The Power of Reordering for Online Minimum Makespan Scheduling

Abstract

In the classic minimum makespan scheduling problem, we are given an input sequence of jobs with processing times. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we do not require that each arriving job has to be assigned immediately to one of the machines. A reordering buffer with limited storage capacity can be used to reorder the input sequence in a restricted fashion so as to schedule the jobs with a smaller makespan. This is a natural extension of lookahead. We present an extensive study of the power and limits of online reordering for minimum makespan scheduling. As main result, we give, for m identical machines, tight and, in comparison to the problem without reordering, much improved bounds on the competitive ratio for minimum makespan scheduling with reordering buffers. Depending on m, the achieved competitive ratio lies between 4/3 and 1.4659. This optimal ratio is achieved with a buffer of size Theta(m). We show that larger buffer sizes do not result in an additional advantage and that a buffer of size Omega(m) is necessary to achieve this competitive ratio. Further, we present several algorithms for different buffer sizes. Among others, we introduce, for every buffer size k isin [1, (m+ 1)/2], a (2 middot 1/(m middot k+ 1))-competitive algorithm, which nicely generalizes the well-known result of Graham. For m uniformly related machines, we give a scheduling algorithm that achieves a competitive ratio of 2 with a reordering buffer of size m. Considering that the best knowncompetitive ratio for uniformly related machines without reordering is 5.828, this result emphasizes the power of online reordering further more.

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The Power of Reordering
for Online Minimum Makespan Scheduling∗
Matthias Englert
Department of Computer Science
RWTH Aachen University, Germany
englert@cs.rwth-aachen.de
Deniz¨Ozmen
Department of Computer Science
RWTH Aachen University, Germany
deniz.oezmen@rwth-aachen.de
Matthias Westermann
Department of Computer Science
RWTH Aachen University, Germany
marsu@cs.rwth-aachen.de
Abstract
In the classic minimum makespan scheduling problem, we
are given an input sequence of jobs with processing times. A
scheduling algorithm has to assign the jobs to m parallel ma-
chines. The objective is to minimize the makespan, which is
the time it takes until all jobs are processed. In this paper, we
consider online scheduling algorithms without preemption.
However, we do not require that each arriving job has to be
assigned immediately to one of the machines. A reordering
buffer with limited storage capacity can be used to reorder
the input sequence in a restricted fashion so as to schedule
the jobs with a smaller makespan. This is a natural extension
of lookahead.
We present an extensive study of the power and limits of
online reordering for minimum makespan scheduling. As
main result, we give, for m identical machines, tight and,
in comparison to the problem without reordering, much
improved bounds on the competitive ratio for minimum
makespan scheduling with reordering buffers. Depending
on m, the achieved competitive ratio lies between 4/3 and
1.4659. This optimal ratio is achieved with a buffer of size
Θ(m). We show that larger buffer sizes do not result in an
additional advantage and that a buffer of size Ω(m) is nec-
essary to achieve this competitive ratio. Further, we present
several algorithms for different buffer sizes. Among oth-
ers, we introduce, for every buffer size k ∈ [1,(m+1)/2],
a (2−1/(m−k+1))-competitive algorithm, which nicely
generalizes the well-known result of Graham.
For m uniformly related machines, we give a scheduling
algorithm that achieves a competitive ratio of 2 with a re-
ordering buffer of size m. Considering that the best known
∗Supported by DFG grant WE 2842/1.
competitive ratio for uniformly related machines without re-
ordering is 5.828, this result emphasizes the power of online
reordering further more.
1. Introduction
In the classic minimum makespan scheduling problem, we
are given an input sequence of jobs with processing times.
A scheduling algorithm has to assign the jobs to m parallel
machines. The objective is to minimize the makespan, which
is the time it takes until all jobs are processed. This problem
isNP-hardinthestrongsense[18]. Inthispaper, weconsider
online scheduling algorithms without preemption. An online
algorithm does not have knowledge about the input sequence
in advance. Instead, it gets to know the input sequence job
by job without knowledge about the future.
Extensive work has been done to narrow the gap between
upper and lower bounds on the competitive ratio for online
minimum makespan scheduling. Increasingly sophisticated
algorithms and complex analyses were developed. Never-
theless, even for the most basic case of identical machines,
in which each job has the same processing time on every
machine, there is still a gap between the best known lower
and upper bounds on the competitive ratio of 1.880 [30] and
1.9201 [16], respectively.
Adding lookahead is common practice to improve the
quality of solutions for online problems. The impact of
lookahead has been studied for various problems, e.g., pag-
ing [26, 32], the list update problem [1], the k-server prob-
lem [8], and bin packing [22]. However, lookahead alone
is not sufficient to improve the quality of solutions for the
minimum makespan scheduling problem. The lookahead
2008 49th Annual IEEE Symposium on Foundations of Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
0272-5428/08 $25.00 © 2008 IEEE
DOI 10.1109/FOCS.2008.46 DOI 10.1109/FOCS.2008.46
603 6030272-5428/08 $25.00 © 2008 IEEE

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window can always be rendered useless by flooding it with
unimportant jobs having arbitrary small processing times.
However, for many problems, including minimum make-
span scheduling, it is reasonable to not only provide a
lookahead to a certain number of future jobs, but addi-
tionally to allow the algorithm to choose one of these jobs
for processing next and, therefore, to reorder the input se-
quence. The paradigm of online reordering is more pow-
erful than lookahead alone and has received a lot of atten-
tion [3, 4, 11, 12, 15]. It has been studied, e.g., by Albers [3]
and Feder et al. [15] for the problem of web caching.
We present an extensive study of the power and limits of
online reordering for minimum makespan scheduling. Over
a decade ago, Kellerer et al. [25] studied the impact of online
reordering for minimum makespan scheduling on two iden-
tical machines. To the best of our knowledge, these are the
only previously known results related to online reordering in
the context of minimum makespan scheduling.
In our model, a reordering buffer can be used to reorder
the input sequence of jobs in a restricted fashion. At each
point in time, the reordering buffer contains the first k jobs
of the input sequence that have not been assigned so far.
An online scheduling algorithm has to decide which job
to assign to which machine next. Upon its decision, the
corresponding job is removed from the buffer and assigned
to the corresponding machine, and thereafter the next job in
the input sequence takes its place.
As main result, we give, for m identical machines, tight
and, in comparison to the problem without reordering, much
improved bounds on the competitive ratio for minimum
makespan scheduling with reordering buffers. Depending
on m, the achieved competitive ratio lies between 4/3 and
1.4659. This optimal ratio is achieved with a buffer of size
Θ(m). We show that larger buffer sizes do not result in
an additional advantage and that a buffer of size Ω(m) is
necessary to achieve this competitive ratio.
More precisely, for m identical machines, we present the
following results.
• We prove a lower bound of rmon the competitive ra-
tio of this problem with m identical machines and a
reordering buffer whose size does not depend on the
input sequence. The precise value of rmis given in
Section 1.1. For example, r2= 4/3 and limm→∞rm=
LambertW−1(−1/e2)/(1 + LambertW−1(−1/e2)) ≈
1.4659.1
• We introduce a fairly simple scheduling algorithm for
m identical machines matching this lower bound with
a reordering buffer of size ?(1+2/rm)·m?+2 ≤ ?2.5·
m?+2.
1LambertW−1(−1/e2) is the smallest real solution to x·ex= −1/e2.
• We show a lower bound of 3/2 > rmon the competitive
ratio of this problem with m identical machines and a
reordering buffer of size at most ?m/2?. This lower
bound improves to 1+1/√2 ≈ 1.7071 for a reordering
buffer of size at most ?m/8? if m ≥ 8.
For m uniformly related machines, i.e., for m machines
with different speeds, we give a scheduling algorithm that
achieves a competitive ratio of 2 with a reordering buffer of
size m. Our algorithm and analysis are extremely simple.
Considering that the best known lower and upper bounds on
the competitive ratio for uniformly related machines without
reordering are 2.438 and 5.828 [9], respectively, this result
emphasizes the power of online reordering further more.
In addition, we present, for m identical machines, several
algorithms for different buffer sizes. In particular, we show
that buffers of size ?(2/3+2/(1+ln3))·m?+1 ≈ 1.6197·
m+1 and m+1 are sufficient to achieve the competitive
ratios 3/2 and 1+rm/2 ≤ 1.733, respectively. For every
buffer size k ∈ [1,(m+1)/2], we introduce a (2−1/(m−
k+1))-competitive algorithm, which nicely generalizes the
well-known result of Graham [20].
In the following table, we compare, for m identical ma-
chines, the competitive ratios of our algorithm and the best
known lower and upper bounds on the competitive ratio for
the case that reordering is not allowed.
mour results
reordering buffer
1.3333
1.3636
1.375
1.4659
lower bounds
no reordering
1.5
1.6667
1.7321
1.8800
upper bounds
no reordering
1.5
1.6667
1.7333
1.9201
2
3
4
∞
Note that our results are tight, i.e., we show matching lower
and upper bounds, in contrast to the problem without re-
ordering for which there are still gaps between the lower and
upper bounds.
1.1. Notations and the value of rm
The processing time or size of a job J is denoted by p(J).
The load L(M) of a machine M is defined as the sum of the
sizes of the jobs assigned to machine M. The total scheduled
load T is defined as the sum of the load of all machines. The
m machines are denoted by M0,...,Mm−1.
We frequently make use of the weight wiof a machine Mi
which is defined as wi:= min{rm/m,(rm−1)/i} or equiva-
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1.32
1.34
1.36
1.38
1.4
1.42
1.44
1.46
5 1015 20 2530
rm
m
Figure 1. The values of rmfor 2 ≤ m ≤ 30.
lently
wi:=
?rm
m,
rm−1
i
if 0 ≤ i <rm−1
ifrm−1
rm
rm
·m
,
·m ≤ i ≤ m−1
.
Now, rm is the smallest positive solution to ∑m−1
?m−m/rm?·rm/m+(rm−1)·∑m−1
ensure that the weights of all machines sum up to 1.
Unfortunately, we do not know a closed-form formula for
rm, but the value can be easily calculated for any given m.
The values of rmfor 2 ≤ m ≤ 30 are depicted in Figure 1.
We can derive limm→∞rm= LambertW−1(−1/e2)/(1 +
LambertW−1(−1/e2)), by using limm→∞?m−m/x?·x/m=
(x−1) and limm→∞∑m−1
rmis monotonically increasing with m. This follows from the
fact that ?m−m/x?·x/m+(x−1)·∑m−1
ically increases with x and monotonically decreases with m.
A more detailed argument will be given in a full version.
i=0wi=
i=?m−m/rm?1/i = 1, i.e., we
i=?m−m/x?1/i=−ln(1−1/x). Further,
i=?m−m/x?1/i monoton-
1.2. Previous work
Kellerer et al. [25] present, for two identical machines, an
algorithm that achieves a competitive ratio of 4/3 with a
reordering buffer of size 2, i.e., the smallest buffer size al-
lowing reordering. They also show that this bound is tight.
Although, we are not aware of any other results related to on-
line minimum makespan scheduling with reordering buffers,
the paradigm of online reordering has been studied for sev-
eral different scheduling problems.
In [12], a reordering buffer of size k is used to minimize
the sum of the distances between consecutive elements in
a sequence of points from a metric space. A randomized
online algorithm is presented that achieves a competitive
ratio of O(log2k ·logn), where n denotes the number of
distinct points in the metric space. A possible application is
the acceleration of rendering in computer graphics [27].
Alborzi et al. [4] consider the similar k-client problem.
In this problem, we are given k clients, each of which gen-
erates an input sequence of requests for service in a metric
space. At each point in time, a scheduling algorithm has to
decide which client’s request to serve next. They present a
deterministic online algorithm that achieves a competitive
ratio of 2k−1.
Web caching with request reordering extends the classic
paging model by allowing reordering of requests under the
constraint that a request is delayed by no longer than a pre-
determined number of time steps (see, e.g., [3, 15]). Albers
[3] presents a deterministic algorithm that achieves an op-
timal competitive ratio of k+1, where k denotes the cache
size. Feder et al. [15] introduce a randomized algorithm
that achieves an asymptotically optimal competitive ratio of
Θ(logk).
Divakaran and Saks [11] consider an online scheduling
problem with job set-ups. Each job has a release time, a
processing time, and a type. Processing a job takes its pro-
cessing time and in addition a job-type specific set-up time.
However, this set-up time is not needed if the previously
processed job was of the same type. The objective is to
minimize the maximum flow time. They present an O(1)-
competitive algorithm for this problem.
Minimum makespan scheduling has been extensively
studied. See the survey by Pruhs, Sgall, and Torng [29]
for an overview.
For m identical machines, Graham [20] shows that the
greedy algorithm, which schedules each arriving job on a ma-
chine with minimum load, is (2−1/m)-competitive. This is
optimal for m ≤ 3 [14]. However, better bounds are known
for larger m. For m = 4, the best known lower and up-
per bounds on the competitive ratio are 1.7321 [31] and
1.7333 [10], respectively. For large m, the best known lower
bound on the competitive ratio was improved from 1.837 [7]
over 1.852 [2] and 1.854 [19] to 1.880 [30]. The first upper
bound on the competitive ratio below 2 was 1.986 [6]. This
upper bound was improved to 1.945 [24], then to 1.923 [2],
and finally to 1.9201 [16].
For uniformly related machines, Aspnes et al. [5] present
the first algorithm that achieves a constant competitive ratio.
Due to Berman, Charikar and Karpinski [9], the best known
lower and upper bounds on the competitive ratio are 2.438
and 5.828, respectively.
In the semi-online variant of the problem, the jobs arrive
in decreasing order of their processing time. To the best of
our knowledge, only the greedy LPT algorithm, which as-
signs each job to a machine with minimum load, was consid-
ered in this setting. For m identical machines, Graham [21]
shows that the LPT algorithm achieves a competitive ratio
of 4/3−1/(3m). For related machines, the LPT algorithm
achieves a competitive ratio of 1.66 and a lower bound of
1.52 on its competitive ratio is known [17]. A detailed and
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tight analysis for two related machines is given by Mireault,
Orlin, and Vohra [28] and Epstein and Favrholdt [13].
2. The algorithm for uniformly related ma-
chines
We start with the algorithm for uniformly related machines,
sincethissimplealgorithmoverviewswellthebasicstructure
of all our algorithms. They consist of two different phases.
Initially, the first k−1 jobs are stored in the reordering buffer
where k denotes the buffer size. Then, the algorithms start
with the iteration phase. As long as new jobs arrive, this
phase is iterated. After all jobs have arrived, the algorithms
schedule the remaining jobs in the final phase.
A generic version of the final phase is to schedule the
k−1 jobs remaining in the buffer optimally on the machines.
However, since the minimum makespan scheduling problem
is NP-hard, it is not known how to perform this generic final
phase efficiently. Although efficiency is usually not consid-
ered for online algorithms, we provide, for all our algorithms
for identical machines, very simple and efficient alternatives
to the generic approach without deteriorating the competitive
ratio. For uniformly related machines, we replace the generic
final phase by the PTAS due to Hochbaum and Shmoys [23].
This deteriorates the competitive ratio from 2 to 2+ε for
any ε > 0.
The algorithm for scheduling a sequence of jobs on m
uniformly related machines M0,...,Mm−1uses a reordering
buffer of size m. For each 0 ≤ i ≤ m−1, let αidenote the
speed of machineMi, i.e., ifloadT isassigned tomachineMi
then the completion time of machine Miis T/αi. Suppose
that α0≤ ... ≤ αm−1. The objective is to minimize the
makespan, i.e., the maximum completion time. The iteration
and final phase are defined as follows.
• Iteration phase: When a new job arrives, store this
new job in the reordering buffer, and remove a job J of
smallest size from the buffer. Let Mibe a machine with
load at most
αi
∑m−1
j=0αj
·(T +m· p(J))− p(J) ,
where T denotes the total scheduled load. (Obviously,
there always exists such a machine.) Then, schedule
job J on machine Mi, i.e., the total scheduled load T is
increased by p(J).
• Final phase: The m−1 remaining jobs in the reorder-
ing buffer are virtually scheduled with the PTAS from
Hochbaum and Shmoys [23] on m empty machines
M?
chine M?
approximation is achieved. Then, for each 0 ≤ i ≤
0,...,M?
m−1, where, for each 0 ≤ i ≤ m − 1, ma-
ihas speed αi. With this scheme an (1+ε)-
m−1, schedule the jobs from M?
Mi.
ion the real machine
Theorem 1. For m uniformly related machines, our algo-
rithm achieves the competitive ratio 2+ε with a reordering
buffer of size m.
Proof. Fix an input sequence of jobs. Let OPT denote the
minimum makespan achieved by an optimal offline algo-
rithm.
At the end of the iteration phase, for each 0 ≤ i ≤ m−1,
the completion time of machine Miis at most
1
∑m−1
j=0αj
·(T +(m−1)· p(Ji)) ,
whereT denotesthetotalscheduledloadattheendoftheiter-
ation phase and Jidenotes the last job scheduled on machine
Miin the iteration phase. Obviously, for each 0 ≤ i ≤ m−1,
1
∑m−1
j=0αj
·(T +(m−1)· p(Ji)) ≤ OPT ,
since m−1 jobs are stored in the reordering buffer at the end
of the iteration phase and the size of each of these jobs is at
least p(Ji).
In the final phase, for each 0 ≤ i ≤ m−1, the completion
time of the machine M?
(1+ε)·OPT, due to the polynomial time approximation
scheme. As a consequence, the makespan of our algorithm
is at most (2+ε)·OPT.
iin the virtual schedule is at most
3. Lower bounds
In this section, we present lower bounds for m identical ma-
chines. As main result, we prove that no online algorithm
can achieve a competitive ratio less than rmwith a reordering
buffer whose size does not depend on the input sequence.
Further, we show that this general lower bound can be im-
proved to 3/2 > rmfor a reordering buffer of size at most
?m/2?, and to 1+1/√2 ≈ 1.7071 for a reordering buffer of
size at most ?m/8? if m ≥ 8.
Theorem 2. For m identical machines, no online algorithm
can achieve a competitive ratio less than rmwith a reorder-
ing buffer whose size does not depend on the input sequence.
Proof. Assume for contradiction that there exists an on-
line algorithm A that achieves a competitive ratio r < rm
with a reordering buffer of size k. Consider the follow-
ing input sequence. At first, (1/ε+k) jobs of size ε ar-
rive. Since only k of these jobs can be stored in the re-
ordering buffer, 1/ε of them have to be scheduled on ma-
chines. Let M0,...,Mm−1denote the m identical machines
with L(M0) ≥ ··· ≥ L(Mm−1). Then, there exists a machine
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Mjwith load at least wj, since otherwise, the total scheduled
load would be strictly less than ∑m−1
We distinguish two different cases.
i=0wi= 1.
• If wj= rm/m, no more jobs arrive. In the optimal
schedule, all jobs are evenly distributed between the
machines. Hence, the optimal makespan is at most
(1+k·ε)/m+ε. As a consequence, the competitive
ratio of A is at least
rm/m
(1+(k+m)·ε)/m=
which is strictly larger than r if ε is chosen appropriately
small.
rm
1+(k+m)·ε
,
• If wj= (rm−1)/j, (m− j) additional jobs of size 1/j
arrive. It is possible, to assign each of the (m− j)
additional jobs to a different machine and to evenly
distribute the remaining (1/ε+k) jobs between the
remaining j machines. Hence, the optimal makespan is
at most (1+k·ε)/j+ε.
IfAschedulestwojobsofsize1/j onthesamemachine,
the competitive ratio of A is at least
2/j
(1+(k+ j)·ε)/j=
which is strictly larger than r if ε is chosen appropriately
small.
Otherwise, i.e., A schedules at least one of the jobs of
size 1/j on a machine that already has load at least
(rm−1)/j, the competitive ratio of A is at least
rm/j
(1+(k+ j)·ε)/j=
which is strictly larger than r if ε is chosen appropriately
small.
2
1+(k+ j)·ε
,
rm
1+(k+ j)·ε
,
This concludes the proof of the theorem.
The above general lower bound can be improved for small
reordering buffers.
Theorem 3. For m identical machines:
• No online algorithm can achieve a competitive ratio
less than 3/2 with a reordering buffer of size at most
?m/2?.
• No online algorithm can achieve a competitive ratio
less than 1+1/√2 with a reordering buffer of size at
most ?m/8? if m ≥ 8.
Proof. The following input sequences are similar to the ones
used by Faigle, Kern, and Tur´ an [14] for lower bounds on
the problem without reordering.
• Consider an online algorithm A with a reordering buffer
of size at most ?m/2?. The input sequence consists of
at most two consecutive phases.
– In the first phase, m jobs of size 1 arrive.
– In the second phase, ?m/2? jobs of size 2 arrive
if A schedules at most one job on each machine
in the first phase.
If the input sequence consists only of the first phase, the
competitive ratio of A is at least 2. Otherwise, at the end
of the first phase, the load on at least m−(?m/2?−1)
machines is 1, and hence, the competitive ratio of A is
at least 3/2.
• Consider an online algorithm A with a reordering buffer
of size at most ?m/8?. Assume that m ≥ 8. The input
sequence consists of at most three consecutive phases.
– In the first phase, m jobs of size 1 arrive.
– In the second phase, m jobs of size 1+√2 arrive
if A schedules at most one job on each machine
in the first phase.
– In the third phase, ?m/4? jobs of size 2+2√2
arrive if the second phase exists and if A schedules
at most two jobs of size 1+√2 and no job of size
1 or at most one job of size 1+√2 and further
jobs of size 1 on each machine in the first two
phases.
If the input sequence consists only of the first phase, the
competitive ratio of A is at least 2. If the input sequence
consists only of the first two phases, the competitive
ratio of A is at least1+2(1+√2)
1+(1+√2)= 1+1/√2.
Otherwise, at the end of the second phase, the load on
at least m−2(?m/8?−1) ≥ m−?m/4?+2 machines
is at least 1+(1+√2), since the load on at least m−
(?m/8?−1) machines is 1 at the end of the first phase
and at least m−(?m/8?−1) jobs of size 1+√2 are
scheduled in the second phase. Hence, the competitive
ratio of A is at least1+(1+√2)+(2+2√2)
2+2√2
this case.
= 1+1/√2 in
This concludes the proof of the theorem.
4. Algorithms for identical machines
In this section, we present scheduling algorithms for m iden-
tical machines M0,...,Mm−1. As main result, we introduce
a fairly simple algorithm that achieves the competitive ra-
tio rm. First, we prove this matching upper bound for a
reordering buffer of size 3m. Then, with a refined anal-
ysis, we improve the buffer size to ?(1+2/rm)·m?+2.
607 607