Nonclairvoyant Scheduling to Minimize the Total Flow Time on Single and Parallel Machines

Transcription

1 Nonclairvoyant Scheduling to Miniize the Total Flow Tie on Single and Parallel Machines LUCA BECCHETTI AND STEFANO LEONARDI University of Roe La Sapienza, Roe, Italy Abstract. Scheduling a sequence of obs released over tie when the processing tie of a ob is only known at its copletion is a classical proble in CPU scheduling in tie sharing operating systes. A widely used easure for the responsiveness of the syste is the average flow tie of the obs, that is, the average tie spent by obs in the syste between release and copletion. The Windows NT and the Unix operating syste scheduling policies are based on the Multilevel Feedback algorith. In this article, we prove that a randoized version of the Multilevel Feedback algorith is copetitive for single and parallel achine systes, in our opinion providing one theoretical validation of the goodness of an idea that has proven effective in practice along the last two decades. The randoized Multilevel Feedback algorith (RMLF) was first proposed by Kalyanasundara and Pruhs for a single achine achieving an O(log n log log n) copetitive ratio to iniize the average flow tie against the on-line adaptive adversary, where n is the nuber of obs that are released. We present a version of RMLF working for any nuber of parallel achines. We show for RMLF a first O(log n log n ) copetitiveness result against the oblivious adversary on parallel achines. We also show that the sae RMLF algorith surprisingly achieves a tight O(log n) copetitive ratio against the oblivious adversary on a single achine, therefore atching the lower bound for this case. Categories and Subect Descriptors: C.4 Perforance of Systes]: perforance attributes; F.2.2 Analysis of Algoriths and Proble Coplexity]: Nonnuerical Algoriths and Probles sequencing and scheduling; G.3 Probability and Statistics]: Probabilistic Algoriths General Ters: Algoriths, Perforance, Theory Additional Key Words and Phrases: Probabilistic analysis, flow tie, ultilevel feedback, randoized algoriths 1. Introduction In this article, we study nonclairvoyant algoriths to schedule a strea of obs released over tie on single and parallel achine systes. Every ob is assigned with a release tie r 0 and a processing tie p. Job ust be globally scheduled for p tie units on parallel identical achines before its copletion. Authors address: Dipartento di Inforatica e Sisteistica, University of Roe La Sapienza, Via Salaria 113, Roe, Italy, e-ail: Luca.Becchetti@dis.uniroal.it. Perission to ake digital or hard copies of part or all of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or direct coercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for coponents of this work owned by others than ACM ust be honored. Abstracting with credit is peritted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any coponent of this work in other works requires prior specific perission and/or a fee. Perissions ay be requested fro Publications Dept., ACM, Inc., 1515 Broadway, New York, NY USA, fax: +1 (212) , or perissions@ac.org. C 2004 ACM /04/ $5.00 Journal of the ACM, Vol. 51, No. 4, July 2004, pp

2 518 L. BECCHETTI AND S. LEONARDI Job preeption is allowed, the execution of a ob can be stopped and resued later on the sae or on a different achine. The copletion tie of ob is denoted by C.Anonclairvoyant scheduling algorith knows very little about the input instance: The existence of a ob is only known at the release tie of the ob; The processing tie of a ob is only known when the ob is copleted. This proble has a nuber of otivating applications. The ost classical one is processor scheduling in a tie-sharing ultitasking operating syste, in which scheduling decisions ust be taken without knowledge of the tie a ob needs to be executed. The obvious goal is to provide a fast response to users. Job preeption is widely recognized as a key factor to iprove the responsiveness of the syste. In ultiprocessor coputer systes, preeption requires a context switching at a processor but this cost is reasonably sall. 1 A widely accepted easure of the quality of service provided to users is the average response tie of the syste. The response tie, or flow tie, of every ob is the tie spent by the ob in the syste between release and copletion, that is, C r for ob. We easure the perforance of a randoized nonclairvoyant scheduling algorith by its copetitive ratio Sleator and Taran 1985; Ben-David et al. 1994], the worst-case ratio between the expected average flow tie of the algorith and the optial average flow tie of an oblivious adversary that generates the input sequence without knowledge of the rando choices of the algorith. This proble has been addressed for a couple of decades in the design of tie sharing operating systes. Windows NT Nutt 1999] and Unix Tanenbau 1992] have the Multilevel Feedback (MLF) algorith at the very basis of their scheduling policies. The basic idea of MLF is to organize obs into a set of queues. Each ob is processed for 2 i tie units if in queue Q i, before being prooted to queue Q i+1 if not copleted. At any tie, the achines process obs in the lowest queues, in each queue giving priority to obs at the front. While this algorith turns out to be very effective in practice, it behaves very poorly with respect to a worst-case analysis, as explained below. A good rule of thub for flow tie iniization is given by the Shortest Reaining Processing Tie (SRPT) first rule. SRPT prescribes the preeption of a ob on execution when a ob with shorter reaining processing tie is released. SRPT is indeed an optial algorith for a single achine Baker 1974] and provides the best known approxiation for parallel achines Leonardi and Raz 1997]. However, a nonclairvoyant scheduling algorith cannot stick to the SRPT rule since it has no knowledge of the processing tie of the obs before they are copleted. As to MLF, it behaves on soe instances very differently fro the SRPT rule in that it ay preept obs in a high queue that are nearly copleted to process newly released obs with large processing tie in lower queues. This ay force obs with sall reaining processing tie to spend a long tie in the syste while other long obs are processed. It has actually been shown that no deterinistic nonclairvoyant algorith can be copetitive at all against a worst case adversary Motwani et al. 1994]. In order to circuvent these difficulties, a randoized version of MLF, called RMLF, was proposed for a single achine by Kalyanasundara and Pruhs 1997]. 1 This in particular holds in systes that support threads Doeppner 1987; IEEE 1994; Mueller 1993; SUN 1993].

