuri, Ashish Goel, Saneev Khanna, and Amit Kumar. Multi-processor scheduling to minimize flow time with epsilon resource augmentation. In STOC, pages , 004. [9] Naveen Garg and Amit Kumar. Better algorithms for minimizing average flow-time on related machines. In ICALP ), pages 8 90, 006. [0] Naveen Garg and Amit Kumar. Minimizing average flow-time : Upper and lower bounds. In FOCS, pages , 007. [] Anupam Gupta, Ravishanar Krishnaswamy, and Kir Pruhs. Scalably scheduling power-heterogeneous processors. In ICALP ), pages 3 33, 00. [] Sungin Im and Benamin Moseley. An online scalable algorithm for minimizing l -norms of weighted flow time on unrelated machines. In SODA, 0. [3] Bala Kalyanasundaram and Kir Pruhs. Speed is as powerful as clairvoyance. In IEEE Symposium on Foundations of Computer Science, pages 4, 995. [4] Hans Kellerer, Thomas Tautenhahn, and Gerhard J. Woeginger. Approximability and nonapproximability results for minimizing total flow time on a single machine. In STOC, pages 48 46,