Worst-case performance of critical path type algorithms

Transcription

1 Intl. Trans. in Op. Res. 7 (2000) 383±399 Worst-case perforance of critical path type algoriths G. Singh, Y. Zinder* University of Technology, P.O. Box 123, Broadway, NSW 2007, Sydney, Australia Received 11 February 2000; accepted 24 February 2000 Abstract The critical path ethod reains one of the ost popular approaches in practical scheduling. Being developed for the akespan proble this ethod can also be generalized to the axiu lateness proble. For the unit execution tie task syste and parallel processors this generalization is known as the Brucker±Garey±Johnson algorith. We characterize this algorith by introducing an upper bound on the deviation of the criterion fro its optial value. The bound is stated in ters of paraeters characterizing the proble, naely nuber of processors, the length of the longest path, and the total required processing tie. We also derive a siilar bound for the preeptive version of the Brucker± Garey±Johnson algorith IFORS. Published by Elsevier Science Ltd. All rights reserved. Keywords: Deterinistic scheduling probles; Parallel identical achines; Precedence constraints; Maxiu lateness; Worst-case analysis 1. Introduction The critical path ethod reains one of the ost popular approaches in practical scheduling. The rst theoretical justi cation of this ethod has been given by Hu (1961). Hu's algorith allows the generalization to the case of the axiu lateness proble. This generalization is known as the Brucker±Garey±Johnson algorith (Brucker et al., 1977). Both the algoriths were developed for the odel with a unit execution tie (UET) task syste, precedence constraints, and parallel processors. Being of considerable practical interest in its own right, this odel provides an iportant insight into the eld of deterinistic scheduling * Corresponding author. Tel.: ; fax: E-ail address: y.zinder@aths.uts.edu.au (Y. Zinder) /00/$ IFORS. Published by Elsevier Science Ltd. All rights reserved. PII: S (00)00023-X

2 384 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 with precedence constraints, and therefore, reains as a subject of intensive research over several decades. The axiu lateness proble for a UET task syste can be stated as follows. Suppose that a set N ˆf1, 2,...,ng of n tasks (jobs, operations) is to be processed on > 1 identical processors (achines) subject to precedence constraints in the for of an anti-re exive, antisyetric and transitive relation on N. If task i precedes task j, denoted as i4j, then the processing of i ust be copleted before the processing of j begins. The processing of the tasks coences at tie t ˆ 0: Each processor can process at ost one task at a tie, and each task can be processed by any processor. Once a processor begins executing a task it continues until copletion (i.e., no preeptions are allowed). The processing tie of task j is denoted by p j : All processing ties are equal, and therefore, without loss of generality, can be taken as the unit of tie. Let C j s be the copletion tie of task j in schedule s. It is necessary to nd a schedule, which iniizes the criterion of axiu lateness L ax s ˆ ax Cj s d j, j2n where d j is a due date associated with each task j. If all due dates are equal to zero the axiu lateness proble converts into the akespan proble. Using the popular three- eld notation ajbjg, where a speci es the processor environent, b speci es the task characteristics, and g denotes the optiality criterion, the above proble can be denoted by Pjprec, p j ˆ 1jL ax : Here, P speci es that there are several parallel identical processors, prec indicates the presence of precedence constraints, and the ter p j ˆ 1 indicates that each task has a unit processing tie. It is convenient to represent a partially ordered set of tasks by an acyclic directed graph, where nodes correspond to the tasks and arcs re ect the precedence constraints. Hence, the scheduling probles with UET task systes and parallel identical processors can be viewed as graph partition probles, where the set of nodes ust be partitioned into several subsets containing no ore than nodes each. The Brucker±Garey±Johnson algorith solves the axiu lateness proble if the corresponding graph is an in-tree. The Pjprec, p j ˆ 1jL ax proble is NP-hard if the corresponding graph is an out-tree (Brucker et al., 1977). Therefore, for general precedence constraints, the Brucker±Garey±Johnson algorith is only an approxiation algorith. This algorith represents the faily of so-called priority algoriths. A priority algorith can be viewed as a two-phase procedure. The rst phase of this procedure associates with each task a nuber, which indicates the priority (urgency) of the task. The second phase allocates tasks for processing in accord to these priorities, i.e. each tie, when a processor becoes available, aong all tasks, which are ready for processing, the algorith selects a task with the highest priority. The idea of a priority algorith can be odi ed to the case of arbitrary processing ties and preeptions. If preeptions are allowed, the processing of any task can be interrupted at any tie and resued later on the sae or another processor. The corresponding axiu lateness proble with preeptions is denoted by Pjprp, precjl ax : In general, the preeptive version of the Brucker±Garey±Johnson algorith provides only an approxiation solution,

