Online makespan minimization for MapReduce scheduling on multiple parallel machines

1 Introduction

MapReduce [1] is a popular model in many big data-processing frameworks such as search indexing, distribution sort, log analysis, etc. In general, MapReduce processing consists of two phases: the map phase and the reduce phase. When a job is submitted, its computation always contains these two phases. In the map phase, there are many map tasks inputting raw data and outputting key-value pairs. These key-value pairs are used as inputs of the reduce phase. In the reduce phase, the machines process the pairs and output the final results.

In this article, we study the online MapReduce scheduling problem where jobs are released by over-list. Each job consists of map tasks and reduce tasks, and the processing times of both tasks become known to the decision-maker once the job is revealed. For each job, we assume: (1) the job’s reduce tasks can be processed after finishing its all map tasks; (2) the map task is fractional, i.e., it can be arbitrarily split and processed between the machines simultaneously, while the reduce task is not fractional; (3) we assume preemption on reduce task is allowed, i.e., any reduce task may be interrupted and resumed at a later time during processing. The problem can be formally described as follows. A set of jobs J = { J 1 , J 2 , … , J n } arrive one by one and must be allocated into m uniform machines { σ 1 , σ 2 , … , σ m } . Let the speed of machine σ i be s i . Without loss of generality, we may assume s 1 ≥ s 2 ≥ … ≥ s m . Each job J j consists of a set of map tasks M j = { m j 1 , m j 2 , … , m j v j } and a set of reduce tasks R j = { r j 1 , r j 2 , … , r j t j } , namely, each job has v j map tasks and t j reduce tasks. Our goal is to minimize the makespan, i.e., the completion time of the last job that finishes. Adopting the classical threefold notion, we denote the aforementioned problem as Q m ∣ M R ( p m t n ) , online ∣ C max , and P m ∣ M R ( p m t n ) , online ∣ C max when s i = 1 , i = 1 , 2 , … , m .

The quality of an online algorithm A is normally measured by its competitive ratio. An algorithm A is called ρ -competitive if, for any instance I , C A ( I ) ≤ ρ ⋅ C * ( I ) holds, where C A ( I ) denotes the objective function value produced by A and C * ( I ) denotes the optimal objective value.

In recent years, the scheduling problem has been one of the active research topics in MapReduce because of the need to guarantee the performance of the system like response times, system utilization, etc. A number of empirical works [2–5] have been done to provide some new scheduling strategies or heuristics and give the experimental results for different models. Besides, several theoretical works [6–9] have also emerged. Most of these works focus on the offline scheduling problem, where job arrivals are known beforehand. However, in practice, job scheduling is often decided without all the information in advance. Thus, the theoretical research of online scheduling needs to be solved urgently. To the best of our knowledge, there are few studies that consider the online scheduling problem in the MapReduce system [6,7,9,10,11,12].

For the online MapReduce scheduling problem, Moseley et al. [6] model the MapReduce system as the two-stage classical flexible flow shop problem and the objective of the problem is to minimize the total flow-time. Then they present an online 1 + ε -speed O ( 1 ε 2 ) -competitive algorithm for this problem, where 0 < ε ≤ 1 . However, there is no guarantee of a competitive ratio without resource augmentation. To solve this problem, Zheng et al. [7] construct a slightly weaker criteria called efficiency ratio because no online algorithm can achieve a constant competitive ratio for nonpreemptive tasks. Then they provide an online algorithm called available shortest remaining processing time (ASRPT) with a very small (less than 2) efficiency ratio and show that it outperforms the state-of-the-art schedulers. Chang et al. [10] focus on minimizing the total completion time, and design an online algorithm that achieved 30% shorter completion time of all jobs compared to the original FIFO-based scheduling via simulation. For minimizing makespan, Jiang et al. [13] consider the problem on two uniform machines, i.e., Q 2 ∣ M ( f r a c ) R ( p m t n ) ∣ C max . They provide an optimal online algorithm with a competitive ratio of s 2 + 2 s + 5 + 1 − s 2 , where s ≥ 1 is the speed ratio of two machines. Under the assumption that a job’s reduce tasks are unknown until its map tasks are finished, Luo et al. [9] present online optimal algorithms with the same competitive ratio of 2 − 1 m for both the preemptive and non-preemptive reduce tasks. When jobs are released over time, Chen et al. [14] present a non-preemptive algorithm MF-LPT with a competitive ratio 2 − 1 m and an optimal preemptive algorithm for two machines.

