Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions

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1 OR Spectrm 06 38: DOI 0.007/s REGULAR ARTICLE Worst-case analysis of the LPT algorithm for single processor schedling with time restrictions Oliver ran Fan Chng Ron Graham Received: Janary 05 / Accepted: Janary 06 / Pblished online: 9 Janary 06 Springer-Verlag erlin Heidelberg 06 Abstract We consider the following schedling problem. We are given a set S of jobs which are to be schedled seqentially on a single processor. Each job has an associated processing time which is reqired for its processing. Given a particlar permtation of the jobs in S, the jobs are processed in that order with each job started as soon as possible, sbject only to the following constraint: For a fixed integer, no nit time interval [x, x + is allowed to intersect more than jobs for any real x. There are several real world sitations for which this restriction is natral. For example, sppose in addition to the jobs being exected seqentially on a single main processor, each job also reqires the se of one of identical sbprocessors dring its exection. Each time a job is completed, the sbprocessor it was sing reqires one nit of time to reset itself. In this way, it is never possible for more than jobs to be worked on dring any nit interval. In ran et al. J Sched 7: , 04a it is shown that this problem is NP-hard when the vale is variable and a classical worst-case analysis of List Schedling for this sitation has been carried ot. We prove a tighter bond for List Schedling for 3 and we analyze the worst-case behavior of the makespan τ LPT S of LPT longest processing time first schedles where we rearrange the set S of jobs into non-increasing order in relation to the makespan Oliver ran o.bran@mwelt-camps.de Fan Chng fan@csd.ed Ron Graham graham@csd.ed Trier University of Applied Sciences, Environmental Camps irkenfeld, 5576 irkenfeld, Germany University of California, San Diego, USA

2 53 O. ran et al. τ o S of optimal schedles. We show that LPT ordered jobs can be processed within a factor of / of the optimm pls and that this factor is best possible. Keywords Schedling Worst-case analysis Time restrictions LPT longest processing time first algorithm Problem description We are initially given a set S ={S, S,...,S n } of jobs. Each job S i has associated with it a length s i. The jobs are all processed seqentially on a single processor. Only one job can be worked on at any point in time. A job S i will be processed dring a semi-open time interval [α, α + s i for some α 0. A special type of job S i, called a zero-job, has length s i = 0. y convention, each zero-job is processed at some particlar point in time. Frthermore, only one job can be worked on at any point in time except in the case of zero-jobs, where it is allowed for several zero-jobs to be processed at the same point in time, provided the constraint below is not violated. Given some permtation π of S, sayπs = T = T, T,...,T n with corresponding job lengths t, t,...,t n, the jobs are placed seqentially on the real line as follows. The initial job T in the list T begins at time 0 and finishes at time t. In general, T i+ begins as soon as T i is completed, provided the following constraint is always observed: For every real x 0, the nit interval [x, x + can intersect at most jobs. Constraint reflects the condition that each job needs one of additional resorces for being processed and that a resorce has to be renewed after the processing of a job has been finished. The preceding procedre reslts in a niqe placement or schedle of the jobs on the real line. We define the finishing time τt to be the time at which the last job T n is finished. A natral goal might be for a given job set S, to find those permtations T = πs which minimize the finishing time τt. In ran et al. 04a it has been shown that the problem is NP-hard in general the athors polynomially redce the NP-hard problem PARTITION Garey and Johnson 979 to a special case of or schedling problem. Let s denote by τ w S the largest possible finishing time for any permtation of S, and let τ o S denote the optimal i.e., the shortest possible finishing time for any permtation of S. The following bonds from ran et al. 04a are similar inspirit to some of the bonds in the very early worst-case analysis literatre of schedling algorithms Graham 966, 969: For = and any set S, τ w S 4 3 τ os. For 3 and any set S, τ w S for 3 are best possible. In fact, the constant 3 can be replaced by τ o S 3. The factors 4 3 for = and and that bond is tight.

3 Worst-case analysis of the LPT algorithm for single processor Theorem For 3 and any set S, τ w S This bond is best possible. τ o S +. We give the proof in Sect. 4. Note that the worst-case analysis of the finishing time of any permtation of S exactly corresponds to the classical worst-case analysis for the List Schedling algorithm Graham 966, 969 where the jobs can be placed in any order on the machine. We need this reslt for the analysis of the LPT longest processing time first algorithm. For the LPT algorithm, we rearrange the job set S of jobs into non-increasing order to form the permtation L = L, L,...,L n where l i denotes the length of L i and l i l i+ for i n. We write τ LPT S for the makespan of an LPT schedle. In what follows we assme that all the lengths l i of the jobs L i satisfy l i for all i, since if any L i had l i = + ɛ>, then by decreasing l i to, we decrease both τ LPT S and τ o S by ɛ, thereby increasing the pper bond. In Sects. and 3, we show the following theorems: Theorem For and any set S, τ LPT S τ o S +. This bond is best possible. Theorem 3 For = and any set S, { + l l n l n if n is odd, τ LPT S τ o S l + l l n l n if n is even. This bond is best possible. Analysis of the LPT algorithm for In this section, we are going to proof the general bond for the worst-case behavior of the makespan of LPT schedles in relation to the makespan of optimal schedles for arbitrary as stated in Theorem. First, we establish an pper bond for LPT schedles. Lemma If the jobs are ordered according to the LPT rle and have only processing times l i, i =,...,n, then we have: τ LPT S l i + n

