Abstract
Nowadays, parallel and distributed based environments are used extensively; hence, for using these environments effectively, scheduling techniques are employed. The scheduling algorithm aims to minimize the makespan (i.e., completion time) of a parallel program. Due to the NPhardness of the scheduling problem, in the literature, several genetic algorithms have been proposed to solve this problem, which are effective but are not efficient enough. An effective scheduling algorithm attempts to minimize the makespan and an efficient algorithm, in addition to that, tries to reduce the complexity of the optimization process. The majority of the existing scheduling algorithms utilize the effective scheduling algorithm, to search the solution space without considering how to reduce the complexity of the optimization process. This paper presents a learner genetic algorithm (denoted by LAGA) to address static scheduling for processors in homogenous computing systems. For this purpose, we proposed two learning criteria named Steepest Ascent Learning Criterion and Next Ascent Learning Criterion where we use the concepts of penalty and reward for learning. Hence, we can reach an efficient search method for solving scheduling problem, so that the speed of finding a scheduling improves sensibly and is prevented from trapping in local optimal. It also takes into consideration the reuse idle time criterion during the scheduling process to reduce the makespan. The results on some benchmarks demonstrate that the LAGA provides always better scheduling against existing wellknown scheduling approaches.
1. Introduction
The parallel processor revolution is underway. Move toward the use of parallel structures is one of the greatest challenges for software industries. In 2005, Justin Rattner, chief technology officer of Intel Corporation, said “We are at the cusp of a transition to multicore, multithreaded architectures, and we still have not demonstrated the ease of programming the move will require…” [1].
A good scheduling of a sequential program onto the processors (or computers) is critical to effective utilization of the computing power of a multiprocessor (or multicomputer) system [2]. To perform scheduling, a sequential program is represented by a Directed Acyclic Graph (DAG), which is called a task graph. In this graph, a node represents a task which is a set of instructions that get a set of inputs to produce a set of outputs and must be executed serially in the same processor. Computation cost associated with each node indicates the amount of computation. The edges in the DAG correspond to the communication messages and precedence constraints among the tasks as they show the communication time from one task to another. This number is called the communication cost and is denoted by [3]. A sample DAG is shown in Figure 1. In a task graph, a node is not ready to run until all its required data are received from its parent nodes. The communication cost between two nodes which are allocated to the same processor is considered to be zero.
Scheduling has been used in two different areas in performance evaluation in the literature: multiprocessors distributed systems and cloud computing. The purpose of the scheduling on multiprocessors systems is to minimize the completion time of DAG on a set of homogeneous or heterogeneous processors. This helps to speed up a sequential program. Cloud computing also utilizes scheduling techniques, but with different goals such as response time, energy consummation, throughput, and RAM efficiency. Some papers that use scheduling in this regard are [4–7]. This paper aims to present an algorithm for scheduling a sequential onto multiprocessors systems, not cloud computing.
In the scheduling problem, the completion time is the time required for a program to be completed. Figure 2 shows a sample timeline (scheduling) in the form of a Gantt chart on two processors for task graph shown in Figure 1. As can be seen in this figure, the completion time for this task graph is 140.
The performance of a scheduling algorithm is usually measured in terms of completion time and the running time of the algorithm. There is, usually, a tradeoff between these two performance parameters; that is, efforts to obtain a minimum completion time often incur a higher time complexity.
Due to the NPhardness of the scheduling problem, the use of searchbased and evolutionary approaches such as genetic algorithm is to be more reasonable than other approaches [2]. The searching procedure in searchbased algorithms can be performed in two ways: global search and local search. Global searchbased methods usually consider the whole search space to find a good solution. These methods have an operator to explore new regions in search space. The genetic algorithm is one of the main search methods that use the global search strategy. This algorithm has no exploitation ability, and also the speed of convergence is not sufficient while searching for global optima. Local search algorithms, such as Hill Climbing and Learning Automata, utilize exploitation ability to improve the quality of the already existing solutions as well as the rate of convergence.