Many scholars have conducted in-depth research on the MapReduce task scheduling problem in a heterogeneous environment. Jeyaraj et al. [15] devised two methods, a roulette wheel scheme and constrained 2-dimensional bin packing, for a batch of heterogeneous MapReduce jobs on heterogeneous virtual machine capacities, to improve makespan and resource utilization. Li et al. [16] discussed the problem of scheduling MapReduce tasks to heterogeneous geo-distributed data centers, where tasks have different deadlines. The goal is to minimize the total tardiness. They provided a Task Scheduling on Heterogeneous GeoDistributed Data Centers algorithm (TSGC) and demonstrated through experiments that it is effective for the considered problem. Wang et al. [17] studied the task scheduling problem with throughput as the objective in a heterogeneous environment. They provided a heterogeneous throughput-driven task scheduling algorithm (HTD), which can quickly obtain a reasonable sequence of job execution to ensure that the job set can be completed in the shortest possible time in a heterogeneous environment.

In this article, for problem Q m ∣ M R ( p m t n ) , o n l i n e ∣ C max , we give a lower bound γ * , where γ * is the positive root of the following quadratic equation ∑ i = 2 m s i γ 2 + ( s 1 − ∑ i = 2 m s i ) γ − ∑ i = 1 m s i = 0 . Note that γ * = s 2 + 2 s + 5 + 1 − s 2 when m = 2 , where s ≥ 1 is the speed ratio of two machines, so it is a general result for the problem in in the study by Jiang et al. [13]. Then we devise an 2 − 1 m -competitive online algorithm for P m ∣ M R ( p m t n ) , o n l i n e ∣ C max .

The rest of this article is organized as follows: In Section 2, we derive a lower bound for the problem Q m ∣ M R ( p m t n ) , o n l i n e ∣ C max . In Section 3, we present an approximate algorithm for the problem P m ∣ M R ( p m t n ) , o n l i n e ∣ C max . Finally, we conclude the article in Section 4.

2 Notations and lower bound for Q m ∣ M R ( p m t n ) , o n l i n e ∣ C max

Throughout the rest of this article, the following notations are used.

M j : the set of the map tasks of the job J j .

R j : the set of the reduce tasks of the job J j .

r j i : the reduce task in R j or the length of this task.

p ( S ) : the total length of the tasks in the set S .

C A ( J ) : the makespan produced by algorithm A after scheduling all jobs in the sequence of jobs J .

C * ( J ) : the optimal makespan after scheduling all jobs in the sequence of jobs J .

Throughout the rest of this article, we will not use another notation to denote the task length. For example, the r j i can denote a reduce task or the length of this task. We assume that r j 1 ≥ r j 2 ≥ … ≥ r j t j for every 1 ≤ j ≤ n . Let P j = ∑ i = 1 j ( p ( M i ) + p ( R i ) ) be the total length of the first j jobs and S k = ∑ i = 1 k s i .

We now give a lower bound of the preemptive version Q m ∣ M R ( p m t n ) , o n l i n e ∣ C max below.

We claim that for any 1 ≤ i ≤ m − 1 ,

Figure 1

Definition of x i and the schedule of J 2 m ‒ 1 .

Since the algorithm A is an online algorithm, the schedule for J 2 m − 1 − i is obtained from the schedule for J 2 m − 2 by removing the last i − 1 jobs. At the time x i , there are at least i jobs running according to the definition of x i , even after removing i − 1 jobs from the schedule at least one job running at time x i remains, and thus, C A ( J 2 m − 1 − i ) ≥ x i .

Noting that the competitive ratio of the algorithm A is γ , we must have C A ( J 2 m − 1 − i ) ≤ γ C * ( J 2 m − 1 − i ) for any 1 ≤ i ≤ m − 1 . By computing the job sequence J 2 m − 1 − i , we can obtain that C * ( J 2 m − 1 − i ) = m ( S m ) m − i − 2 ( S m − s 1 ) m − i − 1 . By combining it with ( 3 ) , we obtain that for any 1 ≤ i ≤ m − 1 ,

After scheduling all the jobs in J 2 m − 1 − i , the last job J 2 m − 1 = { M 2 m − 1 , R 2 m − 1 } arrives, where M 2 m − 1 = { S m x 1 − ∑ i = 1 m s i x i } and R 2 m − 1 = m ( S m ) m − 2 s 1 ( S m − s 1 ) m − 1 .