4 534 O. ran et al. Proof y and becase of l i+ l i,wehave τ LPT S + l + + l + + +l n τ LPT S + l + + l + + +l n... τ LPT S + l + + l l n Conseqently, this implies τ LPT S l i + n which in trn implies τ LPT S l i + n. Note: This holds for n = m, m. The cases n = m+,...,n = m+ can be shown in a similar way. Now we are ready to prove Theorem. We separate the permtation L ={L,..., L n } generated by the LPT rle into two sets V ={V,...,V v } and U ={U,...,U }, where v i = l i denotes the length of V i and i = l i+v < denotes the length of U i, n = + v. In what follows we se the abbreviations β := v v i for the sm of the processing times of the job set V of large jobs, and γ := i for the sm of the processing times of the job set U of small jobs. Case : τ o S + + γ The jobs in the large job set V can be schedled withot any gap, so we have τ LPT V = β. Set U contains the small jobs with processing times <. We know from Lemma that in this case τ LPT U γ +. So we have for the makespan of the LPT ordered job set: τ LPT S τ LPT V + τ LPT U β + γ + β + γ + γ + γ τ o S + τ os + + γ + γ γ τ o S + τ o S + γ γ +

5 Worst-case analysis of the LPT algorithm for single processor τ o S τ o S Case : >τ o S + + γ Again, we have for the makespan of the LPT ordered job set: τ LPT S τ LPT V + τ LPT U β + γ + β + β + β + + γ + γ γ For the axiliary job set T with job lengths t i =,ifs i s i >,wehaveτt = β + γ = β + and with 8to τ LPT S τt + γ τ o S <τ o S τ LPT S τt γ +, and t i = s i,if. Together, we come so to + γ + τ os + + γ + γ 3 τ o S + τ o S + We give the following example that shows that the bond is tight. Consider the set S consisting of t jobs of length and t + zero-jobs for some integer t. We denote a job of length by and a zero-job by 0. The optimal permtation 0 [ 0 ] t 0 has finishing time τ o S = t. On the other hand, the LPT order t 0 t + has a finishing time τ LPT S = t + t + / = /t +. Ths, τ LPT S = /τ o S +.

6 536 O. ran et al. 3 Refined analysis of the LPT algorithm for = Or goal in this section will be to prove Theorem 3. A first version of this reslt has been presented by the athors in ran et al. 04b. As sal see ran et al. 04a, we let sl i denote the starting time of L i when L is schedled, and we let f L i denote the corresponding finishing time. We start with the following observation. Lemma If the jobs are ordered according to the LPT rle and have only processing times l i, i =,...,n, then we have: Proof It follows from that sl i = f L i +, i + sl i f L i +, i = +,...,n. 3 In fact, we claim that 3 holds with eqality for all i if the jobs are given in LPT order. This is certainly tre for i = + since l,...,l. The only reason that we cold have sl i > + f L i for some vale i + is if f L i > f L i +. 4 t we know by indction that f L i = sl i + l i = f L i l i. Hence, by 4, f L i l i > f L i +, i.e., f L i + + l i > f L i = sl i + l i. However, sl i f L i +. Therefore, sl i + l i > sl i + l i,orl i > l i, which is a contradiction. Ths, we have sl i = + f L i for all i. We know from ran et al. 04a that the optimal schedle for S satisfies τ o S s + s n + s i + n l n + l n + l i + n 5 note that l n and l n are the lengths of the two smallest jobs in the LPT order. Now we consider two cases: i n = m +. From Lemma we see that m+ τ LPT S = l + + l l m + + l m+ = l i + m.

7 Worst-case analysis of the LPT algorithm for single processor ecase of l i+ l i for i n, we have conseqently m τ LPT S l + + l + + +l m + + l m = l + l i + m. Ths, τ LPT S l + = l + m+ l i + m l i + n. Hence, by 5, τ LPT S τ o S + l l n l n. ii n = m. From Lemma we see that m τ LPT S = l + l + + l 4 + +l m + + l m = l + l i + m. ecase of l i+ l i for i n, we have conseqently m τ LPT S l +l ++l l m 3 + +l m = l + l i +m. Ths, τ LPT S l + l + = l + l + m l i + m l i + n. Hence, by 5, τ LPT S τ o S l + l l n l n. It follows immediately that the makespan of LPT schedles is at most one nit interval longer than the makespan of optimal schedles. We give the following example that shows that the bond is tight. Consider the set S consisting of t = m jobs of