Hill climbing algorithm starts from an initial solution and then iteratively moves from the current solution to a neighbor solution (to be defined) in the search space by applying local changes. There are no learning concepts in this algorithm. Learning is an ability of a system to improve its responses based on past experience. Reinforcement learning techniques are used to choose the best response based on the rewards or punishments taken from the environment. Learning automata is an automatonbased approach for learning. Learning automata learn the optimal action through past experiences and repeated interactions with its stochastic environment. The actions are chosen according to a specific probability distribution which is updated based on the environmental response.
The problem addressed in this paper is the scheduling of a task graph on homogeneous processors for minimizing the completion time to achieve a higher performance in the execution of parallel programs. To achieve this aim, this paper presents a new learning schema inspired by the behaviors of both hill climbing algorithm and learning automata. We, then, propose an efficient search approach which adds a new learning function to the evolutionary process of the genetic algorithm for scheduling. This new algorithm combines global search (genetic algorithm) and local search (using the concepts of penalty, reward and neighbors) strategies for scheduling of a task graph. This causes the evolutionary process speed, i.e., convergence rate, to increase considerably and be prevented from trapping in local optimal. Hence, the algorithm, at the same time with other algorithms, can search more searchspace of the problem and find the solution with higher quality. Also, the proposed algorithm can be improved by employing the optimal reuse of idle time on processors and the use of the critical path of the task graph for constructing the initial population. This technique enables the GA to schedule the task graph better than previous algorithms. Results of the conducted experimental demonstrate that LAGA is superior to the Parsa, CPGA, TDGA, CGL, BCGA, DSC, MCP, LC, and LAST algorithms.
The contributions of this paper are summarized as below:(1)Proposing a new learning schema inspired by the behaviors two local searchbased algorithms namely hill climbing algorithms and learning automata;(2)Improving the genetic algorithm evolutionary process using the new learning schema;(3)Proposing optimal reuse idle time heuristic, aiming to reuse idle times on the processors;(4)Proposing new reward and penalty function which is based on the earliest start time for task graph scheduling.
The remainder of this dissertation is structured as follows. In Section 2, we provide an overview of related work. In Section 3, a new algorithm for scheduling a task graph is proposed. In Section 4, the results of the proposed algorithm are evaluated. Finally, Section 5 concludes the paper.
2. Background and Previous Works
The need for accurate task graph scheduling has been widely recognized by the research community. In the literature, there are many works about the task graph scheduling algorithms. The Scheduling techniques are typically categorized into two classes; the first class is homogeneous [12] and the second one is heterogeneous [13, 14]. In homogeneous systems, the processing power of all processors is considered similar. In heterogeneous type, processors have different processing power; therefore, for each task, time on every processor should be given separately. In this paper, scheduling is assumed as homogeneous.
Generally, all task graph scheduling algorithms can be grouped into two categories: static and dynamic. Static scheduling is performed offline; in other words, desirable parallel program characteristics such as computation costs, communication cost, and data dependencies are known before execution [11]. Dynamic scheduling algorithms can be further grouped into two categories: accomplished quickly mode and batch processing mode algorithms. In the accomplished quickly mode, a task is mapped as soon as it arrives. In the batch mode, tasks are not mapped as they arrive; instead, they are collected into a list for scheduling and mapping [12]. The work in this paper is solely concerned with the static scheduling.
Static task graph scheduling algorithms are classified into two categories: Heuristicbased algorithms and guided random searchbased algorithms [14]. Heuristicbased algorithms have acceptable performance with a low complexity. These algorithms are further classified into three classes: listbased algorithms, clusteringbased algorithms, and task duplication algorithms.
Listbased scheduling approaches, such as HEFT [15] and CPOP [14], consist of two phases of tasks prioritization and processor selection (to assign the task to the processor). Task duplication approaches aim, such as STDS [16], HCNF [17], is to run a task on more than one processor to reduce the waiting time for the dependent tasks. The main idea behind this approaches is to use the processors time gap.