3 Nonclairvoyant Scheduling to Miniize Total Flow Tie 519 The idea is to try to approxiately behave like SRPT by having a large fraction of obs with a reaining processing tie that is still a sufficiently large share of the initial processing tie when entering their respective queues of copletion. Of course, this requireent can only be satisfied with soe probability. Kalyanasundara and Pruhs 1997] show that RMLF is O(log n log log n) copetitive for iniizing the total flow tie on a single achine against the on-line adaptive adversary Ben- David et al. 1994]. The on-line adaptive adversary ay decide its strategy at tie t based on the knowledge of the rando choices of the algorith up to tie t. However, the processing tie of a ob ust be fixed by the adversary at release tie of the ob. In this article, we present a randoized version of the Multilevel Feedback algorith working for any nuber of parallel achines. The algorith we present is an evolution of the one presented in Kalyanasundara and Pruhs 1997] for a single achine, for this reason we also call RMLF our algorith. We present two ain results for RMLF: (1) We show that RMLF has copetitive ratio O(in{log n log n, log n log P}) against an oblivious adversary, where P is the ratio between the largest and the shortest processing tie of a ob. This is the first o(n)-copetitive result for nonclairvoyant average flow tie iniization on parallel achines. This copares with the (log n ) and (log P) lower bounds against the oblivious adversary given in Leonardi and Raz 1997] for the case in which the processing tie of a ob is known at release tie. (2) For the case of a single achine, we show that our version of RMLF surprisingly atches the (log n) lower bound against an oblivious adversary of Motwani et al. 1994]. These two results are obtained as an outcoe of a unified analysis of RMLF. They are proved with a probabilistic analysis that, unlike Kalyanasundara and Pruhs 1997], does not require any high probability arguent. In turn, our analysis requires a considerable strengthening and siplification of the tools previously developed in the study of algoriths for iniizing the average flow tie Leonardi and Raz 1997; Awerbuch et al. 2002] RELATED RESULTS. Nonclairvoyant scheduling to iniize average flow tie has been first studied by Motwani et al. 1994]. The authors prove that any deterinistic nonclairvoyant algorith is (n 1/3 )-copetitive where n is the nuber of obs released, and that every randoized nonclairvoyant algorith is (log n)- copetitive in the case of a single achine. The authors also present copetitive algoriths for the static case where all obs are released at tie 0. Kalyanasundara and Pruhs 1997] give the first o(n) copetitive nonclairvoyant scheduling result for a single achine. They prove that a randoized version of MLF is O(log n log log n) copetitive against an on-line adaptive adversary Ben-David et al. 1994]. They also clai an (P) randoized lower bound for the proble, where P is the ratio between the axiu and the iniu processing tie of a ob, on an instance with a nuber of obs that is exponential in P. In a previous paper, Kalyanasundara and Pruhs 1995] study a different odel in which the nonclairvoyant algorith is equipped with a faster processor than the adversary. They prove in this case that shortest elapsed tie first is a constant copetitive algorith for a single achine. More recently, Bansal et al. 2003] have applied resource augentation

4 520 L. BECCHETTI AND S. LEONARDI to nonclairvoyant scheduling to iniize average stretch on parallel achines, proposing an O(1)-speed, O(log 2 P)-copetitive algorith, P being the ratio between the largest and sallest ob sizes. They also propose an (log P) lower bound, which is tight when all obs are released at tie 0. Bansal and Pruhs 2003], finally, aong other results also prove that the Shortest Elapsed Tie First heuristic is O(1 + ɛ)-speed, O(1/ɛ (2+2/p) )-copetitive to iniize the L p nor of the flow tie, and that with constant speed-up it is poly-logarithically copetitive (with respect to P) for the L p nor of the stretch. Related to nonclairvoyant scheduling is the work of Bender et al. 2002], where the authors show that a constant-factor copetitive ratio for average stretch is achievable even if the processing ties (or reaining processing ties) of obs are known only to within a constant factor of accuracy. Related results are also concerned with the ore classical average flow tie iniization proble when the processing tie of a ob is known at release tie. Also in this case, if preeption is not allowed, achieving reasonable perforances on-line is not possible. If preeption is allowed, the proble can be optially solved on 1 achine by the well-known SRPT (Shortest Reaining Processing Tie) heuristic Baker 1974]. Leonardi and Raz 1997] proved that SRPT is O(log(in{n/, P}))- copetitive for 2 identical parallel achines. No better approxiation, even off-line, is known for this proble. They also prove an (log n ) and an (log P) lower bound on the copetitive ratio of any randoized algorith for the proble. These lower bounds clearly extend to the nonclairvoyant case on 2 parallel achines. Other related work is concerned with the issues of using preeption without allowing ob igration Awerbuch et al. 2002] and with related obective functions like the stretch etric Acharya et al. 1999; Bender et al. 1998; Gehrke et al. 1999; Becchetti et al. 2004] OUR WORK. We prove that RMLF achieves an O(log n log n ) and an O(log n log P) randoized copetitive ratio against the oblivious adversary for any nuber of parallel identical achines. We recall that P is the ratio between the largest and the shortest processing tie of a ob. Observe that one of the two bounds ay be strictly better than the other depending on the values of P, n and. We do not contradict the (P) randoized lower bound claied in Kalyanasundara and Pruhs 1997] since it is obtained with a nuber of obs exponential in P. For a single achine, we show an O(log n) bound, which is tight. Our algorith requires the knowledge of the iniu processing tie of a ob that is released. This knowledge is essential for an MLF-like algorith to establish how long it should execute a ob in the lowest queue when it enters the syste. A wrong estiation of the iniu processing tie of a ob ay leave any sall obs waiting uch longer than their processing ties. This arguent can be extended to show that in absence of this inforation it is not possible to be copetitive at all. However, we do not need any knowledge of the axiu processing tie of a ob that is released. Moreover, we do not need to know the nuber n of obs released over the sequence. Our analysis of RMLF contains a set of new tools of analysis. We entioned in the introduction that RMLF tries to approxiately follow SRPT with the goal of avoiding the preeption of obs that are nearly finished. We prove that (Leas 6 and 7) every ob, apart fro a logarithic nuber, with constant probability, has a reaining processing tie that is still a large share of the initial processing