8 538 O. ran et al. length and t zero-jobs for some integer t. We denote a job of length by and a zerojob by 0. The optimal permtation 0[] t [0] t has finishing time τ o S = 3t /. On the other hand, the LPT order t 0 t has a finishing time τ LPT S = 3t/. Ths, τ LPT S τ o S =. For the case t = m + one has to add one more zero-job to achieve the given worst-case bond. 4 Refined analysis of the List Schedling algorithm for 3 In this section, we analyze the worst-case behavior of the makespan τ w S of any permtations of S in relation to the makespan τ o S of optimal schedles as stated in Theorem. In ran et al. 04a the athors assme that the permtation S, S,...,S n is the optimal permtation for the set of jobs S = {S, S,..., S n }, so that this permtation has finishing time τ o S. Then the athors create an axiliary job set T ={T, T,...,T n } as follows. The size of t i = T i is defined by: t i = { Observe, that for any permtation T of T, τt = τt = if s i, s i if s i >. 6 t i τ w S. 7 The athors next examine the schedle for S. They replace each job S i that has s i by a zero-job S i placed on the time axis at the starting time of S i the lengths of the jobs with s i > remain nchanged. This certainly cases no violations of the -constraint. Then they assign a weight of size to each zero-job in this modified job set S. Ths, the schedle for S consists of the original large jobs S i of length s i > and a nmber of point masses of weight, all placed so that condition still holds, i.e., no nit interval intersects more than of these jobs. The sm of all the weights of the jobs in S is jst eqal to the sm of all the lengths of the jobs in T, which by 7 is an pper bond on τ w S. In addition since some of the jobs have been replaced by zero-jobs, all of the jobs in S still fit in the interval [0,τ w S. In that way the problem has been redced to that of finding an pper bond W on the total weight of an arbitrary assignment of semi-open intervals of length s i > and point masses of weight so that condition is satisfied and, of corse, so that positive length intervals are disjoint, and point masses can only intersect a positive interval at its starting point. In particlar, we can conclde that τ w S τt W. Next, in ran et al. 04a the athors define blocks U i =[i, i + for 0 i N, where N = τ o S. They define the content cu i to be the sm of all the weight or mass in U i. In other words, all the contribtions of the point masses in U i together with all the portions of those S k that happen to lie within U i are added p.

9 Worst-case analysis of the LPT algorithm for single processor In what follows, the athors analyze varios possibilities for the contents of the blocks U i. The interval U i is named bad if cu i > θ, otherwise U i is good. y analyzing the possible nmbers of zero-jobs in a block U i the athors conclde that the total gain of the sm of the contents of the blocks U i over the average vale for details we refer the reader to [ran et al. 04a, p.40]. That is, the total weight N i=0 cu i of the blocks of S satisfies is at most τ w S τt = N t i = N i=0 + cu i We observe that to achieve this pper bond the last block U N in the optimal schedle S mst be a bad block, i.e., the last block U N mst have content. This means that there has to be a job of size in the last block U N. And that implies that N actally eqals τ o S. So we can follow that τ w S τt τ o S + which shows that the worst-case example for 3asgiveninran et al. 04a is a tight worst-case example. In this worst-case example the job set S consists of t + zero-jobs and t + + jobs of length 0 for a positive integer t, t mst be large enogh. 8 5 Conclding remarks For the single processor schedling problem with time restrictions it is never possible for more than jobs to be worked on dring any nit interval. Another view on the problem is to consider parallel processors that need one additional resorce tool for processing. After a processor has sed this resorce that processor needs one timenit to reset itself e.g., for cleaning, cooling down, etc. whereas the resorce is immediately available again. The single processor schedling problem with time restrictions was stdied for the first time by ran et al. 04a where the athors show that this problem is NP-hard when the vale is variable and where they provide a detailed worst-case analysis of List Schedling. We analyze the worst-case behavior of LPT schedles where we rearrange the set S of jobs into non-increasing order and prove a tight bond for List Schedling for 3. We show that LPT ordered jobs can be processed within a factor of / of the optimm pls and that this factor is best possible. There might be algorithms other than LPT that achieve a better worst-case behavior than LPT. An easy improvement for the case = is a heristic H where we start the schedle with the smallest job and then perform LPT. In that way the bond τ H S τ o S / can be achieved. It wold be interesting to know if or problem remains NP-hard if is fixed. The development of a mathematical

10 540 O. ran et al. programming model to find the optimal soltions wold also be an interesting topic to investigate. References ran O, Chng F, Graham RL 04a Single processor schedling with time restrictions. J Sched 7: ran O, Chng F, Graham RL 04b onds on single processor schedling with time restrictions. In: Fliedner T, Kolisch R, Naber A eds Proceedings of the 4th International Conference on Project Management and Schedling, pp 48 5 Garey MR, Johnson DS 979 Compters and Intractability: a gide to the theory of np-completeness W.H. Freeman, New York Graham RL 966 onds for certain mltiprocessing anomalies. ell Syst Tech J 45: Graham RL 969 onds on mltiprocessing timing anomalies. SIAM J Appl Math 7:46 49