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Xu, Y., Li, K., He, Ligang, Zhang, L. and Li, K.. (2014) A hybrid chemical reaction
optimization scheme for task scheduling on heterogeneous computing systems. IEEE
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TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. *, NO. *, * 2014 1
A Hybrid Chemical Reaction Optimization Scheme
for Task Scheduling on Heterogeneous Computing Systems
Yuming Xu
1,2
, Kenli Li
1,2,*
, Ligang He
3,1
, Longxin Zhang
1,2
, and Keqin Li
1,2,4
, Senior Member, IEEE
1
College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China
2
National Supercomputing Center in Changsha, Hunan, Changsha, 410082, China
3
Department of Computer Science, University of Warwick, United Kingdom
4
Department of Computer Science, State University of New York, New Paltz, NY 12561, USA
Scheduling for directed acyclic graph (DAG) tasks with the objective of minimising makespan has become an important problem
in a variety of applications on heterogeneous computing platforms, which involves making decisions about the execution order of
tasks and task-to-processor mapping. Recently, the chemical reaction optimization (CRO) method has proved to be very effective in
many fields. In this paper, an improved hybrid version of the CRO method called HCRO (hybrid CRO) is developed for solving
the DAG-based task scheduling problem. In HCRO, the CRO method is integrated with the novel heuristic approaches and a
new selection strategy proposed. More Specifically, we make the following contributions. (1) A Gaussian random walk approach is
proposed to search for optimal local candidate solution. (2) A left or right rotating shift method based on the theory of maximum
Hamming distance is used to guarantee that our HCRO algorithm can escape from local optima. (3) A novel selection strategy based
on the normal distribution and a pseudo-random shuffle approach are developed to keep the molecular diversity. Moreover, an
exclusive-OR (XOR) operator between two strings is introduced to reduce the chance of cloning before new molecules are generated.
Both simulation and real-life experiments have been conducted in this paper to verify the effectiveness of HCRO. The results show
that the HCRO algorithm schedules the DAG tasks much better than the existing algorithms in terms of makespan and speed of
convergence.
Index Terms—Chemical reaction optimization, Hamming distance, Hybrid scheduling, Normal distribution, Pseudo random shuffle.
I. INTRODUCTION
A
N application consisting of a group of tasks can be rep-
resented by a node- and edge-weighted directed acyclic
graph (DAG), in which the vertices represent the computations
and the directed edges represent the data dependencies as well
as the communication times between the vertices. DAGs have
been shown to be expressive for a large number of and a
variety of applications. Task scheduling is one of the most
thought-provoking NP-hard problems in general cases, and
polynomial time algorithms are known only for a few restricted
cases [1]. Hence, it is a challenge on heterogeneous computing
systems to develop task scheduling algorithms that assign the
tasks of an application to processors in order to minimize
makespan without violating precedence constraints. Therefore,
many researchers have proposed a variety of approaches to
solving the DAG task scheduling problem. These methods
are basically classified into two major categories: dynamic
scheduling and static scheduling. In dynamic scheduling, the
information, such as a task’s relation, execution time, and
communication time, are all not previously known. The sched-
uler has to make decisions in real time. In static scheduling,
all information about tasks are known before hand. Static
scheduling algorithms by using different techniques to find a
near optimal solution are universally classified into two major
groups: heuristic scheduling and meta-heuristic scheduling.
Heuristic scheduling algorithms are based on the specif-
Manuscript received ****, 2014; revised ****, 2014. Corresponding author:
Kenli Li (email: lkl@hnu.edu.cn).
ic characteristics of an application, and the quality of the
solutions obtained by these algorithms is heavily dependent
upon the effectiveness of the heuristics. These algorithms
give near-optimal solutions but with low time complexity
(polynomial time) and acceptable performance, since the at-
tempted solutions are narrowed down by greedy heuristics to
a very small portion of the entire solution space. Therefore,
it is not likely to produce consistent results on a wide range
of problems, especially when the DAG scheduling problem
becomes complex.
Meta-heuristic scheduling techniques use guided search
strategies to explore the search space more effectively. They
often focus on some promising regions of the search space,
which generate optimal solutions with exponential time com-
plexity. Meta-heuristic methods begin with a set of initial so-
lutions or an initial population, and then, examine step by step
a sequence of solutions to reach the desirable solution. These
methods have demonstrated their potential and effectiveness
in solving a wide variety of optimization problems than many
traditional algorithms. Well-known examples of meta-heuristic
scheduling techniques include ant colony optimization (ACO)
[2], genetic algorithm (GA) [3], [4], simulated annealing (SA)
[5], etc.
Very recently, a chemical reaction optimization (CRO)
method has been proposed. The CRO algorithms are a kind of
the population based meta-heuristic algorithms developed by
Lam and Li in 2010 [6]. The method has been applied to solve
the task scheduling problems, which encodes the solutions
as molecules, and mimics the interactions of molecules in
TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. *, NO. *, * 2014 2
chemical reactions to search the optimal solutions. To date,
the CRO method has only been used to encode the scheduling
of independent tasks on heterogeneous computing platform-
s. Scheduling independent tasks involves mapping tasks to
heterogeneous computing processors. Scheduling DAG tasks
on heterogeneous computing platforms is more complicated,
which involves making decisions about both execution order
of tasks (i.e., task priority) and task-to-processor mapping. It
is a challenge of applying CRO to handle DAG scheduling
problems [7]. In order to solve the dependent task scheduling
problem, we have been investigating the problem of applying
CRO to address the DAG scheduling. Our previous work
presented in [8] applied the conventional CRO framework to
address the DAG scheduling. The DAG scheduling algorithm,
in the works of both [8] and this paper, is divided into two
phases: 1) determining the execution order of the tasks in a
DAG, and 2) mapping the tasks to computing processors. In
[8], we developed a double molecular structure-based CRO
(DMSCRO), in which a set of CRO operations are designed
and applied to both phases of the DAG scheduling.
Although the developed DMSCRO is able to consistently
produce good scheduling solutions, the time overhead, i.e.,
time spent in finding a good solution, is high. This paper
aims to develop a CRO-based DAG scheduling algorithm with
similar performance but reduced overhead. In this paper, we
develop a hybrid CRO scheduling algorithm, called HCRO, by
integrating CRO with the heuristic approaches.
Hybrid algorithms have received significant interest in re-
cent years and are being increasingly used to solve real-world
problems [9], [10], [11], [12], [13], [14], [15], [16]. A hybrid
algorithm is an algorithm that combines two or more other
algorithms that solve the same problem, either choosing one
(depending on the data), or switching between them over the
course of the algorithm. This is generally done to combine
desired features of each, so that the overall algorithm is better
than the individual components [17].
In the HCRO algorithm presented in this paper, the CRO
operations are applied to the first phase of the DAG task
scheduling, i.e., determining the execution order of the tasks,
while heuristic algorithms are designed and applied to the
second phase, i.e., mapping the tasks to processors. A careful
balance is struck between makespan and speed of convergence
in the hybrid CRO. As a result, the hybrid CRO can achieve
similar scheduling performance in terms of makespan while
reducing the scheduling overhead, compared with the exist-
ing guided meta-heuristic algorithms. Moreover, it can also
achieve better performance than heuristic algorithms.
We conducted both simulation experiments over a large set
of randomly generated graphs and real experiments using two
well-known real applications: Gaussian elimination and molec-
ular dynamics application. We compared the proposed HCRO
with two well-known heuristic algorithms and a pure meta-
heuristic method. The theoretical and experimental results
show that the proposed HCRO can achieve better performance
than the heuristic algorithms, and can achieve a very close
performance to that of the pure meta-heuristic algorithm with
much faster convergence speed (therefore with much lower
overhead).
Further, we investigated the CRO parameters that affect
the efficiency of the DMSCRO algorithm presented in our
previous work [8], such as the size of the initial population and
the particular elementary reactions. Hence, in this paper, a new
and novel way is designed to encode the scheduling solutions
of the initial molecular population. Different reactions are
developed to be performed on the encoded solutions and
generate increasingly better solutions.
In summary, the major contributions of this paper are listed
below.
• We adopt some novel approaches to generating the initial
molecular population and to accelerating the convergence
of searching for good scheduling solutions. Because a
high-quality solution, obtained from a heuristic technique,
can help HCRO to find better solutions faster than it can
from a random start, and with a good uniform coverage.
The molecules are well spread out to cover the whole
feasible solution space. Moreover, the molecular diversity
of the molecular population can help the HCRO be able
to reach part of the feasible solution space as large as
possible.
• We present a Gaussian random walk approach to search-
ing for local optimal candidate solution in the operator of
on-wall ineffective collision. On the other hand, for the
purpose of obtaining a global optimum or a near-optimal
solution, we employ a left or right rotating shift strategy
according to the theory of maximum Hamming distance,
aiming to help the operator of decomposition to escape
from local optima.
• We propose a novel selection approach based on the
normal distribution, and use an exclusive-OR (XOR)
operator between two strings to reduce the chance of
cloning before new molecules are generated. Then a
pseudo-random shuffle approach is employed to generate
new molecules to help the operator of inter-molecular
ineffective collision to keep the molecular diversity, and
on the contrary, to realize the operator of synthesis to
eliminate close relatives of the molecules.
• We demonstrate through the experiments over a large
set of randomly generated graphs as well as the graphs
for real-world problems with various characteristics that
our proposed HCRO algorithm outperforms several relat-
ed heuristic-based list scheduling algorithms and meta-
heuristic algorithms in terms of the schedule quality.
The remainder of this paper is organized as follows. Section
II describes the system and workload model. In Section III, a
heuristic chemical reaction optimization algorithm is presented
for DAG scheduling, aiming to minimize the makespan on
heterogeneous computing systems. Section IV compares the
performance of the proposed algorithm with three existing
algorithms. Finally, Section V concludes the paper.
Due to space limitation, a review of related work is moved
to Supplemental File in Section I.
II. MODELS AND WORKLOAD
This section firstly presents the system model and workload
considered in this paper, and then presents an example and
TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. *, NO. *, * 2014 3
motivations of our devised algorithm. In Section II-A, a
heterogeneous computing system consisting of a set P of
m heterogeneous processors is presented, which are fully
interconnected with a high-speed network. In Section II-B,
we introduce an application which is represented by a DAG
graph, with the graph vertexes representing tasks and edges
between vertexes representing execution precedence between
tasks. Then a workload formula is presented. In Section II-C,
we explain the motivation to develop a hybrid approach by
integrating CRO with the heuristic approach and make a
trade-off between the quality of makespan and the speed of
convergence. A list of notations and their definitions used in
the paper is provided in Appendix A of Supplemental File.
A. System Model
The heterogeneity model of a computing system can be
divided into two categories, i.e., processor-based heterogeneity
model (PHM) and task-based heterogeneity model (THM). In
a PHM model, a processor executes the tasks at the same
speed, regardless of the type of the tasks. In a THM model,
how fast a processor executes a task depends on how well
the heterogeneous processor architecture matches the task
requirements and features.
In this paper, we assume a THM model, where the main
characteristics are given in Table I. The heterogeneous comput-
ing system consists of a set P of m heterogeneous processors,
P
1
, P
2
, ..., P
m
, which are fully interconnected with a high-
speed network. Each task in a DAG application can only be
executed on a processor. The communication time between
two dependent tasks should be taken into account if they are
assigned to different processors.
TABLE I
The THM Heterogeneous Computing System
Notation Definition
The amounts of computing power available at each node 0.1-2.0
The maximum number of processors 32
The level (degree) of heterogeneity of the systems
1+h%
1−h%
Note that when the level of heterogeneity is 1 (h=0),
the computing system is homogeneous. Furthermore, as we
change the value of h, i.e., the level of heterogeneity, the
average computing speed remains unchanged.
We also assume that a static computing model in which
the dependency relations and the execution orders of tasks are
known a priori and do not change over the process of the
scheduling. In addition, all processors are fully available to
the computation on the time slots they are assigned to.
B. Application Model
In this work, an application is represented by a DAG graph,
with the graph vertexes representing tasks and edges between
vertexes representing execution precedence between tasks [6].
P red(T
i
) and Succ(T
i
) denote the set of predecessor tasks
and successor tasks of task T
i
, respectively. The entry task
T
entry
is the starting task of the application without any
predecessors, while the exit task T
exit
is the final task with no
successors. The vertex weight, denoted as W
d
(T
i
), represents
the amount of data to be processed in the task T
i
, while the
edge weight, denoted as C
d
(T
i
, T
j
), represents the amount
of communication between task T
i
and task T
j
. The DAG
topology of an exemplar application model is shown in Fig.
1.
Fig. 1. A simple DAG application model containing 10 tasks
In this paper, the execution speeds of the processors in
the heterogeneous computing system is represented by a
two-dimensional matrix, S, in which an element S(T
i
, P
k
)
represents the speed at which processor P
k
to execute task
T
i
. The computation cost of task T
i
running on processor P
k
,
denoted as W (T
i
, P
k
), can be calculated by Eq. (1):
W (T
i
, P
k
) =
W
d
(T
i
)
S(T
i
, P
k
)
. (1)
The average computation cost of task T
i
, denoted as W (T
i
),
can be calculated by Eq. (2):
W (T
i
) =
1
m
m
k=1
W (T
i
, P
k
). (2)
The communication bandwidths between heterogeneous
processors are represented by a two-dimensional matrix B.
The communication startup costs of the processors are repre-
sented by an array, C
s
, in which the element C
s
(P
k
) is the s-
tartup cost of processor P
k
. The communication cost C(T
i
, T
j
)
of edge(T
i
, T
j
), which is the time spent in transferring data
from task T
i
(scheduled on P
k
) to task T
j
(scheduled on P
l
),
can be calculated by C(T
i
, T
j
) = C
s
(P
k
) +
C
d
(T
i
,T
j
)
B(P
k
,P
l
)
and
C(T
i
, T
j
) = C
s
+
C
d
(T
i
,T
j
)
B
, where C
s
=
1
m
m
k=1
C
s
(P
k
)
is the average communication startup cost over all proces-
sors, and B =
1
m
2
m
k=1
m
l=1
B(P
k
, P
l
) is the average
communication cost per transferred unit over all processors
[6]. When T
i
and T
j
are scheduled on the same processor,
the communication cost is regarded as 0. That is to say,
a communication cost is only required when two tasks are
assigned to different heterogeneous processors. It is assumed
that the inter-computing-node communications are performed
at the same speed (i.e., with the same bandwidths) on all links.
For simplicity, we assume B(P
k
, P
l
) = 1 and C
s
(P
k
) = 0 in
our DAG task scheduling model.
Assume that the DAG has the topology as shown in Fig. 1.
An example of the processor heterogeneity and the computa-
tion costs of the tasks are shown in Table II. Note that there
are two numbers in each vertex in Fig. 1. The number at the
top is the task id and the one at the bottom is the average
computation cost as calculated in Table II.
TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. *, NO. *, * 2014 4
TABLE II
Processor Heterogeneity and Computation Costs
Speed W (T
i
, P
j
) Cost Ave. Cost
T
i
W (T
i
) P
0
P
1
P
2
P
0
P
1
P
2
W (T
i
).