5 Nonclairvoyant Scheduling to Miniize Total Flow Tie 521 tie when it enters the queue in which it will be copleted. This allows, roughly speaking, to concentrate only on those obs that are called big. We bound at any tie t the difference between the nuber of big obs of alost equal size that are uncopleted in RMLF and in the optiu, by liiting their corresponding volue, that is, their overall reaining processing tie. We then prove (Lea 13) that at any tie t, the expected nuber of obs not copleted by RMLF is at ost O(log n) ties a constant fraction of the obs uncopleted by the adversary plus an additive ter O(log n( k), where k is related to the nuber of nonepty queues at tie t in RMLF. The O(log n log P) result for parallel achines then follows since there are at ost O(log P) queues in the syste. The proof of O(log n log n ) copetitiveness for parallel achines and of O(log n) copetitiveness for a single achine requires bounding the additional flow tie due to the contribution of the O(log n(k) ter over tie in a different way. This contribution is partly originated by the fact that on 2 achines RMLF ay keep achines idle for ore than the optiu. We will bound this contribution (Section 4.4) by at ost O(log n log n ) ties the optiu for 2 achines while for a single achine we will be able to liit this contribution to O(log n) ties the optiu. These two results, however, are obtained as consequences of a unified analysis of RMLF STRUCTURE OF THE ARTICLE. In Section 2, we forally define our scheduling proble. Section 3 presents the RMLF algorith. In Section 4, we provide the analysis of the behaviour of RMLF. In particular, in Section 4.1, we provide soe definitions and basic facts. In Section 4.2, we present soe tools and preliinary results. Section 4.3 is devoted to proving the O(log n log P) copetitive ratio, while Section 4.4 concerns the O(log n log(n/)) copetitive ratio of RMLF and the O(log n) upper bound for the single achine case. Open probles are discussed in Section Proble Definition We are given a set J of n obs and a set of identical achines. Each ob is assigned with a pair (r, p ) where r is the release tie of the ob and p is its processing tie. We order obs by increasing release ties and we assue the first ob is released at tie r 1 = 0. In the preeptive odel, a ob that is running can be preepted and continued later on any achine. The algorith decides which of the uncopleted obs should be executed at each tie. A ob cannot be processed before its release tie. A ob cannot be executed in parallel on ultiple achines, while only a single ob can be processed by a achine at any tie. For any given schedule, we define C to be the copletion tie of ob in that schedule. The flow tie of ob is F = C r. The total flow tie for the input instance J is F(J) = J F. The goal of the scheduling algorith is to iniize the total flow tie or equivalently the average flow tie. The arrival of a ob is unknown to the nonclairvoyant algorith until the ob is released. The processing tie of a ob is unknown to the nonclairvoyant algorith until the copletion tie of the ob. We assue the iniu processing tie of a ob that is released to be known in advance to the algorith, and the convention in J p = 1, while we do not assue any knowledge of the axiu processing tie of a ob.

6 522 L. BECCHETTI AND S. LEONARDI We copare the randoized nonclairvoyant algorith with an oblivious adversary that decides in advance, without knowledge of the rando choices of the algorith, the nuber n of obs of the sequence together with the release tie and the processing tie of each ob. The adversary is charged with the optial flow tie for the sequence. Denote by A the randoized on-line algorith, and denote by OPT the optial adversary. The generic (deterinistic) execution of A over an input sequence J is defined by a set of rando choices σ. We denote by Pr(σ ) the probability of sequence σ. A randoized on-line algorith A is c-copetitive against an oblivious adversary if for any input J, E σ Fσ A (J)] cfopt (J), where the expectation is taken over any possible outcoe of the rando choices σ of the algorith for input instance J. The instance J will be oitted in the following when clear fro the context. 3. The RMLF Algorith We present our randoized version of the Multilevel Feedback algorith for parallel achines, called RMLF in the following. A ob is said active or alive at tie t if released but not copleted. Denote by x the tie ob has been processed until tie t. Denote by y = p x the reaining processing tie of ob at tie t. β,i is a rando variable with distribution Prβ,i x] = 1 exp( γ x ln ). In 2 our algorith, we choose γ = 4/3. i Active obs are partitioned into a set of Priority Queues Q 0, Q 1,... Within each queue, the priority is deterined by the Earliest Release Tie first rule. For any two queues Q i, Q k, Q i is said lower than Q k if i < k. Job is assigned with a target processing tie T,i when it enters queue Q i. At any tie t, RMLF behaves as follows: (1) Schedule alive obs on the achines proceeding fro the lowest to the highest queue, within any queue in order of priority. (2) Job released at tie t enters queue Q 0 with T,0 = ax{1, 2 β,0 }. (3) For a ob in a queue Q i 1 at tie t, ifx A=T,i 1 < p, ob enters Q i with T,i = ax{2 i, 2 i+1 β,i }. (4) For a ob in a queue Q i at tie t,ifx A= p T,i, assign C = t and discard fro Q i. The Earliest Release Tie first rule assigns a static priority to obs, rather than following the dynaic priority defined by the tie of arrival in a queue. The reason of this choice is to fix, independently fro the specific execution of the algorith, the relative priority between any two obs located in the sae queue. This will turn out to be very useful in the probabilistic analysis of the algorith. We would also like to coent on the dependence of the probability distribution of target T,i on both i and. This is necessary since the nuber n of obs released in the sequence is not known in advance. If n was known, we could siply replace with n in the probability distribution of T,i. However, this would not iprove the worst case perforance of RMLF.

7 Nonclairvoyant Scheduling to Miniize Total Flow Tie 523 A very natural question at this point of the exposition is: why the exponential distribution rather than the unifor distribution? The following exaple, due to Kirk Pruhs (2000 personal counication), shows why a unifor distribution of target T,i 2 i, 2 i+1 ) in queue Q i does not work, even against the oblivious adversary. Consider a single achine and n obs released at tie 0 with processing tie 2 i (1 + 2 n ), with n = 2 i 1. With probability at least n every ob will enter Q i+1 with a reaining processing tie at least n and at ost 2 n. This guarantees that at tie t = n2 i (1 + 2 n ) cn, for soe constant c, the algorith will have ( n) uncopleted obs in queue Q i+1 with high probability, while the optial solution will have terinated all obs with the exception of a constant nuber, for a total flow tie of O(n 3 ). Starting a tie t, a ob of size 1 is released for each of n 3 tie units. The expected flow tie of the algorith will be (n 3 n), while the optiu is O(n 3 ). 4. Analysis 4.1. PRELIMINARIES. In the analysis, we use the following notation. We denote by δ the nuber of active obs at tie t. The volue V is the su of the reaining processing ties of the obs that are active at tie t. L denotes the total work done prior to tie t, that is the overall tie the achines have been processing obs until tie t. For any function f (δ, V and L), we denote by fσ A and f OPT the value of function f at tie t, respectively, for a generic outcoe σ of the rando choices of RMLF and for the optial solution OPT. By σ f= fσ A fopt, we denote their difference. In the analysis, without loss of generality, we restrict to input instances in which r 1 = 0 and r 1 i=1 p i, > 1, since otherwise every input instance can be split into a set of input instances of this for that we separately analyze. In our analysis, we classify active obs into classes according to their processing ties. A ob is of class i, i 0, if its processing tie is in 2 i, 2 i+1 ). For a generic function f (V, V, δ, δ,lor L) the notation f =k will denote the value of function f at tie t when restricted to obs of class exactly k; weuse f h, k to denote the value of f at tie t when restricted to obs of classes between h and k. Denote by P = ax J p in J p the ratio between the axiu and the iniu processing tie of a ob. The axiu class of a ob is log P +1. Moreover: FACT 1. created. During the execution of the algorith at ost log P queues are We start by observing that the total flow tie is the integral over tie of the nuber of active obs (see, e.g., Leonardi and Raz 1997]): FACT 2. F = J F = t δdt. Moreover, we have the following basic lower bound to the total flow tie: FACT 3. F = J F J p. The following fact holds for the algorith: FACT 4. A ob of class i ends either in queue Q i or in queue Q i+1.