3 since the akespan proble with an arbitrary nuber of processors, general precedence constraints, arbitrary processing ties, and preeptions is also NP-hard (Ullan, 1975). The perforance guarantees are viewed as an essential coponent of the characterization of approxiation algoriths. For the scheduling probles with parallel identical processors any results of this type were obtained for the akespan proble (see, for exaple, Parker, 1995; Lawler et al., 1993). Interest in such perforance guarantees is inspired not only by the fact that they provide an iportant insight into the nature of the corresponding approxiation algoriths, but also by the role which these results play in the developent of various algoriths requiring calculation of lower bounds, in particular, in the developent of branchand-bound algoriths. In spite of the existence of perforance guarantees for a nuber of algoriths for the akespan proble, uch less is known about the worst-case perforance of the approxiation algoriths for the axiu lateness proble. Thus, it was shown by Zinder and Roper (1998), that L ax s Z R 2 2 L ax s 1 2 ax d a r, 1 a2n where s Z is a schedule constructed by the approxiation algorith, s is an optial schedule for the axiu lateness proble, and ( 3 r ˆ for odd for even: 2 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± The establishent of siilar results for other known approxiation algoriths can be viewed as a logical next step in the theoretical study in the eld. The result in Zinder and Roper (1998), as well as any perforance guarantees for the akespan proble presents an upper bound on the value of the criterion corresponding to the approxiation algorith expressed in ters of its optiu value. In contrast, in this paper we present two perforance guarantees in the for of bounds on the deviation of the criterion value fro its optiu expressed in ters of the paraeters characterizing the partially ordered set of tasks. This for provides ore inforation on the worst-case perforance of the respective algoriths and is convenient for the use in various exible self-tuned algoriths that select an appropriate strategy by analyzing the proble on hand. A siilar for of perforance guarantee for the classical Hu's algorith is presented in Singh and Zinder (2000). In Section 2 we consider the Pjprec, p j ˆ 1jL ax proble and present a tight upper bound on the di erences L ax s BGJ L ax s, where s BGJ is the schedule constructed by the Brucker±Garey±Johnson algorith and s is an optial schedule for Pjprec, p j ˆ 1jL ax proble. In Section 3 we consider the Pjprp, precjl ax proble and present a tight bound for the preeptive counterpart of the Brucker±Garey±Johnson algorith. 2. The worst-case perforance of the Brucker±Garey±Johnson algorith for the Pjprec, p j ˆ 1jL ax proble The priority associated with each task by the Brucker±Garey±Johnson algorith is called a

4 386 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 odi ed due date, and the tasks are arranged in a list in the non-decreasing order of their odi ed due dates. The tie points t ˆ 0, t ˆ 1, t ˆ 2, and so on, are considered sequentially. For each tie point the list is scanned fro left to right and the rst encountered task available for processing is assigned to an idle processor and eliinated fro the list. The algorith continues to scan the list and to allocate tasks for processing until at least one of the following events occurs: the end of the list has been reached or all processors have tasks assigned to the. Then the allocation of the reaining tasks is ade for subsequent tie points until all tasks have been assigned. We will denote the odi ed due date associated with task j by dj 0: Modi ed due date d j 0 depends on the odi ed due dates of the iediate successors of task j. Task i is an iediate successor of task j if j4i and there is no task k such that j4k and k4i: We will denote the set of all iediate successors of task j by K( j ). In Brucker et al. (1977), the algorith coputing the odi ed due dates was given for in-trees. The algorith below is a straightforward extension of this algorith to the case of an arbitrary partially ordered set. 1. For each task j that does not have successors, set dj 0 ˆ d j : 2. Select a task j which has not been assigned its odi ed due date dj 0 and whose all iediate successors have been assigned their odi ed due dates. If no such task exists, then stop. 3. Set dj 0 ˆin i2k j fd j, di 0 p i g, where p i is the processing tie of task i, and return to step 2. We will refer to this algorith as the due date odi cation algorith. Note that this algorith without any changes will be also used in the preeptive case, where the processing tie p i can be any positive nuber. In order to analyze the worst-case perforance of the Brucker±Garey±Johnson algorith, we observe that the replaceent of the original due dates by the corresponding odi ed due dates does not a ect the axiu lateness. Indeed, since dq 0Rd q for any task q, we have ax C q s d q 0 rax Cq s d q ˆ Lax s : q2n q2n To show that ax C q s d q 0 RL ax s, q2n 2 aong all tasks j satisfying ax q2n C q s d q 0 ˆ C j s d j 0 select a task with the largest copletion tie. Let it be task b. Ifd b ˆ db 0, then Eq. (2) holds. On the other hand, if d b > db 0, then according to the due date odi cation algorith there is an iediate successor of task b, say task v, such that db 0 ˆ d v 0 p v: Since C v s rc b s p v,we have