By the value of M 2 m − 1 , we can conclude that the completion time of M 2 m − 1 is at least x 1 . Noting that the job J 2 m − 1 has only one reduce task with size of m ( S m ) m − 2 s 1 ( S m − s 1 ) m − 1 , its completion time is at least x 1 + m ( S m ) m − 2 ( S m − s 1 ) m − 1 . Hence, we have C A ( J 2 m − 1 ) ≥ x 1 + m ( S m ) m − 2 ( S m − s 1 ) m − 1 . It is not difficult to obtain that

By (4), we obtain

i.e.,

Since f ( 0 ) = − 1 < 0 , the quadratic equation f ( γ ) = 0 has a positive root and a negative root. Denote by γ * the positive root and thus we have γ ≥ γ * .

3 An approximate algorithm for P m ∣ M R ( p m t n ) , o n l i n e ∣ C max

In this section, we provide an approximate online algorithm A with a competitive ratio of 2 − 1 m for the problem P m ∣ M R ( p m t n ) , o n l i n e ∣ C max . Let l j i denote the completion time of machine σ i at the moment right after the job J j has been scheduled, l j i ′ denote the completion time of machine σ i at the moment right after the map tasks of the job J j has been scheduled, i = 1 , 2 , … , m . Denote by C j A and C j * the makespan produced by algorithm A and the optimal makespan for the first j jobs, respectively.

Before presenting our algorithm, we introduce a procedure P ( M j , R j ) to schedule the job J j = ( M j , R j ) for the case, where l j − 1 1 = l j − 1 2 = … = l j − 1 m .

Procedure P ( M j , R j )

1. Assign all map tasks in M j evenly to m machines.

2. Use McNaughton’s wrap-around rule to schedule all the reduce tasks in R j .

3. Reindex machines such that l j 1 ≥ l j 2 ≥ … ≥ l j m .

Now we present our algorithm A as follows:

Algorithm A

1. If p ( M j ) < ( m − 1 ) l j − 1 1 − ∑ i = 2 m l j − 1 i ,

   • 1.1 Schedule all map tasks in M j between machines to finish them as early as possible,

   • 1.2 Take the longest reduce task r j k in R j not yet processed,

   • if l j − 1 i ′ + r j k ≥ l j − 1 1 ′ , where l j − 1 i ′ is the least loaded processor, partition r j k into two parts r j k 1 and r j k 2 such that l j − 1 i ′ + r j k 1 = l j − 1 1 ′ . Then schedule r j k 1 on σ i , let l j − 1 i ′ = l j − 1 i ′ + r j k 1 ;

   • else schedule r j k on σ i , let l j − 1 i ′ = l j − 1 i ′ + r j k ;

   • 1.3 Repeat the aforementioned steps until the reduce tasks in R j are scheduled once, or the completion time of each machine is l j − 1 1 ′ reindex machines such that l j − 1 1 ′ ≥ l j − 1 2 ′ ≥ … ≥ l j − 1 m ′ . Then use McNaughton’s rule to schedule the leftover reduce tasks in R j from the time l j − 1 1 ′ .

   • 1.4 Reindex machines such that l j 1 ≥ l j 2 ≥ … ≥ l j m .

2. If p ( M j ) > ( m − 1 ) l j − 1 1 − ∑ i = 2 m l j − 1 i ,

   1. schedule the portion p ( M j i ) = l j − 1 1 − l j − 1 i + 1 ( i = 1 , 2 , … , m − 1 ) of map tasks on σ i + 1 such that l j − 1 i + 1 + p ( M j i ) = l j − 1 1 ( i = 1 , 2 , … , m − 1 ) and denote the remainder of map tasks as M j ′ = M j \ ⋃ i = 1 m − 1 M j i .

   2. Run procedure P ( M j ′ , R j ) for the job J j ′ = ( M j ′ , R j ) .

4 Conclusion

We study online scheduling on m parallel machines in a MapReduce-like system where the map tasks can be arbitrarily split and processed in parallel on multiple machines.

There are some related problems that deserve further study. A natural question is to study whether the lower bound of the non-preemptive version Q m ∣ M R ∣ C max is the same as that of Q m ∣ ∣ C max . Can we apply the algorithm for Q m ∣ ∣ C max to tackle the problem Q m ∣ M R ∣ C max and obtain an online algorithm? It is also interesting to design a preemptive online algorithm.