8 524 L. BECCHETTI AND S. LEONARDI PROOF. Job, with processing tie p 2 i, 2 i+1 ), has target T,i 1 < 2 i in Q i 1 while it has target T,i+1 2 i+1 in queue Q i+1. Definition 1. Job of class i is unlucky in a given execution of the algorith if it has processing tie p 2 i + 2 i 2 and it ends in queue Q i+1. A ob that is not unlucky is said lucky. We denote by δσ l and by δu σ respectively the nuber of lucky and unlucky obs that are active at tie t for a specific execution σ of RMLF. Definition 2. At any tie t, a lucky ob in queue Q i is said big if it entered Q i at tie t t and y p, it is said sall otherwise. ln Observe that, according to this definition, a lucky ob of class i is always big while in queues Q i 1 or lower. We denote by δσ b the nuber of big (lucky) obs that are alive at tie t for a specific outcoe σ of the rando choices of RMLF. Finally, we define a set of rando variables that will be used in the analysis of the algorith. More in detail, for each ob X l has value 1 if ob is lucky, while X l = 0if is unlucky. Observe that this variable does not depend upon tie but only on the specific execution of the algorith. Moreover, for each ob and for each tie t, four binary variables are defined: X l, X, X l and X b. The value of X is1ifob is alive at tie t, 0 otherwise. X l is defined in ters of Xl and X, naely, X l = Xl X. Finally, X b = 1ifXl =1 and ob is big at tie t, X b = 0 otherwise. The following gives a basic property of the RMLF algorith: FACT 5. At any tie t and for any i, at ost obs, alive at tie t, have been executed in queue Q i but have not been prooted to Q i+1. PROOF. Let 1, 2,...be the obs in Q i at tie t ordered by decreasing priority. Let k be the largest index such that k has been processed while in queue Q i. Let t t be the last tie that k was processed in Q i. We show below that all obs h, h < k, are also executed at tie t. Since at ost obs are executed at the sae tie, this iplies k and the clai follows. Two cases are possible for a ob h, h < k: (i) if h was in queue Q i at tie t, it would be on execution since it precedes ob k in the earliest released tie first order; (ii) if at tie t ob h was in a queue Q l, l < i, then it would also be on execution since a ob is executed in a higher queue Q i. We first present the probabilistic stateents we use in the analysis. Then we give the analysis of the copetitive ratio for RMLF PROBABILISTIC ANALYSIS. We start by showing that a ob of class i with processing tie close to 2 i is likely to be copleted in queue i, therefore overcoing the counterexaple for the unifor distribution described in Section 3. As illustrated in Figure 1(a), we prove that the probability that target T,i falls in the interval 2 i, 2 i + 2 i 2 ) is at ost 1/. This is forally proved in the following lea: LEMMA 6. The expected nuber of unlucky obs along the execution of the algorith is H n.

9 Nonclairvoyant Scheduling to Miniize Total Flow Tie 525 FIG. 1. Function PrT,i x] is the probability distribution of T,i. a) The probability that a ob of class i is unlucky is at ost 1/. b)p 1 and P 2 P 1 are the probabilities that a ob is respectively big or sall. They differ for at ost a constant factor. PROOF. A ob of class i with processing tie p 2 i + 2 i 2 is unlucky if it enters queue Q i+1. This happens with probability at ost 1/ as shown by: Pr X l = 0 ] = PrT,i < p ] Pr T,i < 2 i + 2 i 2] = Pr β,i > 32 i 2] = exp( ln ) = 1, since γ = 4/3. The expected nuber of unlucky obs is then bounded by n =1 1 = H n. In the reainder of this section, we present a second fundaental property of our algorith. As pointed out in the introduction we would like our algorith to approxiately follow the SRPT heuristic, iplying that ost of the obs are big, that is, they enter the queue of copletion with a reaining processing tie that is at least a logarithic fraction of the original processing tie. We prove that, for any ob of class i and for any tie t, the probability that ob is alive and big at tie t is at least a constant fraction of the probability that ob is lucky and alive at the sae tie. This property iplies that, at any tie t, a constant fraction of lucky obs that are alive at t are also big. This will be used to liit the difference between the expected nuber of obs in the schedule of RMLF and in the optial schedule. Observe that to ensure the property stated above we need that when dividing the interval 2 i, 2 i+1 ) into log n subintervals of size 2i, for every l, the probability log n that the target falls in the first l intervals is at least a constant fraction of the probability that it falls in the first l + 1 intervals. It is straightforward to conclude that an exponential decreasing probability distribution is necessary to this purpose. The proof of this property is straightforward if, at tie t, a ob of class i is in a queue Q k, k < i, while it needs soe work if ob is in one of the queues of possible copletion, naely Q i or Q i+1. Assue for instance ob is in queue Q i+1, the fact that ob is big when it enters queue Q i+1 depends on the value of target T,i. In Figure 1b) it is pictorially shown that for a ob with p 2 i + 2 i 2 (a lucky ob by definition), if T,i < p (the ob entered queue Q i+1 ), then with constant probability T,i is also saller than p (1 1/ ln ), therefore the ob is big. However, we restrict our attention to those executions in which ob is alive a tie t. This ight restrict the set of favourable executions, hence the arguent above cannot be applied directly. The foral proof for our case is given in the following lea:

10 526 L. BECCHETTI AND S. LEONARDI LEMMA 7. There exists a constant α such that, for every ob α and tie t it holds PrX b = 1] exp( 4γ ) PrXl = 1]. PROOF. Since X b = 1 iplies Xl = 1, we have: PrX b = 1] = Pr X b = 1 Xl = 1] = Pr X b = 1 Xl = 1] Pr X l = 1]. The proble is to bound PrX b = 1 Xl = 1] for a ob of class i. We need to characterize the set of specific executions σ of the algorith for which X l,σ = 1. Denote by X the generic assignent of all rando variables with the exception of T,i 1 (T,i ). We denote by i ( i+1 ) the set of all executions σ such that: (i) all rando variables with the exception of T,i 1 (T,i ) are assigned according to X ; (ii) at tie t ob is in queue Q i (Q i+1 ); (iii) X l,σ = 1. Every execution in i ( i+1 ) is denoted in the following by σ =< X, T,i 1 > (σ =< X, T,i >). Hence, the Lea is proved if PrX b = 1 σ i ] exp( 4γ ) and PrX b = 1 σ i+1 ] exp( 4γ ), since a ob of class i is always big in queues Q i 1 or lower. The following, technical Lea proves that, for any execution σ with an assignent T,i 1 = T (T,i = T ) such that is lucky and alive at tie t, is also lucky and alive at t, for any other execution differing fro σ only in the assignent T,i 1 = T < T (T,i = T < T ). We will use this arguent to coplete the proof. LEMMA 8. Consider a ob of class i and a specific execution σ =< X, T,i 1 = T > i (σ =<X,T,i =T > i+1 ). Then, for every T 2 i 1, T )(T 2 i, T )) it holds σ =< X, T,i 1 = T > i (σ =< X,T,i =T > i+1 ). PROOF. We prove the clai for a ob of class i and σ i. The proof for σ i+1 proceeds exactly the sae, with straightforward changes, by replacing i with i + 1 and is therefore oitted. Intuitively, we prove that if is alive, lucky and in queue Q i at tie t in accordance with execution σ, the sae happens with execution σ, when the target T,i 1 is reduced fro T to T < T. Let t denote the tie at which is processed for T units in queue Q i 1 according to execution σ. The schedules for σ and σ are identical up to tie t, hence we shall always assue t > t. We first prove (Fact a) that, for any ˆt (t, t] such that is alive at ˆt in execution σ,wehavey h,σ (ˆt) y h,σ (ˆt) for every ob h. Observe that at least one such tie instant ˆt when ob is alive exists since by assuption, T < T and σ =< X, T,i 1 = T > i. The clai above intuitively eans that, by reducing the target of ob in queue Q i 1 to T, during interval (t, t], every ob other than receives at least the sae processing tie in the new execution. We consider obs in Q l, l < i 1, and in queues Q i 1, Q i and Q i+1 separately. The processing of obs in queues lower than Q i 1 is by no way affected by the processing in queues Q i 1 or higher, hence the clai is true for any ˆt (t, t] and for any ob h in a queue lower than Q i 1 at tie ˆt.

11 Nonclairvoyant Scheduling to Miniize Total Flow Tie 527 As to obs in queues Q i 1, Q i and Q i+1, the proof is by contradiction. Assue there exists at least one ob h in Q i 1, Q i or Q i+1, such that y h,σ (ˆt) > y h,σ (ˆt), for soe ˆt (t, t]. Let t t, ˆt) be the last tie when y h,σ y h,σ for all h. There has to be a ob h such that y h,σ = y h,σ while y h,σ (t + ɛ) > y h,σ (t + ɛ) for every ɛ sufficiently sall. The contradiction follows by proving that the set of obs that have priority over h in σ at tie t contains the set of obs that have priority over h in σ at tie t. Recall that a ob has priority over a ob if is in a lower queue than or if they are in the sae queue but has been released earlier. At this point of the proof, we use the Earliest Release Tie First priority rule in order to have a fixed priority between obs independently of the tie they are prooted into a queue. In fact, if obs priorities were deterined by their arrival ties in the queue, the relative priority between two obs prooted to a queue Q i could change if the target of ob in queue Q i 1 were reduced. Assue ob prooted to queue i after ob in execution σ. By stopping the execution of ob at tie t in execution σ we ay anticipate the processing of soe ob that could therefore be prooted to queue i before ob. In deterining the set of obs that have priority over h, we consider and all obs other than or h separately. According to execution σ, at tie t, ob is in queue Q i, while at the sae tie it is either in queue Q i 1 or Q i in execution σ.onthe other side, since y h,σ = y h,σ, at this in the sae queue in the two executions. Therefore, if has priority over h in σ, so it does in σ. For all obs other than h and, we observe that in execution σ and at tie t they have a reaining processing tie that is saller or equal than in σ, by definition of t. Therefore, in execution σ and at tie t, they are in the sae queue or in a lower queue than in execution σ, for which, if at t and in execution σ they have priority over h, the sae happens with execution σ. This proves that the set of obs that have priority over h in σ at tie t contains the set of obs that have priority over h in σ at tie t. But this contradicts the assuption above and hence (Fact a) y h,σ (ˆt) y h,σ (ˆt) for every ob h and for any ˆt (t, t] such that is alive at ˆt. As a corollary of (Fact a), we have (Fact b) for any ˆt (t, t] such that is alive at ˆt, wehave δσ A (ˆt) δ σ A(ˆt). We can now prove that X l,σ = 1 iplies Xl,σ = 1. Assue this is not the case and consider the last ˆt (t, t] such that is alive at tie ˆt. (Fact b) iplies that the overall work done by the achines during interval (t, ˆt] in execution σ is not ore than in σ. This consideration and (Fact a) together iply y,σ (ˆt) y,σ (ˆt), which contradicts the assuption. As a consequence, is alive at tie t in execution σ and it is obviously lucky, since T,i 1 has been reduced, while T,i was left unchanged, that is X l,σ = 1 (for the case σ i+1, observe that X l,σ = 1 iplies p > 2 i + 2 i 2 ). Finally, since y,σ (ˆt) y,σ (ˆt), is in queue Q i at tie t, according to execution σ. This, and the fact that X l,σ = 1, together iply σ i. COROLLARY 9. Consider a ob of class i and a specific assignent X of all rando variables with the exception of T,i 1 (T,i ). Consider the set i ( i+1 ) of all sequences with assignent X where ob is in queue Q i (Q i+1 ) at tie t, such that X l,σ = 1. Then, there exists a value x 2i 1, 2 i )(x 2 i, 2 i+1 )) such that for every T,i 1 2 i 1, x) (T,i 2 i, x)) the corresponding sequence