5 C b s d 0 b ˆ C b s d 0 v p v ˆ Cb s p v d 0 v RC v s d 0 v, which contradicts the selection of task b. Let s BGJ be a schedule constructed by the Brucker±Garey±Johnson algorith. Fro the set of all tasks j such that C j s BGJ d 0 j ˆ L ax s BGJ G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± choose a task p with the sallest copletion tie. If C p s BGJ ˆ1, then s BGJ is optial. If C p s BGJ > 1, then we construct a sequence of tasks a h,...,a 1 and a sequence of sets of tasks M h,...,m 1 using the following procedure. Select task p as a 1, and denote by ~M 1 the set coprised of task p and all tasks j such that dj 0Rd p 0 and C j s BGJ < C p s BGJ : If for any positive integer t < C a1 s BGJ at least two tasks fro ~M 1 are processed in s BGJ on the tie interval t 1, t], then M 1 ˆ ~M 1 and the procedure terinates. Otherwise, we select the largest nuber aong all positive integers t < C a1 s BGJ such that only one task fro ~M 1 is processed on the tie interval t 1, t]. Denote this task by a 2 and set M 1 equals the set of all tasks j 2 ~M 1 with C j s BGJ > C a2 s BGJ : Let ~M 2 be the set of all tasks j such that dj 0Rd a 0 2 and C j s BGJ RC a2 s BGJ : Suppose that C a2 s BGJ ˆ1, or C a2 s BGJ > 1 and for any positive integer t < C a2 s BGJ at least two tasks fro ~M 2 are processed in s BGJ on the tie interval t 1, t]. Then M 2 ˆ ~M 2 and the procedure terinates. Otherwise, we select the largest nuber aong all positive integers t < C a2 s BGJ such that only one task fro ~M 2 is processed on the tie interval t 1, t]. Denote this task by a 3 and set M 2 equals the set of all tasks j 2 ~M 2 with C a2 s BGJ rc j s BGJ >C a3 s BGJ : We continue in the sae anner until for soe h the procedure terinates because either C ah s BGJ ˆ1, or C ah s BGJ > 1 and for any positive integer t < C ah s BGJ at least two tasks fro ~M h are processed in s BGJ on the tie interval t 1, t]. Lea 2.1. Let j be an arbitrary task fro M i, for soe 1RiRh 1: Then a i 1 4j: Proof. Note that any task q, satisfying the conditions dq 0Rd a 0 i and C q s BGJ < C ai s BGJ, belongs to ~M i : Therefore, on the tie interval C ai 1 s BGJ 1, C ai 1 s BGJ ], according to the schedule s BGJ, only one processor processes a task with the odi ed due date less than or equal to da 0 i : Let j be an arbitrary task fro M i : Since dj 0Rd a 0 i, according to the Brucker±Garey±Johnson algorith, task j ust have a predecessor, say task j', with C j 0 s BGJ ˆC ai 1 s BGJ : Since j 0 4j, we have dj 0 < d 0 0 j Rd a 0 i, but on the tie interval C ai 1 s BGJ 1, C ai 1 s BGJ Š only task a i 1 has the odi ed due date less than or equal to da 0 i : Therefore, a i 1 4j: Let M ˆ Sh iˆ1 M i: For any positive integer t < C p s BGJ, we say that the tie slot t is coplete, if exactly tasks fro M are processed on the tie interval t 1, t]. Otherwise, the tie slot t is said to be incoplete. Let w be the nuber of all incoplete tie slots. If w > 0, we will denote incoplete tie slots by t 1,...,t w, where t 1 <...< t w < C p s BGJ : Lea 2.2. The tie slot C p s BGJ 1 is coplete and any task that is processed in this tie slot does not precede task p. Proof. Suppose that there exists a task j such that C j s BGJ ˆC p s BGJ 1 and j4p: Then