In the literature, there are several heuristicbased (nonevolutionary) scheduling algorithms such as PEFT [13], ETF [18], DSC [19], DLS [20], HLFET [20], LAST [21], ISH [22], MCP [23], LC [24], EZ [25], and MTSB [26]. Most of these algorithms consider several simple assumptions about the structure of a task graph [2, 12]. For example, ignoring task communication cost, or considering the same computational cost for all tasks [2], or assuming a tree structure for a task graph [8] is some of these simple assumptions. In general, nowadays many recent algorithms are designed to deal with any graphs. Moreover, scheduling on a limited number of processors, called Bounded Number of Processors (BNP), or the availability of unlimited number of processors, called Unbounded Number of Clusters (UNC), is another assumption of scheduling algorithms.
Since the task graph scheduling problem is an NPHard problem [2, 12]; even simplified versions of the task scheduling problem are NPHard. Let P and T indicate the number of processors and the number of tasks, respectively, thus, the possible number of task to processor allocation is P^{T} and the possible number of task orderings is T!; then, the overall search space is . Considering this huge search space, obtaining an optimal schedule by a comprehensive search of all possible schedules is not achievable; hence, an evolutionary approach such as genetic algorithms may be applied to solve such problems, effectively [8, 27].
Using genetic algorithms to solve task graph scheduling have received much attention. The main difference among genetic algorithms is the encoding scheme used to represent a schedule. The structure of encoding significantly impacts the complexity of the genetic process to the convergence of an optimal schedule. For example, Wang and Korfhage [10] presented a matrix representation of schedules which records the execution order of the tasks on each processor. In this encoding, the crossover and mutation operators do not prevent the production of invalid solutions. Although exist the repair operations to correct these solutions. However, the operations consume additional time. Parsa et al. [8] proposed a new representation of schedules based on the string representation. In this new encoding scheme, a scheduling solution is represented by an array of pairs (, ), where indicates the task number assigned to the processor . Using this representation, both the execution order of the tasks on each processor and the global order of the tasks executions on the processors are determined in each chromosome. Hou et al. [9] proposed a variablelength representation which orders the tasks on each processor. Their proposed string representation prevents the production of invalid solutions. Whereas this representation cannot span the complete space of possible schedules, it may be impossible for the genetic algorithm to converge to an optimal solution.
Omara and Arafa [11] proposed two geneticbased algorithms with a new encoding scheme, namely, Critical Path Genetic Algorithm (CPGA) and TaskDuplication Genetic Algorithm (TDGA). The CPGA algorithm tries to schedule the tasks placed on the critical path on the same processor aiming to reduce the completion time. The second algorithm, TDGA, is based on the principle of task duplication to minimize the communication overhead. One of the weaknesses of this method is to generate invalid chromosomes. The runtime of these two algorithms is very high for finding a scheduling because it uses some internal algorithms for scheduling such as critical path and task duplication. Also, the performance of the algorithm is not clear on benchmarked DAGs.
Daovod et al. [28] proposed a new scheduling algorithm, which is called the Longest Dynamic Critical Path (LDCP), to support scheduling in heterogeneous computing systems. The LDCP is a listbased scheduling algorithm, which effectively selects the tasks for scheduling in heterogeneous computing systems. The accurate and better selection of tasks enables the LDCP to provide a high quality in scheduling the heterogeneous systems. The function of this algorithm is compared with two wellknown HEFT and DLS algorithms in this context. One of the weaknesses of this algorithm is its greediness, which will not respond well in the case of largescale DAGs.
Bahnasawy et al. [29] proposed a new algorithm, named Scalable Task Duplication Based Scheduling for static scheduling in computing systems with heterogeneous multiprocessors. Considering the priority between the tasks, this algorithm divides the graph into levels so that the tasks at each level are arranged in a list based on their size. The tasks based on their priority are assigned to the first processor found in the list. One of the weaknesses of this algorithm is its greediness, which will not function well in solving largescale DAGs.
Naser et al. [30] proposed a scheduling algorithm, called Communication Leveled DAG with Duplication (CLDD), in the heterogeneous multiprocessor systems. This algorithm consisted of three phases: level arrangement, tasks prioritization, and processor selection.
Wen [31] proposed a hybrid algorithm of genetics and variable neighborhood search (VNS) [32] to minimize the tasks scheduling in the heterogeneous multiprocessor systems. This algorithm has derived a variety of features from GA and VNS.
NSGAII [33] is a geneticbased algorithm for task graph scheduling. In this algorithm does not exist the dependency among the tasks, and elitism operator is used to protect the good solutions in the evolutionary process of the genetic algorithm. To provide diversity in solution, this algorithm uses crowding distance technique.
TSB algorithm [34] belongs to a class of scheduling algorithms called BNP. TSB utilizes two queues, namely, ready task queue and not ready task queue to perform scheduling. Then, it uses breath first search traversal algorithm for selecting tasks. The MTSB algorithm [26], also, belongs to the BNP class, with the difference that, unlike the algorithm TSB, it does not consider communication time between tasks and processors.
Akbari et al. [35] proposed a geneticbased algorithm for static task graph scheduling in heterogeneous computing systems. Their proposed algorithm presents a new heuristic to generate an initial population and also proposed new operators aiming to guarantee the diversity and convergence. Akbari and Rashidi [36] employed cuckoo optimization algorithm for static task graph scheduling in heterogeneous computing systems. Because this algorithm works on continuous space, hence, they proposed a discretization of this algorithm to solve the task graph scheduling problem.
Moti Ghader et al. [37] proposed a learning automatabased algorithm for static task graph scheduling. This algorithm uses learning automata to represent different possible scheduling. Since this algorithm is a localbased search algorithm, thus, it traps in the local optima.
A significant performance assessment and comparison of the addressed algorithms is a complex task and it is necessary to consider a number of issues. First, there are several scheduling algorithms based on various assumptions. Second, the performance of the most existing algorithm is evaluated on small sized task graphs, and, hence, it is not clear their performance on largescale problem size [38]. Due to the NPhardness of the task graph scheduling, the rate of convergence when searching for global optima is still not sufficient. For this reason, there is a need for methods that can improve convergence speed, to achieve a good scheduling.
3. The New Algorithm
In this section, a new genetic algorithm is presented for the task graph scheduling problem which is based on learning. In this algorithm, each solution is represented by a chromosome. Each chromosome’s cell is called a gene. The number of genes depends on the number of tasks. We use two different chromosome structures in our algorithm named classic and extended chromosome structure. In classic chromosome, the structure of each chromosome includes two rows where the first row is task number and the second row is processor number assigned to the task (see Table 1).
The extended chromosome structure is utilized for the learning process. This representation has four rows where the first row is task number, the second row is processor number assigned to the task, the third row is the depth of each gene, and the fourth row is selection probability of each task needed to perform learning. The total probability of them, initially, is equal (Table 2).
In extended chromosome, structure a chromosome is defined as a tuple as follow:(i): is a set of genes (r is number of the tasks).(ii): is a set of processors in the chromosome.(iii): This is the outcome of evaluation of a gene considering interactions with its stochastic environment and indicates that a gene must be fined or rewarded.(iv): Let N and R denote the depth allocated to a gene and the number of genes, respectively, this shows a set of internal states for a gene.(v): This is a mapping function. This function based on evaluating a gene determines the next action for a gene.(vi): is a chromosome’s genes probability. This vector (fourth row in Table 2) shows the selection probability of each gene to perform reward or penalty operations. Based on the fact that the penalty is taken or rewarded, these probabilities are updated. If no prior information is available, there is no basis in which the different genes can be distinguished. In such a case, all gene probabilities would be equal  a “pure chance” situation. For an rgene chromosome, the gene probability is given by:(vii) indicates the parameters required for a learning.
In Algorithm 1, the steps of our proposed task graph scheduling algorithm are given. Different parts of this algorithm are further described in the following sections.

The objective of the new task scheduling algorithm is to minimize the completion time. Some terms are used; the proposed algorithm is defined below.
Definition 1. In a task graph, the tlevel of node i is the length of the longest path from an entry node to i (excluding i). Here, the length of a path is the sum of all the node and edge weights along the path [39].
Definition 2. The blevel of a node is the length of the longest path from the node to an exit node [39].
Definition 3. A Critical Path (CP) of a task graph is a path from an entry node to an exit node as the sum of execution time and communication cost is the maximum. The nodes located on the CP can be identified with their values of (tlevel + blevel); i.e., tlevel + blevel for that node should be maximized [39].
Definition 4. — Earliest Start Time of task i and processor j: The EST refers to the earliest execution start time of the node n_{i} on processor p_{j}.
3.1. Initial Population
The initial population is constructed as shown in Algorithm 2.

The initial population is generated in such a way that the overall communication time is minimized. To do this, the algorithm identifies all tasks located on the critical path and then tries to shorten the critical path in the DAG by assigning the tasks on the critical path into the same processor. By reducing the length of the critical path, the minimum completion time of the scheduling is reduced.
3.2. Computing Fitness of Chromosome
Quality of each the chromosome depends on the completion time of the scheduling. The completion time of a scheduling represents its fitness. The objective is to minimize the completion time. In the proposed algorithm, first, the EST for all tasks are calculated. The value of an entry , referring to task i and processor j, will be given bywhere FT_{j} is the time that machine j finishes the execution of all previous tasks of task i and ETT_{ij} indicates the time that all the data required to run task i on processor j is ready. To calculate this time, the finish time and time required to transfer data between processors of all immediate predecessors of task i in the task graph are considered.
The fitness function used to guide the evolutionary process of the genetic algorithm for finding a scheduling is based on the total finish time of the schedule, which is obtained bywhere and denote the number of processor and the finish time of the last task in the processor , respectively.In our method the length of an obtained schedule and makespan is computed as Algorithm 3.

In the LAGA, we employed the roulette wheel selection operator to select potentially useful chromosomes for the reproduction process. The fitness value is used to associate a selection probability for each chromosome. Let and N denote the fitness value of the chromosome i (which is calculated by (4)) and the number of chromosomes in the population, respectively. The selection probability of this chromosome, , is calculated as follows: Using (5), a proportion of the wheel is assigned to each chromosome. The higher the fitness value of the chromosome, the higher its opportunities of being elected for the reproduction process.
3.3. Reuse Idle Time
A reuse idle time criterion can improve the quality of a schedule [40]. In this section, we present a reuse idle time greedy algorithm to control the original system and improve the quality of returned results. For a randomly selected chromosome in the population, in each generation, we assign some tasks to idle time slots. For example, Figure 3 shows a sample of the Gantt scheduling chart related to Figure 1. According to Figure 3 (left), the time gap in the P1 processor is as large as 10 units and the completion time for the processor P1 is equal to 148. Also, the weight of the task t10 is equivalent to 10. By applying changes, the t10 can start from the time 73, and the completion time reduces from 148 to 138 units with this reuse, as can be seen in Figure 3 (left).
The algorithm of this heuristic is shown in Algorithm 4; while all tasks and all slots are not addressed, this algorithm will be repeated. Let idle start time and idle end time of slot i denoted by and , respectively. Also assume DAT(, ) and w() denote data arrival time of task on processor and execution time of task , respectively. Task allocated to slot when

For example, assume the processor P_{1} has an idle slot; and the SIT_{1} =73 and EIT_{1} =83. On the other hand, w(t_{10})= 10 and DAT (t_{10}, P_{1})=59. In our algorithm t_{10} is scheduled to start time 93. By applying the reuse idle algorithm, t_{10} can be rescheduled to start at time 83.
We need a list that maintains all of SIT and EIT for each slot. The algorithm tries to fill the slots as much as possible. Our aim is to select the maximum tasks that meet the above requirement. For doing this task, first, we sort the tasks ascending considering their execution time, i.e., ; then, using Algorithm 4, we assign the tasks in slots.
The reuse idle time heuristic more minimizes the makespan; however, this algorithm for largescale task graphs makes an overhead cost in terms of time. Therefore, we limit reuse idle time heuristic just to 30% of best chromosomes. This means that, among the chromosomes of a generation, 30% of them with the shortest completion time are selected to apply reuse idle time heuristic.
3.4. Learning Operators
The evolutionary process of the LAGA is enhanced using reward and penalty operators. These operators are utilized for learning. In the LAGA, in addition to evaluating the chromosome, the genes of a chromosome are assessed based on their outcome on chromosome’s fitness. Thus, the most fitting location for genes within chromosomes and the most proper chromosome inside the population is regularly discovered during the evolutionary process of the genetic algorithm. Intuitively, the algorithm selects a gene a chromosome and evaluates it based on certain criteria. If the result of the evaluation is good, this gene will be rewarded. This means that the gene in the chromosome is in the right place and should be protected from future changes. But if the evaluation result is not good, the gene must be penalized. In this case, if the gene has not already been rewarded or the amount of penalties and rewards are the same, the whole possible neighbors (to be defined) of the gene are searched for a penalty action. The neighbor with the highest quality will be the replacement of this gene. But if the number of rewards received by the gene is greater than the number of received penalties, one of the numbers of rewards is reduced.
Formally, the penalty and reward operators are employed in the LAGA as follows.
during the scheduling process, the LAGA selects gene in a chromosome and assesses it, if it gets a promising feedback, the probability, , of this gene increases and the probability of other genes decreases. If it gets an inappropriate feedback, the probability, , of this gene decreases and probability of other genes increases. In this paper, we utilize a linear learning scheme for learning. This scheme was considered first in mathematical psychology [41] but was later independently conceived and introduced into the engineering literature by Shapiro and Narendra [41]. Letting and be two functions, for a linear learning scheme with multiple genes, we haveIn (6) and (7), m, , and are the number of genes of a chromosome, reward, and penalty parameters, respectively. The parameters and are used to control the rate of convergence of a chromosome. According to (6) and (7), if and are equal, the learning algorithm will be known as linear reward penalty. If , the learning algorithm will be known as linear reward epsilon penalty and if b=0, the learning algorithm will be a linear reward inaction. Depending on whether penalty or reward has been made, the probability of genes is updated as follows.
A promising feedback for gene ibutsoandAn inappropriate feedback for gene ibutsoandThese probabilities cause good genes to find their place in the chromosome over time. The main intention of these probabilities is to exploit the previous behavior of the system in order to make decisions for the future, hence, learning occurs. To perform learning, the number of internal states of each gene must be determined. These internal states for each gene maintain the number of failures and successes. How to select a gene from a chromosome is based on calculating the cumulative probability of each gene as follows ( is gene probability): A random number is generated between 0 and 1. If generated random number is higher than and lower than , the corresponding gene to is selected to evaluate for reward and penalty. Obviously, genes with higher probabilities are preferred for reward or penalty.
In each iteration step of the algorithm, a gene of each chromosome is selected based on its probability ((8), (11), (12), and (15)) and this gene is evaluated using Algorithm 5.

Generally, a gene is penalized or rewarded as follows.
Suppose that a chromosome includes R genes () and has RN internal states (). Internal states of are related to , are related to , and are related to . Therefore,so that G is a mapping function.
If gene is in the internal state on the chromosome and this gene gets a penalty then the internal state changes as follows:If the gene gets the reward then the internal state changes as follows:If gene is in internal state and it gets a penalty then the internal state changes as follows: If the gene gets the reward then transferring of state performs as follows:In Figure 4, the internal states of each gene are considered to be N. In this figure, N and 2N indicate the boundary states of the first gene and second gene, respectively. After evaluating a gene, if it gets a positive feedback, it moves to internal states, and if an inappropriate feedback is taken, it moves to a boundary state. If a gene reaches boundary state and gets an undesirable feedback, then it is displaced with another gene in the chromosome, so that the overall quality of the chromosome is improved. To perform this, the neighbors of a gene are generated and then from these neighbors the algorithm searches for a processor in the chromosome for displacement so that makespan value in that chromosome is lower than others. If the makespan value of new chromosomes generated is more than the initial chromosome, it remains the same initial chromosome. Indeed, the algorithm continues until there no better solution is obtained. The concept of neighbor scheduling is illustrated in Figure 5. In this figure, A shows the current scheduling and B, C, and D are three neighbor scheduling for that. For example, consider the task a, so that, in B, it moved into processor 4, in C, it moved into processor 2, and in C, it moved into processor 3.
(a) Positive feedback
(b) Inappropriate feedback
In the learning process, several percents of the genes from a selected chromosome are chosen and each chromosome selected is regularly improved by this algorithm (Algorithm 6). The proposed algorithm considers a threshold for determining the number of neighbors for searching. Such threshold could be any percentage in the range of . Using a low value for reduces the algorithm steps prior to converging, and using a high value for increases the algorithm steps prior to converging. However, it is reasonable that a high threshold can often produce a good result. Based on value, we propose to types of learning schema named Steepest Ascent Learning Criterion (SALC) and Next Ascent Learning Criterion (NALC). In Next Ascent Learning Criterion is equal to zero. This means the algorithm stops searching a scheduling when the first scheduling found with a lower makespan (i.e., highquality solution); and in the Steepest Ascent Learning Criterion, is equal to 100%. This causes the algorithm to generate all the neighbors scheduling and examine all of them and choose the one with the lowest makespan.

Theorem 5. The time complexity of LAGA algorithm is O(MaxGen×PopSize× N×V).
Proof. To calculate the time complexity of the algorithm, first, we consider the following assumptions: PopSize: the number of individuals in the population MaxGen: The number of generation I: Number of iterations for reward and penalty operators V: the number of tasks in the task graph N: the number neighbors for a selected gene #P: the number of processors #S: the number of idle slots in a processor v’: the number of tasks allocated into a processor To calculate time complexity for a chromosome, it is necessary to calculate the completion time for all processors and select the maximum time from them as time completion of the chromosome. Thus, the time complexity to calculate the completion time for a chromosome is O(V). The rewarding operation time complexity is O, because only the internal state of a gene changes. But, the penalizing operation time complexity is O(N×V), because it is necessary to find its neighbors and to compute them completion times and then select a neighbor with minimum completion time.
Creating the initial population can be accomplished in linear time, thus, the main computational cost lies in Step 2 (Algorithm 1). The computational complexity of learning schema is O(N×V). The computational complexity of extending a chromosome to classic chromosome is O(PopSize×V). The computational complexity of learning is O(PopSize× N×V). The computational complexity of the selection operator is O(V). The computational complexity of crossover is O(PopSize×V), and that of mutation is O(PopSize×V). The computational complexity of reuse idle time heuristic is O(PopSize × #P × #S × v’logv’). Thus, the overall time complexity is O(V) + O(N×V) + O(N×V) + O(PopSize × #P × #S × v’logv’) + O(V) + O(PopSize×V) + O(PopSize× N×V) + O(PopSize×V) + O(PopSize×V). Therefore, considering the number of generations, the time complexity of LAGA can be simplified as O(MaxGen×PopSize× N×V).
4. Experimental Results
The task graphs used to support the findings of this study have been deposited in the [8, 21] and http://www.kasahara.cs.waseda.ac.jp/schedule/optim_pe.html. In the following, we compare the proposed algorithm with a number of other algorithms on these task graphs.
We, first, compare the LAGA with Standard Genetic Algorithm (denoted by SGA). In SGA, there is no depth and tasks are arranged based on the topological order in each chromosome. The result is obtained by applying LAGA to the task graph offered in [8, 21], where this graph is redrawn in Figure 6. The result indicates considerably improvement LAGA compared to SGA. The parameter setting for the experiments are as follows:(i)Population size: 30 chromosomes,(ii)Number of generations: 500,(iii)Number of processors: 3,(iv)Crossover probability: 80%,(v)mutation probability: 5%,(vi)Depth: 3,(vii)(viii)Reward and penalize probability: 50%.
Table 3 displays the scheduling and completion time obtained for the task graph shown in Figure 6 for both LAGA and SGA algorithms. Also, the stability of the proposed algorithm is evaluated. Stability is a critical issue in evolutionary and heuristic algorithms. It means that the algorithm gives the same or close solutions in different runs. To demonstrate the stability of the LAGA, the results obtained by twenty runs of the algorithm to schedule the task graph shown in Figure 7.
The progression of the chromosomes, while applying LAGA and SGA algorithms, is shown in Figure 8.
Comparison of the Proposed Algorithm with Other Algorithms. Most existing genetic algorithms and other heuristic algorithms have been evaluated on graphs which are randomly produced and do not communication cost. Hence, we select the algorithms for comparison for which their results on the certain graph are known. Therefore, in this section, the LAGA is compared with nine different algorithms on the graphs which are used by them.
In Figure 9, LAGA is compared against Parsa algorithm and six other scheduling algorithms. The comparison is performed by applying LAGA to the task graph drawn in Figure 6 [8, 21] and is used by Parsa and other six algorithms. The results demonstrate that the scheduling obtained by LAGA outperforms the algorithms 3–7 and is the same with Parsa and LC algorithms. However, LAGA obtains the result very swift and stable. In addition, LAGA runs faster when the number of tasks grows. Furthermore, Parsa algorithm is evaluated using small problem sizes, and it is not clear their performance on largescale problem size. In Figure 10, the stability of LAGA is compared with Parsa algorithm.
Fast Fourier Transform Task Graphs (FFT), depicted in Figure 11, are good benchmarks to assess the performance of the scheduling algorithms. Table 4 shows three versions of these graphs (i.e., FFT1, FFT2, and FFT4). The first column in each of these cases shows the computation time of each task and the second column shows the communication time of each task with others.
According to Tables 5 and 6, it has been found that LAGA with always outperforms the genetic and nongenetic based algorithms. Because, considering highspeed of LAGA, at the same time with other algorithms, it can search more statespace of the problem and, as a result, can find the answers with higher quality.
Usually, the efficiency of most algorithms presented for task’s graph scheduling problem is studied on graphs with small size and low communications and their results on graphs with real applications which are graphs with large size and high communications are not clear. The LAGA () is applied to specific benchmark application programs which are taken from a Standard Task Graph (STG) archive (http://www.kasahara.cs.waseda.ac.jp/schedule/optim_pe.html) (see Table 7). The first program of this STG set, Task Graph 1, is a randomly generated task graph, the second program is a robot control program (Task Graph 2) which is an actual application program, and the last program is a sparse matrix solver (Task Graph 3). Also, we considered the task graphs with random communication costs. These communication costs are distributed uniformly between 1 and a specified maximum communication delay.
Table 8 shows the results of comparing among SGA and LAGA (on depth 3 and 5), Parsa, CPGA, and TDGA algorithms on 4 processors for programs listed in Table 7. According to Table 8, the proposed algorithm (LAGA) with depth=3 produces schedule length lower than other algorithms.
Table 9 shows a comparison of three geneticbased algorithms with LAGA on a number of randomly generated graphs. The task execution time and communication time between processors are considered random. The task execution time and communication time are distributed uniformly within (10, 50) and (1, 10) second, respectively. To evaluate the efficiency of the algorithms, we set the running time of the algorithms equally. As can be seen in this table, the LAGA, in all cases and at the same time with them, outperforms better than the rest. This demonstrates that the LAGA is able to search more space than the rest at the same time, and as a result it is efficient.
5. Conclusions
The task graph scheduling problem is an NPHard problem. Hence, to solve this problem, evolutionary approaches are pretty effective. But, their convergence rate when searching for global optima is still not sufficient. For this reason, there is a need for methods that able to improve the convergence speed, to achieve a good scheduling. In this paper, we combined genetic algorithm with an accelerated convergence rate strategy, aiming to achieve higher performance in parallel environments. We have proposed a learningbased genetic algorithm for solving the problem which is to better utilize the processors. Considering highspeed of the proposed approach, the proposed approach can search more statespace of the problem at the same time with other algorithms, as a result, it can find the solution with higher quality.
5.1. Future Works
A number of directions can be explored to extend and improve this work.(1)Finding a useful and effective a reward and penalty function;(2)Using clustering in the scheduling of task graph, instead of, using the critical path;(3)Finding a useful and effective load balancing heuristic for the proposed algorithm.(4)The parameter tuning is one of the major drawbacks of the genetic algorithm. Recently, for solving NPHard problems, using optimization algorithms without parameter tuning, such as Beetle Antennae Search Algorithm [42], has attracted a lot of attention. In the future work, we intend to use this algorithm for task graph scheduling problem.
Also, the proposed method may be extended to solving scheduling in grid computing, cloud computing, and heterogeneous environment; and also it is possible to use our approach to solve other NPHard problems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that they have no conflicts of interest.