12 528 L. BECCHETTI AND S. LEONARDI is in i (σ i+1 ), while it is not in i (σ i+1 ) for every T,i 1 x, 2 i )(T,i x,2 i+1 )). Denoted by x the sallest value for variable T,i 1 (T,i ) such that for the corresponding execution σ we have X l,σ = 0, the proof of Corollary 9 follows by a siple contradiction arguent and is therefore oitted. We can now prove Lea 7. Consider a ob e 5 of class i and consider all sequences σ for which X l,σ = 1. For all sequences in which is in queue Q k, k i 1, is big, since T,i 2 < 2 i 1 and then, if t denotes the tie was prooted to queue Q k, y > p 2 i 1 p /2 p / ln, for 8. As a consequence, the only possibility for a lucky ob of class i to be sall is to enter queue Q i or queue Q i+1 at tie t t, with y < p ln. First, consider any subset i ( i+1 ) of the set of sequences in which ob is alive, lucky and in queue Q i (Q i+1 ) at tie t, such that all sequences in i ( i+1 ) only differ for the value of T,i 1 (T,i ). If we prove a bound on the probability that ob is big when conditioned to sequences σ i (σ i+1 ), then the clai is proved for in Q i ( in Q i+1 ) at tie t. Let x denote the value derived fro Corollary 9. We proceed as follows: Pr T,i 1 p p ] ( ln σ i = Pr T,i 1 p 1 1 ) T,i 1 < x] ln ( Pr T,i 1 p 1 1 ) ] T,i 1 < 2 i ln Pr ( )] T,i 1 p 1 1 ln Pr T,i 1 < 2 i] ( Pr β,i 1 2 i p 1 1 )] ln ( ) ) exp ( γ 2i p 1 1 ln ln 2 i 1 ( = exp γ 2i p ln γ p ) 2 i 1 2 ( i 1 exp γ p ) exp( 4γ ), 2 i 1 where the second inequality follows since x < 2 i, the seventh inequality stes fro p 2 i, while the last inequality fro p < 2 i+1. We then prove the clai for sequences belonging to i+1. Since ob is lucky, p > 2 i + 2 i 2. We show that ob is big when it enters Q i+1 with (conditioned) probability at least e 2γ. Corollary 9 ensures that there exists a value x such that σ =< X, T,i > i+1 for every T,i 2 i, x). If we prove a bound on the probability that ob is big when conditioned to sequences σ i+1, then the clai is proved. We proceed

13 Nonclairvoyant Scheduling to Miniize Total Flow Tie 529 as follows: Pr T,i p p ln σ i+1 ] ( = Pr T,i p 1 1 ) T,i < x] ln ( Pr T,i p 1 1 ) ] T,i < p ln = Pr ( )] T,i p 1 1 ln PrT,i < p ] = Pr ( )] β,i 2 i+1 p 1 1 ln Prβ,i > 2 i+1 p ] e γ 2 i+1 p (1 ln 1 ) 2 i ln e γ 2i+1 p 2 i ln 2 i+1 p = e γ 2 i ln γ p 2 i exp e γ 2i+1 p 2 i ln ( γ p 2 i ) exp( 2γ ). Here, the second inequality follows since x < p (because σ i+1 ), while the fourth inequality holds since T,i, = ax{2 i, 2 i+1 β,i }, since p 2 i + 2 i 2 and p (1 1/ ln ) 2 i when e 5. The last inequality follows fro p 2 i+1. Taking α = e 5 the clai of Lea 7 is proved THE O(LOG n LOG P)-COMPETITIVE RATIO. In this section, we prove that RMLF is O(log n log P) copetitive. Intuitively, the log n factor is the price of nonclairvoyance and it is a direct consequence of Definition 2 (see Eq. (3) in the proof of Lea 13). The log P factor, instead, coes fro the fact that during the execution, at ost log P queues are created and each ight have a nuber of obs that have been initiated and subsequently preepted by the algorith. As pointed out further in Section 4.4, this is a consequence of the fact that the on line algorith in general does not optially balance the load aong the available achines. Recall that, for any possible execution σ, δσ A = δu σ + δl σ, for which the total flow tie of RMLF can be expressed as: ] ] E σ Fσ A ] = E σ δσ A dt = E σ (δσ u + δl σ )dt ] ] = E σ δσ u dt + E σ δσ l dt = E σ δ u σ ]dt + E σ δσ l ]dt. (1)

14 530 L. BECCHETTI AND S. LEONARDI In the reainder of this section, we need to bound the two contributions to Eq. (1). The following lea, based on the clai of Lea 6, bounds the first contribution to the total flow tie. LEMMA 10. E σ δ u σ ] dt = O(log n) p. PROOF. First, observe that E σ δσ u ] = O(log n). This follows fro: E σ δ u σ ] = n Pr X l = 0 X = 1 ] =1 n Pr X l = 0 ] = O(log n). =1 Since the algorith never keeps achines idle, no ob is in the syste after tie p. This, together with the inequality above iplies: E σ δ u σ ] dt 0 t E σ δ u σ ] dt = p 0 t O(log n)dt p = O(log n) p. The next lea uses the clai of Lea 7 to relate the expected nuber of lucky obs alive at tie t to the expected nuber of big obs alive at tie t. LEMMA 11. There exists a constant α such that, at any tie t, the nuber of lucky obs in the syste satisfies the following relation: E σ δ l σ ] α + exp(4γ )E σ δ b σ ]. PROOF. Set α = e 5.We have the following inequalities: E σ δ l σ ] = n =1 α + Pr X l = 1] n =α+1 α + exp(4γ ) Pr X l = 1] n =α+1 α + exp(4γ )E σ δ b σ ], Pr Xb l = 1] where the third inequality follows fro the clai of Lea 7. The next lea bounds the expected tie spent by the algorith while all achines are busy. LEMMA 12. Prδ A ] dt 1 J p.