0 11 1.10 1.00 1.00 10.0 11.0 11.0 10.67
1 9 1.00 0.90 1.13 9.0 10.0 8.0 9.00
2 9 1.13 1.50 1.13 8.0 6.0 8.0 7.33
3 9 0.90 0.90 1.00 10.0 10.0 9.0 9.67
4 13 1.00 1.08 1.00 13.0 12.0 13.0 12.67
5 3 1.00 1.50 0.75 3.0 2.0 4.00 3.0
6 9 0.90 1.13 1.00 10.0 8.0 9.00 9.0
7 2 1.00 1.00 1.00 2.0 2.0 2.00 2.0
8 18 1.00 1.06 1.13 18.0 17.0 16.0 17.00
9 15 1.00 1.07 1.07 15.0 14.0 14.0 14.43
C. Example and Motivation
Fig. 2. A case study of the HCRO algorithm (the number of processors=3,
makespan=61)
Fig. 3. A case study of the DMSCRO algorithm (the number of processors=3,
makespan=58)
Fig. 4. The convergence trace of the average makespan of the population for
the simple DAG task graph (the number of processors=3, 100 independent
runs)
In Figs. 2 and 3, we can observe that DMSCRO can get a
better makespan than HCRO for the simple DAG application
in Fig. 1. In Fig. 4, we can also observe that the convergence
of HCRO is faster than DMSCRO while maintaining a good
makespan of the best individual and the average makespan
of the population. Therefore, in this paper, we try to find a
trade-off between the quality of makespan and the speed of
convergence. We develop a hybrid approach by integrating
CRO with the heuristic approach. We find that being com-
pared with the overhead of pure meta-heuristic algorithms,
such as DMSCRO, this hybrid approach can achieve similar
performance in terms of makespan for DAG scheduling while
reducing the scheduling overhead. We also find that the
hybrid approach can achieve better performance than heuristic
algorithms. All these benefits will be shown in the experiment
section.
III. HYBRID CHEMICAL REACTION OPTIMIZATION
In this paper, a method of integrating CRO and the heuristic
approach, called HCRO, is proposed to schedule DAG tasks
on heterogeneous computing systems. The idea of this method
is to exploit the advantages of both CRO and heuristic algo-
rithms, while avoiding their drawbacks. In HCRO, the CRO
technique is used to search the execution order of the tasks,
while a heuristic algorithm is used to determine a suitable
task-to-processor mapping. In this section, an important factor
for the priority queues of task scheduling is first introduced
in Subsection III-A. Namely, not every permutation of n tasks
forms a legal schedule due to the precedence relations in the
DAG application graph. Secondly, the encoding mechanism
of our task scheduling algorithm is presented in Subsection
III-B to represent the search nodes as molecules. It is desirable
that any molecule can determine a unique schedule. Thirdly,
four molecular operators are proposed for our task scheduling
algorithm in detail in Subsection III-C. Fourthly, the heuristic
method is proposed in Subsection III-D, which tries to map
each task in the solution (in the order of its position in the
queue) to the computing processor that can provide the earliest
finish time. Finally, the time and space complexity of HCRO
is analyzed in Subsection III-E. Supplemental File Section II
provides the background information of CRO.
A. Fundamental Characteristics of Priority Queues
Our aim is to use CRO to find a good execution order of the
tasks in a DAG application. Therefore, the execution order is
the solution that should be optimized by CRO. A solution ω of
the execution order is encoded as an integer queue and an in-
teger represents a task id, i.e., ω = {T
1
, T
2
, · · · , T
i
, · · · , T
n
}.
Here, ω corresponds to a molecule, a task corresponds to an
atom in the molecule, and the execution order of the tasks
corresponds to the molecule structure.
Theorem 1. The size of the solution space (i.e., the number
of possible solutions) of an application containing n tasks
without dependency is n! = n × (n − 1) × ... × 1.
Proof. In the task scheduling, the solution of task scheduling
is the execution order (without consideration of the task-to-
processor mapping). The number of possible execution order
of n tasks is n!. Therefore, the size of the solution space is
n!.
When an application contains n tasks with dependency, the
size of the solution space will be much smaller than n!. This
is because a solution of task scheduling is an execution order
which is a linear order (topological order) of tasks based on
their dependencies. In a DAG application graph, the tasks
are represented by vertices, which may represent tasks to be
performed, and the edges may represent constraints so that one