6 388 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 dj 0Rd p 0 1, and C j s BGJ d j 0 rc p s BGJ 1 d p 0 1 ˆ C p s BGJ d p 0, which contradicts the selection of task p. Suppose that the tie slot C p s BGJ 1 is incoplete. Then in this tie slot at least one processor is either idle or processes a task with the odi ed due date greater than dp 0: Therefore, according to the Brucker±Garey±Johnson algorith, task p ust have a predecessor that is processed in s BGJ in the tie slot C p s BGJ 1: But as has been shown above, this contradicts the selection of task p. Lea 2.3. Let U be the set coprised of task p and all tasks processed in the tie slot C p s BGJ 1: Then for any task b 2 U, there exists a sequence of tasks b 1,...,b w fro M such that b 1 4 4b w 4b, and C bi s BGJ ˆt i, for all 1RiRw: Proof. Let us denote task b by b w 1 : Suppose that for soe krw 1 a subsequence b k,...,b w 1 has been constructed, and b k 2 M i : If t k1 ˆ C ai 1 s BGJ, then by Lea 2.1 a i 1 4b k, and we choose task a i 1 as b k1 : Suppose that t k1 > C ai 1 s BGJ : Observe that any task j such that d 0 j Rd 0 a i and C j s BGJ < C ai s BGJ belongs to ~M i : Hence, in the schedule s BGJ in the tie slot t k1 at least one processor is either idle or processes a task with the due date greater than d 0 b k : Therefore, task b k ust have a predecessor, which is processed in the tie slot t k1 : We denote this predecessor by b k1 : Since b k1 4b k, according to the due date odi cation algorith d 0 b k1 Rd 0 b k 1: Hence, b k1 2 M i : Continuing in the sae anner, we obtain the desired chain. Consider a longest path } in the subgraph corresponding to the set M. Let l(t ) be the nuber of tasks in this path with the copletion tie greater than t. Then the length of } equals l(0). Lea 2.4. If each incoplete tie slot contains exactly one task fro the set M, then L ax s rl 0 1 d 0 p, 3 where s is an optial schedule for the axiu lateness proble. Proof. Lea 2.2 iplies that there exists a positive integer trc p s BGJ such that no task fro } is processed on the tie interval t 1, t]. Let t' be the sallest of these nubers, and let r be the largest value of i satisfying inequality t 0 RC ai s BGJ : Since the tie slot t' does not contain a task fro }, l C ar s BGJ C ar s BGJ rl 0 1: 4 By the due date odi cation algorith, in the path } the l C ar s BGJ th fro the right task has a due date less than or equal to d 0 p l C a r s BGJ 1: Then Lea 2.1 iplies that d 0 a r Rd 0 p l C ar s BGJ : Let j be an arbitrary task fro M r : Since d 0 j Rd 0 a r, we have

7 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± d 0 j Rd 0 p l C ar s BGJ : 5 Suppose that r < h: Then a r 1 belongs to the path }, and each tie slot t, where trc ar 1 s BGJ, contains a task fro }: Hence, task a r 1 cannot be copleted earlier than in the schedule s BGJ, and therefore, C ar 1 s rc ar 1 s BGJ : Because all tie slots t, where C ar 1 s BGJ < t 0 < C ar s BGJ, are coplete, and because by Lea 2.1 a r 1 precedes all tasks fro M r, ax C j s rc ar s BGJ : j2m r 6 Since all tie slots t, where t < C ah s BGJ, are coplete, the inequality Eq. (6) also holds if r ˆ h: Using Eqs. (4)±(6), we have n o n L ax s rax C j s d j 0 rax C j s d p 0 l C ar s BGJ o j2m r j2m r rc ar s BGJ d 0 p l C ar s BGJ rl 0 1 d 0 p : This copletes the proof. Lea 2.5. If w ˆ 0, then s BGJ is optial. Proof. If there are no incoplete tie slots, then the tasks fro M cannot be processed ore quickly by any other schedule. Therefore, for any schedule s that is optial for the axiu lateness proble, C p s BGJ Rax C j s, j2m and since dj 0Rd p 0, for all j 2 M, L ax s BGJ ˆ C p s BGJ d 0 p Rax j2m n o Rax C j s d j 0 j2m Hence, s BGJ is optial. Rax j2n n o C j s d p 0 ˆ ax C j s d p 0 j2m n o C j s d j 0 ˆ L ax s : Theore 2.1. If s is an optial schedule for the axiu lateness proble, then 8 n < nl 1 L ax s BGJ L ax s in 1, o, R l if n lr, 7 : 0, otherwise where n is the nuber of tasks and l is the length of the longest path in the corresponding graph. For any positive integer ^n there is an instance of the axiu lateness proble with nr ^n such