15 Nonclairvoyant Scheduling to Miniize Total Flow Tie 531 PROOF. The clai is derived fro the following relations: Prδ A ] dt = Prσ ] dt σ :δσ A = Prσ ] σ 1 p Prσ ] J = 1 p, J t:δ A σ dt where the third inequality follows since for every σ t:δσ A dt is bounded by J p. 1 The expected cost of RMLF of expression (1) is then rewritten as follows: ] ] E σ F A σ = Eσ δσ A dt = E σ δ u σ ] dt + E σ δ l σ ] dt O(log n) p + 0 t (α + exp(4γ )E σ δ b σ ] )dt p = O(log n) p + α p + exp(4γ ) E σ δ b σ ] dt ] = O(log n) p + exp(4γ )E σ δ b t:δσ A< σ dt + exp(4γ )E σ δ b t:δσ A σ dt = O(log n) p + exp(4γ ) E σ δ b σ δ A ] Prδ A ]dt. (2) The third inequality follows by Lea 10, by Lea 11, and by recalling that no ob is in the syste after tie p, hence δ l = 0, for any t > p. The fifth inequality follows by the linearity of expectation and by partitioning the tie axis, for each possible execution σ, into the instants for which δσ A < and into those for which δσ A. Finally, the last inequality follows by observing that t:δσ A< δb σ dt t:δσ A< δ σ Adt J p, since the overall work done by the achines in any possible execution of the algorith is bounded by the su of the processing ties of the n obs, and by expanding E σ t:δσ A δb σ dt]. We are left to bound the last ter of Eq. (2) and to this purpose we present the critical part of the proof, in which we bound the nuber of big obs at tie t in the generic execution of RMLF deterined by a set of rando choices of the algorith. LEMMA 13. For any outcoe σ of the rando choices of the algorith, for any tie t : δσ A, δb σ, k 1, k 2 2lnn(2( 1)(k 2 k 1 + 1) + 3δ OPT ) + (k 2 k 1 + 2). σ ]

16 532 L. BECCHETTI AND S. LEONARDI PROOF. In the following, we oit σ when clear fro the context. For δ k b 1, k 2 we write the following relations: δ k b 1, k 2 (k 2 k 1 + 2) + ln n V A =i. (3) 2 i The bound follows since every big ob of class i has reaining processing tie at least 2 i / ln n. We should additionally consider those big obs of class i, i = k 1,..,k 2, with a processing tie saller than 2 i / ln n since they have been processed after entering the queue they are in. For a ob of class i this ay happen in queues i 1, i or i + 1. However, for every queue, there are at ost obs that have been processed after entering the queue (Fact 5), for which the total nuber of such obs is bounded by (k 2 k 1 + 2). We continue with: A V =i = 2 i OPT V =i + V =i 2 i V =i 2δ k OPT 1, k i V i V i 1 = 2δ k OPT 1, k i = 2δ k OPT 1, k 2 + V k k V i + V k k 2 2 i+1 2 k 1 2δ OPT k 1, k 2 + δ OPT k 2 k V i 2 i+1 V i 2δ k OPT 2 + 2, (4) 2 i+1 where the second inequality follows since a ob of class i has size less than 2 i+1, while the fifth inequality follows since V k 1 1 V k OPT 1 1 δ OPT 2 k 1 2 k 1 k 1 1. We are left to study the ter k 2 V i. For any t 2 i+1 1 t 2 t, for a generic function f, denote by f t 1,t 2 ] the value of function f at tie t when restricted to obs released between t 1 and t 2, for exaple, L t 1,t 2 ] i is the work done by tie t on obs of class at ost i released between tie t 1 and t 2. Let t 0 be the last tie prior to tie t in the execution deterined by the set of rando choices σ, such that δσ A(t 0) <. Denote by t i < t the axiu between t 0 and the last tie prior to tie t in which a ob was processed in queue Q i+1 or higher in this specific execution of RMLF. Observe that, for i = k 1,...,k 2,t i+1,t) t i,t). At tie t i+1 either the algorith was processing a ob in queue Q i+2 or higher, or t i+1 = t 0, for which at tie t i+1 at ost 1 obs were in queues Q 0,...,Q i+1.

17 Nonclairvoyant Scheduling to Miniize Total Flow Tie 533 At tie t we have: V i V A0,t i+1] i + V (t i+1,t] i V A0,t i+1] i + V A0,t i+1] V A0,t i+1] (t i+1 ) + L A(t i+1,t] L OPT(t i+1,t] ( 1)2 i+2 + L A(t i+1,t] L OPT(t i+1,t]. The first inequality follows since V i is bounded by the su of a first contribution equal to the volue of the obs released until tie t i+1 in RMLF s schedule and a contribution equal to the volue difference of obs released after tie t i+1. The second inequality states that the volue difference is bounded by the tie spent in the interval (t i+1, t] by RMLF processing obs of class bigger than i inus the tie spent by the optial solution processing obs of class bigger than i released in the interval (t i+1, t]. The third inequality follows fro V A0,t i+1] i + V A0,t i+1] V A0,t i+1] (t i+1 ) ( 1)2 i+2. This relation holds since at ost 1 obs are in queues Q 0,..,Q i+1 at tie t i+1. Assue x 1 such obs are of class i and x 2 such obs are of class > i. The reaining volue of the x 1 obs of class i at tie t i+1 is at ost x 1 2 i+1, while the x 2 obs of class > i are processed in t i+1, t) for at ost x 2 2 i+2 tie units. Fro x 1 + x 2 1, the inequality follows. The fact that at ost 1 obs are in the syste in queues Q 0,...,Q i+1 at tie t i+1 is crucial in proving the tightness result for = 1. In fact for = 1, this contribution to the flow tie will disappear. In the following, we adopt the convention t k1 1 = t. Fro the inequality above we write: V i 2 i+1 ( 1)2i+2 (t + L i+1,t] 2( 1)(k 2 i+1 2 k 1 + 1) + L (t i+1,t] (5) 2 i+1 We then concentrate on the ter k 2 A(t L i+1,t] L OPT(t i+1,t] = 2 i+1 =k 1 1 i= L (ti+1,t] 2 i+1 for which we have: i =k 1 1 L A(t +1,t ] L OPT(t +1,t ] 2 i+1, L A(t +1,t ] L OPT(t +1,t ] 2 i+1 where the second equality follows by partitioning the work done on the obs released in the interval (t i+1, t] into the work done on the obs released in the intervals (t +1, t ], = k 1 1,..,i.

