Extension of algorithm list scheduling for a semi-online scheduling problem

Transcription

1 CEJOR (2007) 15: DOI /s x ORIGINAL PAPER Extension of algorithm list scheduling for a semi-online scheduling problem Yong He György Dósa Published online: 20 August 2006 Physica-Verlag 2006 Abstract A general algorithm, called ALG, for online and semi-online scheduling problem Pm C max with m 2 is introduced. For the semi-online version, it is supposed that all job have their processing times within the interval [p, rp], where p > 0, 1 < r m/m 1. ALG is a generalization of LS and is optimal in the sense that there is not an algorithm with smaller competitive ratio than that of ALG. Keywords ratio Analysis of algorithm Scheduling Semi-online Competitive 1 Introduction In the parallel identical machine scheduling problem P C max, we are confronted with a sequence J of independent jobs with positive processing times p 1, p 2,..., p n, that must be scheduled on m parallel and identical machines P 1,..., P m. We identify the jobs with their processing times. The jobs and machines are available at time zero, and no preemption is allowed. The load of a machine is the sum of the processing times of the jobs assigned to that machine. The objective is to minimize the maximum machine load C max, called makespan. A scheduling problem is called on-line if it requires to schedule jobs Y. He (B) Department of Mathematics, Zhejiang University, Hangzhou , People s Republic of China heyong@math.zju.edu.cn G. Dósa Department of Mathematics, Pannon University, Veszprém, Hungary dosagy@almos.vein.hu

2 98 Y.He,G.Dósa irrevocably on the machines as soon as they are given, without any knowledge about jobs that follow later on. If we have full information on the job data before constructing a schedule, this problem is called off-line. If the problem is semi-online with tightly-grouped processing times, then we know in advance that all jobs have their processing times between p and rp (p > 0, r 1). W. l. o. g., we assume that p = 1 by normalization and the jobs come in the order of p 1, p 2,..., p n in this paper. It is allowed that the jobs with processing times p or rp may not come. This semi-online version may have applications in practice. As pointed out in [4, 11], in many cases, jobs are normally disturbed and it is possible to give acceptable lower and upper bounds for the processing time of each job. One wishes to get algorithm with smaller competitive ratio by utilizing the semi-online information. In a worst-case analysis, the performance of an online or a semi-online algorithm is measured by its competitive ratio. For a job sequence J and an algorithm A, letw A (J ) (or shortly w A ) denote the makespan produced by the algorithm A and let w (J ) (or shortly w ) denote the optimal makespan in an off-line version. Then the algorithm A is called c-competitive, if w A (J )/w (J ) c holds for every instance J. An online (or semi-online) scheduling problem has a lower bound c if no online (or semi-online) algorithm can be c -competitive with c < c. An online (or semi-online) algorithm is called optimal if its competitive ratio matches the lower bound of the problem. For the on-line version of the discussed problem, Graham [6] proposed a simple greedy algorithm list scheduling (LS in short). This algorithm always assigns the incoming job to the machine with minimum current load. Graham showed R LS = 2 1/m. Faigle, Kern and Turán [7] observed that LS is the optimal online algorithm for two and three machines. For a large number of machines, several algorithms have been proposed which have a slightly smaller competitive ratio than that of LS algorithm [1, 13], the competitive ratio of an optimal online algorithm is now known to lie in the interval [1.88, ] [2]. The semi-online scheduling problem with tightly-grouped processing times was proposed in [10]. For the two machine case, it has been shown that LS is an optimal semi-online algorithm with competitive ratio (1 + r)/2 for any 1 r 2 and 3/2 for any r > 2. It also can be shown [8] that LS is optimal for any m 3 and 1 r m/(m 1). In a recent paper [9], the authors presented a comprehensive analysis on m = 3. They showed that the competitive ratio of LS is as follows: 2(r 1) 1 +, if 1 r 3 3 2, 2 3 r + 3, if 3 2 < r 3, r + 1 2, if 3 < r 2, 2 1, if 2 < r 3, r 5, if r > 3, 3