8 390 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 that Eq. (7) is an equality. Proof. It is easy to see that schedule s BGJ is optial, and therefore Eq. (7) holds, if either n l <, or n lr and C p s BGJ ˆ1: By Lea 2.5, s BGJ is also optial when n lr, C p s BGJ > 1, and w ˆ 0: Hence, we only need to prove that Eq. (7) holds when n lr, C p s BGJ > 1, and w > 0: In this case we can construct the set M, and let c be the nuber of coplete tie slots t satisfying the inequality t < C p s BGJ : Hence, C p s BGJ ˆ c w 1: Since l(0) is the length of a longest path in the subgraph corresponding to the set M, we have L ax s rl 0 dp 0: Suppose that L ax s rl 0 1dp 0 : By Lea 2.3 l 0 rw 1, and Hence, L ax s BGJ ˆ C p s BGJ d 0 p ˆ c w 1 w 1 1 d 0 p RjMjl 0 l 0 d 0 p : d p 0 1 RjMj w w 1 L ax s BGJ L ax s R jmjl 0 1: 8 Now suppose that L ax s ˆl 0 dp 0 : Then fro Lea 2.4 we conclude that there is an incoplete tie slot containing at least two tasks fro M, and therefore, jmjrc w 2: Consider the set U speci ed in the stateent of the Lea 2.3. Since juj ˆ 1, this lea iplies that ax C j s rw 2: On the other hand, since L ax s ˆl 0 dp 0, we have C j2m j s ˆl 0, and hence, l 0 rw 2: We have ax j2m L ax s BGJ ˆ C p s BGJ d 0 p ˆ c w 2 w 2 w 2 1 d 0 p RjMjl 0 l 0 1 d 0 p, and, subtracting L ax s ˆl 0 dp 0, we again obtain Eq. (8). Since n jmjrll 0, Eq. (8) gives L ax s BGJ L ax s R n l 1: 9 Using Lea 2.3, we conclude that lrl 0 rw 1: Hence, L ax s BGJ ˆ C p s BGJ d 0 p ˆ c w 1 w 1 1 and subtracting the obvious inequality L ax s r jmj d p 0, we obtain d 0 p RjMj 1 l d 0 p,

9 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± L ax s BGJ L ax s R 1 l, which together with Eq. (9) gives Eq. (7). In order to show that Eq. (7) is tight, consider a partially ordered set of tasks presented by the graph in Fig. 1. This graph is coprised of k identical sections and one terinal section. Fig. 1. The partially ordered set of tasks considered in Theore 2.1.

10 392 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 Each task corresponding to a node in the rst (top) row has a due date of one unit of tie. Each task fro any other row has a due date of the previous row plus one. The corresponding optial schedule and a schedule constructed by the Brucker±Garey±Johnson algorith are presented in Figs. 2 and 3, respectively. It is easy to see that ax d j ˆ k 1, L ax s BGJ ˆ j2n k k, L ax s ˆ1 and Eq. (7) is an equality for any k and. 3. The worst-case perforance of the preeptive version of the Brucker±Garey±Johnson algorith for the Pjprp, precjl ax proble The stateent of the axiu lateness proble with preeptions is the sae as for the Pjprec, p j ˆ 1jL ax proble, but now we assue that each task j can have an arbitrary processing tie p j and its processing can be interrupted at any tie and resued later on the sae or another processor. Let p j t be the reaining processing tie for task j at tie t. Since preeptions are allowed and the sae task j can be allocated for processing at di erent points of tie, its priority depends on p j t : As a straightforward generalization of the Brucker± Garey±Johnson algorith, one ay copute the priority of task j at tie t as dj 0 p j t, where dj 0 is the odi ed due date calculated by the due date odi cation algorith presented in Section 2. In this case, a saller value of dj 0 p j t gives a higher priority. The construction of the schedule is based on a sequence of decisions regarding what tasks should be allocated for processing, what aount of processing tie should receive each of these tasks, and how these tasks should be scheduled. Points in tie that correspond to these decisions will be referred to as points of allocation. The rst point of allocation is t ˆ 0: At each point of allocation tasks are assigned for processing as follows. All tasks, which are ready for processing, are split into several subsets. Each subset is coprised of all tasks with the sae priority. If the nuber of tasks of the highest priority is greater than or equal to the nuber of processors, then the corresponding subset will occupy all processors and each task will receive the sae aount of processing tie on the interval between the current and the next points of allocation. If the nuber of tasks of the highest priority is less than the nuber of processors, then these tasks are allocated one task per processor, and the siilar allocation procedure is conducted for the reaining subsets of tasks and the reaining processors. The Fig. 2. An optial schedule.

11 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± allocation terinates if there are either no reaining subsets, or no reaining processors. All processors are released at the next point of allocation, and the allocation procedure repeats. Suppose that D is the length of the tie interval between two successive points of allocation. If a subset coprising u tasks is allocated to k processors, where u > k, then each task in this subset will receive D u k units of processing tie. If u ˆ k, then each task fro this subset will receive D units of processing tie. The next point of allocation is selected in such a way that either soe task is copleted or the priority of tasks in one subset becoes equal to the priority of tasks in another. In other words, suppose that tasks fro a subset N 1 are allocated one task per processor at a point of allocation t, then D Ð the length of the tie interval between the current and the next points of allocation Ð ust satisfy the inequality DRin p j t : j2n 1 Suppose that tasks fro another subset N 2 are allocated to reaining k processors, where k < jn 2 j: Then p j t jn 2 j DRin j2n 2 k and for any j 1 2 N 1 and j 2 2 N 2 d j 0 1 p j1 t D Rd j 0 2 p j2 t Dk : jn 2 j Suppose that a task j is ready for processing at point t, but is not assigned to be processed on the interval between this and the subsequent point of allocation. Then for any j 1 2 N 1 and j 2 2 N 2, d j 0 p j t rd j 0 1 p j1 t D and d j 0 p j t rd j 0 2 p j2 t Dk : jn 2 j The next point of allocation is de ned by the axiu value of D, which satis es these conditions. Fig. 3. A schedule constructed by the Brucker±Garey±Johnson algorith.

12 394 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 The actual scheduling on the tie interval between two successive points of allocation is obtained by applying McNaughton's algorith (McNaughton, 1959). To describe this algorith we assue that tasks fro a subset N' are allocated at point t to k processors, where k < jn 0 j: As before, we denote by D the length of the tie interval between the current point of allocation and the next one. Therefore, each task fro N' will be processed on this interval for Dk jn 0 j tie units. We select a processor and allocate to this processor fro tie t an arbitrary task j 1 2 N 0 : After that we select an arbitrary task j 2 2 N 0 and allocate this task to the sae processor fro tie t Dk jn 0 j : We continue to allocate tasks one after another to the selected processor until we reach a task j r, which cannot be allocated entirely to this processor, then we allocate task j r to the selected processor only till the tie point t D: After that we select another processor and allocate this task to this new processor fro tie t in such a way that the total processing tie for this task on both processors becoes Dk jn 0 j : We continue to allocate tasks to this second processor until we encounter a situation that the next task cannot be allocated entirely to this processor. In this case we allocate this task only till the tie t D, again select a new processor and allocate to this new processor the considered task fro tie t for the reaining processing tie, and so on. To illustrate this algorith consider the following exaple. Let N 0 ˆfj 1,...,j 6 g, D ˆ 3, and k ˆ 4: Then each task fro N' will be processed on the tie interval t, t DŠ for 2 tie units. Fig. 4 depicts the resulting schedule. The algorith described above produces an optial schedule if the corresponding graph is an in-tree (Lawler, 1982). In what follows we will analyze the worst case perforance of this algorith for the general precedence constraints. We will denote the schedule constructed by this algorith by s L : In order to be able to indicate what schedule is considered, we also replace the notation p j t by p j t, s, where p j t, s is the reaining processing tie for task j at tie t in schedule s. As has been shown in Section 2, if preeptions are not allowed, then the replaceent of the original due dates by the corresponding odi ed due dates does not e ect the axiu lateness. It is easy to see that the sae reasonings are valid for the preeptive case and in this case also, for any schedule s, n o L ax s ˆ ax C j s d j 0 : j2n Suppose that schedule s L has been constructed using h points of allocation. Let they be points t 1,...,t h, where 0 ˆ t 1 <...< t h : For each point of allocation t i, let S i be the set of all tasks, which are allocated for processing at point t i : Note that S h ˆ b and the procedure terinates Fig. 4. A preeptive schedule.

13 at this point. For each point of allocation t i, i > 1, let F i be the set of all tasks, which coplete their processing on the tie interval t i1, t i Š at point t i : Since, for any j 2 F i, C j s L rt i p j t i, s L, ax t i p j ti, s L d j 0 RL ax s L : j2f i Consider task q satisfying the equality C q s L d 0 q ˆ L ax s L : Suppose that t i1 < C q s L < t i, for soe i > 1: This eans that at point t i1, task q belonged to a subset, say N', which was allocated to the nuber of processors less than jn 0 j: According to the algorith, there is a task r 2 N 0 that copletes its processing on the tie interval t i1, t i Š at point t i : Hence, C r s L rt i p r t i, s L : Because q and r belongs at point t i1 to the sae subset, they ust have the sae priority at point t i, that is dq 0 ˆd0 r p r t i, s L : We have, C q s L d 0 q < t i d 0 q ˆ t i d 0 r p r ti, s L RC r s L d 0 r, which contradicts the selection of task q. Therefore, C q s L ˆt i, for soe i > 1, and for this point of allocation t i p j ti, s L d j 0 ˆ L ax s L : 10 ax j2f i G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± Aong all t i satisfying Eq. (10) select the sallest, say t i, and select x 2 F i t i p x ti, s L h d x 0 ˆ ax t i p j ti, s L i d j 0 : j2f i such that For each 1 < iri, we denote by M i the subset of all j 2 S i1 such that d 0 j p j ti, s L Rd 0 x p x ti, s L : We will say that an interval t i1, t i ], where 1 < iri, is coplete if jm i jr: Otherwise the interval is said to be incoplete. Lea 3.1. For any task j 2 S i 1 d j 0 p j ti 1, s L ˆ d x 0 p x ti 1, s L, and jm i j > : Proof. Suppose that there is a task j 2 S i 1 such that d j 0 p j ti 1, s L < d x 0 p x ti 1, s L : Then j 2 F i and

14 396 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 p j ti 1, s L ˆ t i t i 1 p j ti, s L : 11 If jm i jr or there is a task q 2 S i 1 such that d 0 q p q ti 1, s L > d 0 x p x ti 1, s L, then Eq. (11) holds for j equals x. Since in both cases we have d 0 j p j ti, s L ˆ d 0 x p x ti, s L, t i 1 p j ti 1, s L d 0 j ˆ t i p j ti, s L d 0 j ˆ t i p x ti, s L d 0 x ˆ L ax s L, which contradicts the selection of t i : Therefore, jm i j > and d 0 j p j t i 1, s L ˆd 0 x p x t i 1, s L, for all j 2 S i 1: We de ne the length of any path in the directed acyclic graph which represents the partially ordered set of tasks as a su of processing ties corresponding to the nodes in this path. Let l be the length of the longest path, w be the total length of all incoplete tie intervals for the schedule s L, and p in ˆ in j2n p j : Lea 3.2. Let t i1, t i Š be an incoplete interval and j be an arbitrary task fro M i 0, where i < i 0 : Then either j is processed on t i1, t i Š during t i t i1 tie units, or there is a task j 0, such that j 0 2 M i, j 0 4j, and j 0 is processed on this interval during t i t i1 tie units. Proof. Since jm i j <, on the interval t i1, t i Š at least one processor is either idle or processes a task with priority lower than the priority of j. Hence, either j is processed on t i1, t i ], or there is a task j 0, which precedes j and is ready for processing at t i1 : In the forer case, since jm i j < and all other tasks fro S i1 have a lower priority than tasks fro M i, task j is processed on t i1, t i Š during t i t i1 tie units. In the latter case, according to the due date odi cation algorith d 0 j 0Rd 0 j p jrd 0 x p x ti, s L : Hence, j 0 2 M i and this task is also processed on the interval t i1, t i Š during t i t i1 tie units. Theore 3.1. Let s be an optial schedule for the axiu lateness proble, then L ax s L L ax s R 1 l p in : 12 For any positive integer ^n there is an instance of the axiu lateness proble with nr ^n such that Eq. (12) is an equality. Proof. Suppose that w ˆ 0: Because on the interval t 1, t i Š all processors are busy, there exists a task j such that j is processed on this interval in schedule s L and C j s rt i p j t i, s L : Since d 0 j

15 p j t i, s L Rd 0 x p x t i, s L, L ax s rc j s d 0 j rt i p j ti, s L d 0 j rt i p x ti, s L d 0 x ˆ L ax s L : Hence, s L is optial, and Eq. (12) holds. Suppose that w > 0: For any j 2[ i iˆ2 M i we denote by i j the largest index i such that j 2 M i : Then for any j 2[ i iˆ2 M i and any trt ij d 0 j p j t, s L Rd 0 x p x ti, s L : On the other hand, because M i 6ˆb, by Lea 3.2, for each incoplete tie interval t i1, t i ], there is a task j 2 M i, which is processed on this interval during t i t i1 tie units. Hence, denoting the total length of all coplete tie intervals by c, we obtain X j2[ i iˆ2 M i p j r X j2[ i iˆ2 M i hp j p j t ij, s L irc wr X Therefore, there exists a task q 2[ i iˆ2 M i such that C q s r c w c w p q, s : j2[ i iˆ2 M i c w p j p j, s : 13 Without loss of generality we can assue that p q c w, s rp q t iq, s L, because otherwise Eq. (13) iplies that there exists a task r 2[ i iˆ2 M i such that p r c w, s > p r t ir, s L, and since p r c w, s > 0, c w We have C r s r c w p r, s L ax s rc q s d 0 q rc w d q 0 rc w Hence, G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± : p x ti, s L d x 0 : c w p q, s d q 0 rc w p q t iq, s L L ax s L ˆ t i p x ti, s L d x 0 ˆ c w p x ti, s L d x 0 ˆ c w 1 w p x ti, s L d x 0 1 R w L ax s : Let t i 0 be the largest t i for which interval t i1, t i Š is incoplete. By Lea 3.1 jm i j >, and since jm i 0j <, according to Lea 3.2 there exist tasks j 0 2 M i 0 and j 2 M i such that j 0 4j: Therefore, Lea 3.2 iplies that lrw p in, and

16 398 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383±399 L ax s L L ax s R 1 1 wr l p in : In order to show that Eq. (12) is achievable, consider the graph depicted in Fig. 5. Nodes in the rst (top) row do not precede any other nodes and the corresponding tasks have the sae due date equals one. Tasks in each subsequent row have due date of the previous row plus one. All tasks have the sae processing tie equals one. It is easy to see that L ax s L ˆ l 1 1 l 2: In the optial schedule s the tasks fro the rst row are processed in parallel with the chain of l 1 tasks preceding all tasks fro the last (botto) row. Since L ax s ˆl2, Eq. (12) is an equality for any value of l. 4. Conclusions This paper presents the perforance guarantees for the non-preeptive and preeptive versions of the Brucker±Garey±Johnson algorith. The Brucker±Garey±Johnson algorith is a straightforward generalization of the classical Hu's algorith. The presented results coplient the results on the worst-case perforance of Hu's algorith (Singh and Zinder, 2000) and its preeptive counterpart known as the Muntz±Co an algorith (La and Sethi, 1977). Each perforance guarantee presented in this paper is in the for of an upper bound on the deviation of the criterion value fro its optiu. The bounds are expressed in ters of the nuber of processors and paraeters characterizing the partially ordered set of tasks, and therefore, establish a relationship between these paraeters and the perforance of the algorith. Another feature of the presented bounds is their achievability for arbitrary large instances of the respective probles. The presented results will be copliented by the Fig. 5. The partially ordered set of tasks considered in Theore 3.1.

17 G. Singh, Y. Zinder / Intl. Trans. in Op. Res. 7 (2000) 383± coputational experients which constitute a part of the ongoing research. The design of approxiation algoriths with better than known perforance guarantees also reains an open proble for both akespan (Schuuran and Woeginger, 1999) and axiu lateness probles. References Brucker, P., Garey, M.R., Johnson, D.S., Scheduling equal-length tasks under tree-like precedence constraints to iniise axiu lateness. Math. Oper. Res. 2, 275±284. Hu, T.C., Parallel sequencing and assebly line probles. Operations Research 9, 841±848. La, S., Sethi, R., Worst case analysis of two scheduling algoriths. SIAM J. Coput. 6, 518±536. Lawler, E.L., Preeptive scheduling of precedence-constrained jobs on parallel achines. In: Dester, M.A.H, Lenstra, J.K., Rinnooy Kan, A.H.G. (Eds.), Deterinistic and Stochastic Scheduling, pp. 101±123. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shoys, D.B., Sequencing and scheduling: algoriths and coplexity. In: Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, P.H. (Eds.), Logistics of Production and inventory. Elsevier, Asterda. McNaughton, R., Scheduling with deadline and loss functions. Manageent Sci. 6, 1±12. Parker, R.G., Deterinistic Scheduling Theory. Chapan and Hall, New York. Schuuran, P., Woeginger, G.J., Polynoial tie approxiation algoriths for achine scheduling: ten open probles. J. Sched. 2, 203±213. Singh, G., Zinder, Y., Worst-case perforance of two critical path type algoriths. Asia Paci c J. Oper. Res. 17(1) (to appear). Ullan, J.D., NP-Coplete scheduling probles. J. Coput. Syste Sci. 10, 384±393. Zinder, Y., Roper, D., An iterative algorith for scheduling unit-tie operations with precedence constraints to iniise the axiu lateness. Annals of Operations Research 81, 321±340.