18 534 L. BECCHETTI AND S. LEONARDI Let i( ),..,k 2 ] be the index that axiizes L A(t +1,t ] L OPT(t +1,t ].We then have: L 2 =k 1 1 (t i+1,t] 2 i+1 =k 1 1 L A(t +1,t ] ( ) δ OPT(t +1,t ] ( ) =k 1 1 i= L A(t +1,t ] ( ) L OPT(t +1,t ] 2 ( ) L OPT(t +1,t ] ( ) 2 i+1 2δ OPT(t +1,t] 2δ k OPT 1. (6) To prove the third inequality, observe that every ob of class bigger than i( ) released in the tie interval (t +1, t ] is processed by RMLF in the interval (t +1, t] for at ost 2 +2 tie units. Order the obs of this specific set by increasing x. Now, observe that each of these obs has reaining processing tie at least 2 i( ) at the beginning of interval (t +1, t] and we give to the optiu the further advantage that it finishes every such ob when processed for an aount x To axiize the nuber of finished obs, the optiu places the work L OPT(t +1,t ] on the obs with saller x. The optiu is then left at tie t ( ) with a nuber of obs δ OPT(t +1,t ] ( ) k 1 L A(t +1,t ] ( ) L OPT(t +1,t ] ( ) 2 +1 for which the third inequality holds. Altogether, fro (3), (4), (5) and (6) we obtain: δ k b 1, k 2 (k 2 k 1 + 2) + 2lnn(δ k OPT 2 + 2( 1)(k 2 k 1 + 1) + 2δ k OPT 1 ) (k 2 k 1 + 2) + 2lnn ( 2( 1)(k 2 k 1 + 1) + 3δ OPT ), for which the clai of the Lea is proved. We then plug the clai of Lea 13 into the expression (2) of the total flow tie of RMLF to achieve our first result: E σ Fσ A ] = O(log n) p + exp(4γ ) E σ δ b σ δ A ] Prδ A ]dt O(log n) p + exp(4γ ) (2 ln n(2( 1)(log P + 2) + 3δ OPT )) Prδ A ]dt,

19 Nonclairvoyant Scheduling to Miniize Total Flow Tie 535 p = O(log n) +O(log n log P) Prδ A ]dt + O(log n) δ OPT dt = O(log n) p + O(log n log P) p + O(log n)f OPT J = O(log n log P)F OPT, where in the second inequality we apply Lea 13 and Fact 1, while in the fourth we apply the clai of Lea 12. We therefore state our first result: THEOREM 14. RMLF is an O(log n log P) copetitive nonclairvoyant randoized algorith for iniizing the total flow tie on parallel achines THE O(LOG n)competitive RATIO FOR = 1, O( LOG n LOG n )COMPET- ITIVE RATIO FOR 2. In this section, we strengthen the analysis to obtain an O(log n) tight bound for a single achine and an O(log n log n ) bound for 2 parallel achines. Again, the log n factor is the price of nonclairvoyance, as already coented at the beginning of Section 4.3. The log n ter, instead, coes fro the fact that the optiu ight work ore than the algorith, due to a better distribution of the overall load aong the achines. In the worst case, the unbalance of MLF with respect to the optiu brings to a ultiplicative factor O(log n log n )inthe copetitive ratio. As expected, this ter vanishes in the single achine case, where instead of a copetitive ratio O(log 2 n), we have a tighter O(log n). In this respect, this contribution is siilar to the one eerging in the analysis of SRPT Leonardi and Raz 1997]. Let k be the lowest class such that less than obs of class k are released. Let T (σ ) ={t:δσ A }be the set of tie instants where all achines are busy in the execution of RMLF defined by σ. Let T (σ ) T (σ ), = 0,.., k, be the set of tie instants where at least one achine is busy with obs in queue Q and no achine is busy with obs in queues higher than Q in the execution of RMLF defined by σ. Let T k+1(σ ) T (σ ) be the set of tie instants when at least one achine is processing a ob in queue Q k+1 or higher and all achines are busy. Observe that for each σ, {T 0 (σ ), T 1 (σ ),...,T k+1(σ)} defines a partition of T (σ ). For the sake of siplicity, in the sequel, with a slight abuse of notation, we use T (σ ) to denote both the above defined set and its size. For the total flow tie of RMLF, the last ter of Eq. (2) can be rewritten using the clai of Lea 13 as follows: E σ δ b σ δ A ] Prδ A ]dt = Pr(σ ) δσ b dt σ t T (σ ) Pr(σ ) + δ σ t T (σ )(2 b σ,, k 1 )dt + Pr(σ ) δσ b dt σ t T k+1(σ ) =0

20 536 L. BECCHETTI AND S. LEONARDI σ + σ = σ + σ + σ Pr(σ ) =0 Pr(σ )2 Pr(σ ) Pr(σ ) Pr(σ ) t T (σ ) t T k+1(σ ) =0 t T (σ ) =0 =0 t T (σ ) (2 ln n(2( 1)( k ) + 3δ OPT ) + ( k + 3))dt dt 4lnn( 1)( k )dt 6lnnδ OPT dt t T (σ ) ( k + 3)dt + 2 = O(log n) Pr(σ ) 1)( k )T (σ ) + σ =0( Pr(σ ) ( k )T (σ ) σ =0 +3 Pr(σ ) T (σ ) + O(log n) Pr(σ ) δ OPT dt + 2 p σ =0 σ t T (σ ) = O(log n) Pr(σ ) 1)( k )T (σ ) + σ =0( σ +O(log n)f OPT. p Pr(σ ) =0 ( k )T (σ ) The second inequality follows by partitioning for every σ, the set {t 0 : δ A σ } into the subsets {T 0(σ ), T 1 (σ ),...,T k+1(σ)}, and by observing that at any tie t T (σ ) there are at ost obs of class less than and 1 obs of class bigger or equal than k. The third inequality follows by the clai of Lea 13 and by observing that for any tie t T k+1(σ ) at ost 2 obs are in the syste since (i) a achine is processing a ob in a queue k + 1 or higher and hence at ost 1 obs of class less than k are in the syste; (ii) at ost 1 obs of class k have been released. The other inequalities follow by rearranging the ters and observing that (i) σ Pr(σ) t T (σ) δopt dt F OPT, (ii) p F OPT, (iii) by applying the clai of Lea 12. As a consequence of the inequalities above and fro Eq. (2), we can write: E σ F A σ ] = O(log n)f OPT + exp(4γ )( 1)O(log n) σ + exp(4γ ) σ Pr(σ ) =0 Pr(σ ) =0 ( k )T (σ ) ( k )T (σ ). (7) We are left to bound, for any σ, the ter F σ (n) = k =0 ( k )T (σ ). We show this in the following Lea: