On-Line Load Balancing of Temporary Tasks on Identical Machines

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1 On-Line Load Balancing of Teporary Tasks on Identical Machines Yossi Azar Tel-Aviv Univ. Leah Epstein y Tel-Aviv Univ. Abstract We prove an exact lower bound of 2 1 on the copetitive ratio of any deterinistic algorith for load balancing of teporary tasks on identical achines. We also show a lower bound of 2 1 for randoized algoriths for sall and 2 2 for general. If in addition, we +1 restrict the sequence to polynoial length, then the lower log log bound for randoized algoriths becoes 2 O( log ) for general. 1. Introduction We consider the proble of non-preeptive on-line load balancing of tasks on identical achines. Tasks (jobs) arrive at arbitrary ties, where each task has a weight and a duration. A task has to be assigned upon its arrival to exactly one of the achines, thereby increasing the load on this achine by its weight for the duration of the task. The duration of each task becoes known only upon its terination (this is called teporary tasks of unknown durations). Once a task has been assigned to a achine it cannot be reassigned to another achine. The goal is to iniize the axiu load over achines and tie. The proble of scheduling tasks on identical achines was first introduced by Graha [11, 12]. He gave a greedy algorith List Scheduling which is 2 1 copetitive, where is the nuber of achines. The upper bound was proved for peranent tasks, i.e., tasks that start at arbitrary ties but continue forever. Nevertheless, his 2 1 analysis of the upper bound holds also for teporary tasks. In this paper we show that his algorith is optial by proving a atching lower bound. We show a lower bound of on the copetitive ratio of any deterinistic on-line 2 1 Dept. of Coputer Science, Tel-Aviv University. Research supported in part by Allon Fellowship and by a grant fro the Israel Science Foundation. azar@ath.tau.ac.il y Dept. of Coputer Science, Tel-Aviv University. lea@ath.tau.ac.il algorith for load balancing of teporary tasks on identical achines. We also show a lower bound of on the copetitive ratio of any randoized on-line algorith for the proble. In fact, for = 2; 3; 4 we can iprove the lower bound to 2 1, which iplies that List Scheduling is also optial in these cases. The randoized lower bound for general requires a sequence of tasks of super-polynoial length in. If we restrict the sequence to have a polynoial length we prove a lower bound of 2 O( ) for any randoized algorith. log log log Recall that Graha [11, 12] considered only peranent tasks. He showed that the greedy algorith List Scheduling does not perfor better than 2 1. For = 2; 3 the algorith is optial [9]. However, the algorith of Graha is not optial (for all 4) [1, 8]. Bartal et al. [5] were the first to show an algorith whose copetitive ratio is strictly below c < 2 (for all ). More precisely, their algorith achieves a copetitive ratio of Later, the algorith was generalized by Karger, Phillips and Torng [13] to yield an upper bound of 1:945. Very recently, Albers [1] designed 1:923 copetitive algorith and iproved the lower bound to 1:852 (the previous lower bound for peranent tasks was 1:837 [6]). The best lower bound known for randoized algoriths is 1:582 (for large ) [7, 14]. For = 2 the randoized copetitive ratio is precisely 4=3 [5]. We show that in contrast to peranent tasks, the siple algorith of Graha turns out to be optial for teporary tasks. Moreover, even randoization cannot reduce the copetitive ratio below 2 o(1). Note that for = 2 our tight randoized lower bound is 3=2. We also prove the sae lower bound for the known duration case. This is in contrast to the copetitive ratio for peranent jobs that is 4=3. To prove our randoized lower bound we introduce a new technique that converts a lower bound for deterinistic algoriths to a lower bound for randoized algoriths. More precisely, we show that a lower bound for deterinistic algoriths that aintains a fixed value for the optial assignent is a lower bound for randoized algoriths. The proble of on-line load balancing of teporary

2 tasks was introduced by Azar, Broder and Karlin [2]. They studied the restricted assignent case, i.e. each task can be assigned only to a achine in a subset which ay depend on the task. They showed an ( p ) lower bound in contrast to the (log ) copetitive ratio for peranent tasks [4]. A atching upper bound was given in [3]. Load balancing of teporary tasks was also studied for the related achines odel. In this odel the increase of the load on a achine is the ratio of the weight of the task and the speed of that achine. An algorith which is 2 copetitive and a lower bound of 3 o(1) were given by [3]. 2. Notations We denote the input sequence by = 1 ; :::; r. Each event i is an arrival or a departure of a job (task). We view as a sequence of ties, the tie i is the oent after the i th event happened. We denote the weight of job j by w j, its arrival tie by a j and its departure tie (which is unknown until it departs) by d j. An on-line algorith has to assign a job upon its arrival without knowing the future jobs and the durations of jobs that have not departed yet. We copare the perforance of on-line algoriths and the optial off-line algorith that knows the sequence of jobs and their durations in advance. Let J i = fjja j i < d j g be the active jobs at tie i. For a given algorith A (on-line or off-line) let A j be the achine on which job j is assigned. Let l A k (i) = X fjja j=k;j2jig w j be the load on achine k at tie i, which is the su of weights of all jobs assigned to k, and active at this tie. The cost of an algorith A is the axiu load ever achieved by any of the achines, i.e., C A = ax i;k lk A (i). The copetitive ratio of A is r if for any sequence C A rc opt where C opt is the cost of the optial off-line algorith. 3. Lower bounds for deterinistic algoriths We start with a siple lower bound of The lower bound holds for the copetitive ratio of any deterinistic algorith even if the optial value is fixed and known in advance. Later we show the tight 2 1 lower bound. Theore 3.1 Any deterinistic on-line algorith for load balancing of teporary tasks has a copetitive ratio of at least 2 2. This is true even if the optial value is +1 known in advance (and is + 1). Proof: We consider the following sequence. First arrive 2 unit jobs (w j = 1). Since there are achines, there is at least one achine (say x) with a load of at least, (see figure 1). Now all jobs depart, except jobs on this achine, (figure 2). Figure 1. The on-line algorith after the arrival of the unit jobs Figure 2. The on-line algorith after the departure of the unit jobs except jobs on one achine Now jobs of weight arrive. Since achine x has load, the on-line has to assign two jobs of weight on one achine, or to assign one job of weight on the achine x. In both cases, the axiu load of the on-line is at least 2. The off-line distributes the large jobs and the unit jobs that reained fro the first phase evenly on the achines, and thus has a load of + 1. The other ( 1) unit jobs of the first phase are also distributed evenly on the achines. The ratio between the costs of 2 the two algoriths is at least +1 =

3 The above lower bound can be iproved as follows: Theore 3.2 Any deterinistic on-line algorith for load balancing of teporary tasks has the copetitive ratio of at least 2 1. Proof: We consider the following sequence. First ( 1) unit jobs arrive (w j = 1). Since there are achines, there is at least one achine (say x) with load at least 1. Then all jobs depart, except 1 jobs on this achine. Next 1 jobs of load 1 arrive. There are two possible cases: 1. The on-line algorith keeps at least one epty achine. In this case, the load of the on-line algorith on soe achine is at least 2( 1) since at least two jobs of load 1 were assigned to one achine, or a job of load 1 was assigned to the achine x, (see figure 3). -1 Figure 4. The off-line algorith after the arrival of the 1 jobs of load 1 Figure 3. The on-line state if it assigns two jobs of weight 1 on one achine At the first phase, the off-line assigns the 1 unit jobs which will not depart on one achine and distributes the ( 1) 2 unit jobs that will depart evenly on the other 1 achines. Then it assigns one job of weight 1 on each of the other achines (see figure 4), to have the axiu load of 1. In this case, the ratio between the costs of the on-line and the off-line algoriths is at least 2( 1)=( 1) = The on-line algorith does not keep an epty achine, i.e., there is now one job of weight 1 on -1 each achine except x, on which there are 1 units jobs. Now one additional (and final) job of weight arrives. The on-line ust assign it on one of the achines (figure 5), which results in a load of 2 1. We show that the off-line can assign the jobs, having a axiu load of. At the first phase, the unit jobs that do not depart, are assigned each to a different achine, the other jobs are distributed so that the load on each achine is exactly 1. At the second phase only 1 unit jobs, each on a different achine, have reained, and the jobs of load 1 are added to those achines yielding load of and keeping one achine epty. At last the job of load is assigned to the epty achine (figure 6). In this case, the copetitive ratio is at least (2 1)= = 2 1=. In both possible cases, the copetitive ratio is at least 2 1, and thus any on-line algorith has at least this ratio. 4. Lower bounds for randoized algoriths In this section we prove lower bounds for randoized on-line algoriths. We start by proving a general theore that converts lower bounds for deterinistic algoriths with a fixed value for the optial load to general lower bounds for randoized algoriths. We represent any lower bound for a deterinistic algorith by a tree. Each path in the tree is one possible lower bound sequence. Each node in the tree is a subsequence, and a child node of a node is one possible way to continue the sequence (see figure 7). The

4 -1-1 Figure 5. The on-line assigns the job of weight above a job of weight 1 Figure 6. The off-line assignent if the job of weight arrives size jt j of a tree T is defined to be the nuber of leaves in T (the nuber of possible sequences). We consider both the unknown duration case and the known duration case. In the first case the duration of a job is known only when it departs where in the second one the duration of a job is known upon its arrival. Theore 4.1 Let r 1 (r 2, respectively) be a deterinistic lower bound for teporary jobs with unknown (known, respectively) durations, when the optial value of the load is known in advance. Then, r 1 (r 2, respectively) is also a lower bound for randoized algoriths for teporary jobs with unknown (known, respectively) durations. Proof: Consider a lower bound tree T for deterinistic algoriths with a fixed optial load which is known in advance. We show how to convert it into a lower bound for randoized algoriths. A lower bound for teporary jobs with unknown durations, is converted into a lower bound for randoized unknown durations, and a lower bound for known durations is converted into a lower bound for randoized known durations. We first slightly odify the lower bound tree as follows: each possible sequence 2 T, is followed by the events that all the existing jobs depart. This can be done for unknown durations and also for known durations. Note that the new tree T satisfies jt j = jt j. Next we recall the adaptation of Yao s theore for on-line algoriths. It states that a lower bound for the copetitive ratio of deterinistic algoriths on any distribution on the input is also a lower bound for randoized algoriths and is given by E(C on =C opt ). The ain idea of the proof is to construct sequences in which on one hand the new value Copt is the sae as the known optial value C opt of the lower bound of the original tree T and on the other hand with high probability the new value of Con is also the sae as the value C on for T. To construct the lower bound we choose unifor at rando a leaf of the tree T that corresponds to a sequence. Define this short sequence as a segent. Repeat the choice of segents jt jk ties, and concatenate the sequences to one long sequence. This defines a distribution on the set of possible long sequences. Since the off-line costs of all possible segents are the sae, the off-line cost of every resulting sequence is C opt as well. For any deterinistic algorith, there exists a leaf in T that has the on-line cost C on. With probability at least 1 jt, the cost of the on-line algorith on a spe- j cific segent (and thus for the whole sequence) is C on. The probability that the cost C on would not be achieved in one segent is at ost (1 1 jt j )jt jk e k and thus with probability at least 1 e k the copetitive ratio is C on =C opt, where and otherwise it is at least 1. We calculate E C on Copt Copt,and Con, are respectively, the off-line and on-line costs of the long sequence. E C on C opt Since this is true for every k, (1 e k ) C on C opt + e k : E C on C opt C on C opt ;

5 leave -1 jobs on one achine (-1) unit jobs arrive o o o ((-1)) possibilities We can extend this lower bound for the known duration case, i.e., the duration of each job is known upon its arrival. Theore 4.4 Any randoized on-line algorith for load balancing of teporary tasks on two achines has a copetitive ratio of at least = 3 even if the durations of 2 the jobs are known upon its arrival. -1 jobs of weight -1 arrive two were placed on one achine (-1) ain possibilities the load on each achine is -1, a job of weight arrives cases can be reduced into two Figure 7. The tree for the lower bound in Theore 3.2 which is exactly the copetitive ratio of the lower bound of the tree T. Corollary 4.2 Any randoized on-line algorith for load balancing of teporary tasks has a copetitive ratio of at least Proof: The proof follows fro Theores 3.1 and 4.1. In this case, there are ( 2 ) = 2 different possibilities for the 2 unit jobs to be placed, and there are 2 leaves in the lower bound tree. We now iprove the lower bound for sall nubers of achines. Theore 4.3 Any randoized on-line algorith for load balancing of teporary tasks on two achines, has a copetitive ratio of at least = 3 2. Proof: We show a lower bound for deterinistic algoriths, for which the value of the optial load is 2. Theore 4.1 iplies the sae lower bound for randoized algoriths. We consider the following sequence. First three unit jobs arrive (w j = 1). The possible behaviors of the algorith can be divided into two cases. 1. All jobs are on one achine. In this case C on = 3 and thus the copetitive ratio is at least 1:5. 2. There is at least one job on each achine. In this case, there is one achine with two jobs. We pick one of these jobs and let it depart. Thus, we are left with one job on each achine. Now a final job of weight 2 arrives, and the current load on the achine on which it is assigned is 3. Again the copetitive ratio is 1:5. Proof: We again show a lower bound for deterinistic algoriths with fixed optial load which is 2. The randoized lower bound follows fro Theore 4.1. We consider the following sequence. At tie a unit job of duration 8 arrives. At tie 1 a unit job of duration 3 arrives. Now there are several cases. If the two jobs are on different achines, a job of weight 2 and duration 1 arrives at tie 2. Between tie 2 and tie 3 all the three jobs are present and the on-line load is 3, which yields the copetitive ratio of 3=2. Otherwise, the two jobs are on one achine. In this case a third unit job of duration 5 arrives at tie 2. If it is assigned to the sae achine, the on-line load is already 3. If it is assigned to the other achine, we wait to tie 5 that one unit job has already departed, and we have one unit job on each achine. At that tie a final job of weight 2 and duration 1 arrives, and the on-line load is 3. It is easy to check that the optial load in each case is 2 and thus the copetitive ratio is 3=2. We can extend the lower bound of Theore 4.3 for 3 and 4 achines. Theore 4.5 Any randoized on-line algorith for load balancing of teporary tasks on = 3; 4 achines has a copetitive ratio of at least 2 1, i.e., 5=3 for 3 achines, and 7=4 for 4 achines. Proof: We show lower bounds for deterinistic algoriths with fixed optial load as before. Let = 3. Consider the following sequence that aintains optial off-line load of 3. First eight unit jobs arrive. 1. If there is a achine with at least five jobs then the on-line load is at least 5. Thus we can assue that there are at ost four jobs on each achine. 2. If there are at least two jobs on each achine then we continue the sequence by a departure of two jobs so that exactly two jobs reain on each achine. Now one final job of size 3 arrives and the on-line load is If there is at least one achine with at ost one job then there ust be at least three jobs on each of the other two achines, since there are at ost four jobs on each achine. Now two jobs depart, so that there is one epty achine, and two achines, each with three jobs. Now one job of size 2 arrives.

6 If it is assigned to one of the non-epty achines then the on-line load is 5. If it is assigned to the epty achine then two unit jobs, one fro each of the other two achine, depart. This results in a load 2 on all three achines. One final job of weight 3 causes the on-line the load of 5. In each case the optial algorith can aintain axiu load 3 and thus the copetitive ratio is at least 5 3. We use the sae idea to prove the lower bound for = 4. Consider the following sequence that aintains an optial off-line load of 4. First 16 unit jobs arrive. There are several cases: 1. If there is a achine with at least seven jobs then the on-line load is at least If there are at least three jobs on each achine then four jobs depart, so that there are exactly three jobs on each achine. Then a final job of weight 4 arrives, which causes the load of 7 to the on-line. 3. If there is at least one job on each achine then we continue as follows. First note that there is at least one achine with four jobs or ore. Four jobs on this achine and one job on each of the other achines reain and all other jobs depart. Now three jobs of weight 3 arrive. If two jobs of weight 3 assigned to one achine, or a job of weight 3 assigned to the achine with load 4 then the on-line load is at least 7. Otherwise, there are three achines with the sae configuration: two jobs, one of which has weight 3, and the other has weight 1. The fourth achine contains four unit jobs. Now four unit job depart, one fro each achine, which results in load 3 to all achines. A final job of weight 4 arrives and creates on-line load of If there is one epty achine and there are at ost six jobs on each achine then there are three achines each with at least four jobs. Next, four jobs depart so that there are exactly four jobs on each of the three achines. Now one job of weight 3 arrives. It should be assigned to the epty achines in order not to create load 7. Now three unit jobs depart, one fro each of the achines with load 4. Consequently, every achine has load 3. Finally, one job of weight 4 arrives and creates load 7 for the on-line algorith. One can easily see that the optial off-line algorith can aintain load 4 in all the cases and thus the copetitive ratio is at least Sequences of polynoial length Theore 4.1 provides lower bounds that require jt j k large sequences. To get short sequences, we use the sae ethods, but we exaine the tree ore carefully. We first consider a base sequence. Let h be an integer h. 1. h unit jobs (w j = 1) arrive. 2. h unit jobs chosen uniforly at rando depart. Thus a rando set S of exactly unit jobs h reains. Note that all possible sets of reaining jobs are equally likely. 3. Now jobs of weight h arrive. 4. All jobs depart. The off-line can distribute evenly the unit jobs of S and the larger jobs on the achines. Thus it aintains load of h + 1 at the end of phase 3 and can easily aintain a load of h at the end of phase 1. Let us calculate the expected axiu load of the on-line algorith. In the first step, there are h unit jobs, thus there is (at least) one achine that contains at least h jobs. Denote the set of the first h jobs on that achine as T. If S T the load for the on-line algorith is at least 2h, at the end of phase 3. In this case, the ratio between the axiu load of the on-line and the off-line is 2h h+1 = 2 2. Note that S is a rando variable h+1 and T is a fixed set. Clearly, the probability of the event h h S T is h = h. We use this sequence to prove the polynoial length lower bound for randoized algoriths. Theore 5.1 Any randoized on-line algorith for load balancing of teporary tasks has a copetitive ratio of at log log least 2 O( ) even when the input sequence is of log polynoial length in. Proof: We choose h = ( log log ) such that 2hh and repeat the base sequence k ties. Let us calculate the probability that S never contains T. The probability that S does not contain T in one base sequence is 1 h h h h log (h h)!! = 1 ( h)! (h)! 1 1 h h h h 1 1 h h h 1 1 2h h : 2

7 The probability that S never contains T is at ost (1 1 2h h )k (1 1 )k e k : In this case the ratio is at least 1, otherwise the ratio is 2h h+1 = 2 2. This iplies as before that h+1 log log : E Con C opt 2 O log Note that the length of the sequence is polynoial in as required. 6. Concluding rearks We proved 2 o(1) lower bounds on the copetitive ratio of load balancing of teporary tasks. Note that there is a sall gap between the randoized lower bound and the optial deterinistic one. One would like to know the exact bound for randoized algoriths or at least if randoization helps at all to reduce the copetitive ratio in these probles. It follows fro our results that randoization ay help slightly only for 5. It is open if knowing the durations of the tasks can help in reducing the copetitive ratio strictly below 2 both for deterinistic and randoized algoriths. [7] B. Chen, A. van Vliet, and G. Woeginger. A lower bound for randoized on-line scheduling algoriths. Inforation Processing Letters, 51: , [8] B. Chen, A. van Vliet, and G. Woeginger. New lower and upper bounds for on-line scheduling. Operations Research Letters, 16:221 23, [9] U. Faigle, W. Kern, and G. Turan. On the perforance of online algoriths for partition probles. Acta Cybernetica, 9:17 119, [1] G. Galabos and G. Woeginger. An on-line scheduling heuristic with better worst case ratio than graha s list scheduling. Sia Journal on Coputing, 22(2): , [11] R. Graha. Bounds for certain ultiprocessor anoalies. Bell Syste Technical Journal, 45: , [12] R. Graha. Bounds on ultiprocessing tiing anoalies. SIAM J. Appl. Math, 17: , [13] D. Karger, S. Phillips, and E. Torng. A better algorith for an ancient scheduling proble. In Proc. 5th ACM-SIAM Syp. on Discrete Algoriths, [14] J. Sgall. On-line scheduling on parallel achines. Technical Report Technical Report CMU-CS , Carnegie- Mellon University, Pittsburgh, PA, USA, Acknowledgents We would like to thank Allan Borodin for any helpful discussions. References [1] S. Albers. Better bounds for on-line scheduling. In Proc. 29th ACM Syp. on Theory of Coputing, To appear. [2] Y. Azar, A. Broder, and A. Karlin. On-line load balancing. In Proc. 33rd IEEE Syp. on Found. of Cop. Science, pages , Oct [3] Y. Azar, B. Kalyanasundara, S. Plotkin, K. Pruhs, and O. Waarts. On-line load balancing of teporary tasks. In Proc. Workshop on Algoriths and Data Structures, pages , Aug [4] Y. Azar, J. Naor, and R. Ro. The copetitiveness of online assignent. In Proc. 3rd ACM-SIAM Syp. on Discrete Algoriths, pages 23 21, [5] Y. Bartal, A. Fiat, H. Karloff, and R. Vohra. New algoriths for an ancient scheduling proble. In Proc. 24th ACM Syposiu on Theory of Algoriths, pages 51 58, To appear in Journal of Coputer and Syste Sciences. [6] Y. Bartal, H. Karloff, and Y. Rabani. A better lower bound for on-line scheduling. Inforation Processing Letters, 5: , 1994.