3 Extension of algorithm LS for a semi-online scheduling problem 99 It can conclude that LS is optimal only for r [1, 1.5], [ 3, 2] and [6, + ). Optimal or improved algorithms for the intervals where LS is not optimal were designed in the same paper. The motivation of this note is the following: Due to their simpleness, the classical approximation algorithms for scheduling problems such as LPT, Multifit, LS, etc, are still considered worthwhile in applications. Hence the relaxation and generalization of these algorithms which have the same performance guarantee are of interest. Goldberg and Shapiro [5] considered the relaxation of off-line algorithm LPT. They created a class of algorithms which have the same worstcase ratio 4/3 1/(3m) as that of LPT. In this note, we consider how to relax online algorithm LS. An (semi-) online algorithm, called ALG is presented. Its competitive ratio is not greater than that of LS, and it reserves more freedom in allocating jobs. This property may be useful if there are some secondary objectives. The algorithm which is presented uses the same idea like the algorithm of Karger et al. [12]. The idea is to choose one of the machines which do not hurt the competitive ratio. In the remainder of this note, denote by load(p i ) the current load of machine P i in heuristic. 2 On-line version On-line algorithm ALG 1. Let k = 1, load(p i ) = 0, i = 1,..., m. 2. Let C = ((2m 1)/m)C 1 (k), where C 1 (k) = max{max i=1,...,k p i, k i=1 p i /m} 3. Let I = {i load(p i ) + p k C, 1 i m}.ifi =, then stop. 4. Let i 0 be an arbitrary index in I. Allocate job k to the machine P i0,let load(p i0 ) = load(p i0 ) + p k. 5. k =. If k > n then halt, otherwise go to 2. Theorem 1 ALG cannot be halted at Step 3, and thus the competitive ratio of ALG cannot be greater than (2m 1)/m. Proof Suppose that ALG halts at Step 3. Without loss of generality, it can be assumed that job p n is the first one which cannot be scheduled by ALG. Thus load(p i ) + p n > C, for all 1 i m. Summarizing these inequalities, it follows that n i=1 p i + (m 1) p n > mc. Then dividing it by m, we get ( n i=1 p i /m) + ((m 1)/m)p n > C. It implies that C = ((2m 1)/m)C 1 (n) = C 1 (n) + ((m 1)/m)C 1 (n) ( n i=1 p i /m) + ((m 1)/m)p n > C, a contradiction.

4 100 Y.He,G.Dósa 3 Semi-online version with tightly-grouped processing times The following theorem is from [10]. Theorem 2 Suppose that all jobs have their processing times within interval [1, r], where 1 r m/(m 1). Then applying LS to the problem Pm C max, we have w LS /w 1 + (((m 1)(r 1))/m) and LS is optimal. Suppose that assumption of Theorem 2 holds in the following. We will show that if some further consideration on general algorithm ALG is done, then we can get a generalized semi-online algorithm, which is still an optimal semionline algorithm and furthermore the competitive ratio can be sharpened in some special cases. Let T l be the first l jobs in the sequence. In the following description of ALG, let C 1 (l) denote a lower bound of the optimal makespan for T l, and q(l) denote an upper bound of the competitive ratio if the algorithm is applied for T l. Their value will be determined later. Modified ALG algorithm in case p i [ 1, r ],1 r m/(m 1) 1. Let l = 1, load(p i ) = 0fori = 1,..., m. 2. Let C = q (l) C 1 (l). 3. Let I = {i load(p i ) + p l C, i {1,..., m}}.ifi =, then stop. 4. Let i 0 be an arbitrary index in I. Allocate job p l to the machine P i0,let load(p i0 ) = load(p i0 ) + p l. 5. l = l + 1. If l > n then halt. 6. Update the value of C 1 (l) and q (l), and go to 2. Lemma 3 Let T ={p 1, p 2,..., p n }. Suppose n = mk + jforsome1 j m. 1. Let t i be the ith smallest job in {p 1, p 2,..., p n }. Then C 1 = (1/j) j(k+1) i=1 t i is a valid lower bound of the optimal makespan. 2. Consider an arbitrary algorithm which schedules the first n 1 jobs without introducing any idle time between consecutive two jobs. We force the algorithm to schedule the job p n to the machine with minimum current load s. Then the completion time of this machine is w = s + p n. We have w C 1 () (r 1)((j 1)/j), if2j m + 1; and w C 1 () (r 1)((m j)/()), if2j < m + 1. Proof 1. In any schedule, there are j machines processing totally at least j () jobs, the total processing time of these jobs is at least j(k+1) i=1 t i, thus its makespan is at least C 1 = (1/j) j(k+1) i=1 t i. 2. To prove it, a further notation is required. For every α satisfying 1 α j, denote S α = α(k+1) i=(α 1)(k+1)+1 t i, and for every α satisfying j + 1 α m, denote S α = j(k+1)+(α j)k i=j(k+1)+(α j 1)k+1 t i. Then by this notation, we have

5 Extension of algorithm LS for a semi-online scheduling problem 101 C 1 = 1 j j S α. (1) It is clear that if 1 α j, S α () r holds, and if j + 1 α m, k S α kr holds. Consider an arbitrary algorithm which schedules the first n 1 jobs without introducing any idle time between consecutive two jobs. Before allocating the last job p n, there are j 1 machines which have been already assigned at least (j 1)() jobs. The total processing time of these jobs is at least j 1 α=1 S α, thus the total processing time of jobs being assigned to the other machines is at most m S j + S α p n S j + (m j) kr p n. (2) α=j+1 Thus there is a machine among them such that its current load is not greater than (1/())(S j + (m j)kr p n ). Since w = s + p n, we get α=1 1 ( ) w Sj + (m j) kr p n + pn 1 ( = Sj + (m j) kr ) + m j p n 1 ( Sj + (m j) kr ) + m j r 1 = S (m j)() j + r. (3) From (1), (3) and (1/()) j 1 α=1 S α j 1, it follows that w C 1 1 S j + (m j)() r 1 j j α=1 (m j)() 1 = r + S j 1 j S α j α=1 = () j j (m j) r + (2j m 1) S j () j S α () j 1 S α α=1 { j (m j) r + (2j m 1) S j ()(j 1) If 2j m 1 0, i.e., 2j m + 1, then substituting S j /() r into (4), we get w C 1 ()(r 1) ((j 1)/j). }. (4)

6 102 Y.He,G.Dósa Otherwise 2j m 1 < 0, i.e., 2j < m + 1, substituting (S j /()) 1into (4), we get w C 1 ()(r 1) ((m j)/()). Now we are ready to prove the following theorem: Theorem 4 Denote T l the first l jobs, and suppose l = mk+jforsome1 j m. Let t i be the i-th smallest job in {p 1, p 2,..., p l }, and let q (l) = 1 + q (l) = 1 + C 1 (l) = 1 j j(k+1) i=1 t i, (r 1)(j 1), if 2j m + 1, j (r 1)(m j), if 2j < m + 1, then ALG schedules all jobs. Remark 5 1. C 1 (l) is a step by step actualized lower bound and C 1 (l + 1) can be computed from C 1 (l) by easy calculation. Hence computing C 1 (l) requires O(n) times. On the other hand, computing q (l) requires only O(1) time. 2. If Theorem 4 holds, the competitive ratio of the algorithm cannot be greater than q (l). Furthermore q (l) < 1 + ((m 1)(r 1))/m, ifj = 1 and j = m, and q (l) = 1 + ((m 1)(r 1))/m, ifj = 1orj = m. 3. Usually the current job can be allocated not only to one machine in Step 4, thus there is some freedom in choosing the machine to which the current job is allocated. It may be useful if there are some secondary objectives. 4. In Step 2, we may use a C which is smaller than q (l) C 1 (l) to replace q (l) C 1 (l) as an upper bound although sometimes algorithm may stop at Step 3. In this situation we simply increase value of C and follow the algorithm. If algorithm allocates all jobs with some C < q (n) C 1 (n) then the practical performance guarantee is certainly smaller than q (n). Proof It is enough to prove that ALG cannot be halted at Step 3. Suppose that the algorithm halted at Step 3. We can assume that p n is the first job that is not scheduled. Let the minimum current machine load right before assigning p n by ALG be s. By allocating job p n to the minimum loaded machine, we get a schedule with makespan w = s + p n > C, where C = q (n) C 1 (n). Note that C 1 (n). Applying Lemma 3, if 2j m + 1, we get w C 1 (n) = 1 + w C 1 (n) () (r 1)(j 1) C 1 (n) jc 1 (n) (r 1)(j 1) j, (5)

7 Extension of algorithm LS for a semi-online scheduling problem 103 and if 2j < m + 1, we get w C 1 (n) = w C 1 (n) () (r 1)(m j) C 1 (n) () C 1 (n) (r 1)(m j). (6) Thus we have that w q (n) C 1 (n) = C < w, a contradiction. It is clear that w C 1 (n), we obtain Corollary 6 w ALG w 1 + (r 1) j 1, if n = mk + j and 2j m 1 0, j w ALG m j w 1 + (r 1), if n = mk + j and 2j m 1 < 0, the overall competitive ratio of ALG is 1 + (((m 1)(r 1))/m) and thus is optimal. Proof We only need to show that the example given in [7] proves the optimality of ALG. For completeness, we do it as follows. In fact, note that the competitive ratio of the algorithm is smaller than 1 + (r 1) (m 1)/m,ifj = 1, and j = m, thus for any sharp counterexample n = mk+j, j = 1, or j = m must hold. Let for any m 2 the job sequence is T m = {1, 1,...,1,r, r,..., r}, where the number of jobs with processing time 1 is exactly m, and the number of the other jobs with processing time r is (m 1) 2. (In this case j = 1.) For example T 2 = {1, 1, r}, and T 3 = {1, 1, 1, r, r, r, r}. The jobs are coming in this order. It is clear that the optimal makespan is m because r m/(m 1). On the other hand, for any schedule, if the first m jobs are allocated to at most m 1 machines, then there is not more incoming job, and the competitive ratio is at least 2. If the 1 s are allocated to different machines, then the next (m 1) 2 = m (m 2) + 1 jobs are allocated to m machines, thus there is a machine to which at least m 1 further jobs are allocated. Thus the makespan is at least 1 + (m 1) r, and the competitive ratio is at least (1 + (m 1) r)/m = 1 + ((m 1)(r 1))/m. References Albers S (1999) Better bounds for on-line scheduling. SIAM J Comput 29(2): Albers S (2002) On randomized online scheduling. In: Proceedings of the 34th ACM symposium on theory of computing. Montreal, pp Coffman EE Jr, Garey MR, Johnson DS (1981) An application of bin-packing to multiprocessor scheduling. SIAM J Comput 7:1 17 Dósa G, He Y (2004) Semi-Online Algorithms for Parallel Machine Scheduling Problems. Computing 72: Goldberg RR, Shapiro J (2001) Extending Graham s result on scheduling to other heuristic. Oper Res Lett 29: Graham RL (1969) Bounds on multiprocessor timing anomalies. SIAM J Appl Math 17:

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