DeepAI

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03/05/2022

### Worst-Case Analysis of LPT Scheduling on Small Number of Non-Identical Processors

The approximation ratio of the longest processing time (LPT) scheduling ...
06/14/2010

### Semi-Partitioned Hard Real-Time Scheduling with Restricted Migrations upon Identical Multiprocessor Platforms

Algorithms based on semi-partitioned scheduling have been proposed as a ...
02/11/2020

### A polynomial time parallel algorithm for graph isomorphism using a quasipolynomial number of processors

The Graph Isomorphism (GI) problem is a theoretically interesting proble...
02/28/2018

The sporadic task model is often used to analyze recurrent execution of ...
12/12/2012

### Feasibility Tests for Recurrent Real-Time Tasks in the Sporadic DAG Model

A model has been proposed in [Baruah et al., in Proceedings of the IEEE ...
05/18/2021

### Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval ...
09/08/2018

### Dependency Graph Approach for Multiprocessor Real-Time Synchronization

Over the years, many multiprocessor locking protocols have been designed...

## 1 Introduction

On a uniprocessor, checking the feasibility for an implicit-deadline task set is simple and well-known: the timing constraints are met by EDF if and only if the total utilization is at most [27]. Moreover, if every task on the processor is with , it is not difficult to see that testing whether the total utilization is less than or equal to is also a necessary and sufficient schedulability test. This can be achieved by considering a more stringent case which sets to for every . Hence, this special case of arbitrary-deadline task sets can be reformulated to task sets with implicit deadlines without any loss of precision. However, determining the schedulability for task sets with constrained or arbitrary deadlines in general is much harder, due to the complex interactions between the deadlines and the periods, and in particular is known to be co-hard or co-complete [17, 19, 18].

In this paper, we consider partitioned scheduling in homogeneous multiprocessor systems. Deciding if an implicit-deadline task set is schedulable on multiple processors is already -complete in the strong sense under partitioned scheduling. To cope with these -hardness issues, one natural approach is to focus on approximation algorithms, i.e., polynomial time algorithms that produce an approximate solution instead of an exact one. In our setting, this translates to designing algorithms that can find a feasible schedule using either (i) faster or (ii) additional processors. The goal, of course, is to design an algorithm that uses the least speeding up or as few additional processors as possible. In general, this approach is referred to as resource augmentation and is used extensively to analyze and compare scheduling algorithms. See for example [29] for a survey and motivation on why this is a useful measure for evaluating the quality of scheduling algorithms in practice. However, such a measure also has its potential pitfalls as recently studied and reported by Chen et al. [12]. Interestingly, it turns out that there is a huge difference regarding the approximation factors depending on whether it is possible to increase the processor speed or the number of processors. As already discussed in [11], approximation by speeding up is known as the multiprocessor partitioned scheduling problem, and by allocating more processors is known as the multiprocessor partitioned packing problem. We study the latter one in this paper.

Formally, an algorithm for the multiprocessor partitioned packing problem is said to have an approximation factor , if given any task set , it can find a feasible partition of on processors, where is the minimum (optimal) number of processors required to schedule . However, it turns out that the approximation factor is not the best measure in our setting (it is not fine-grained enough). For example, it is -complete to decide if an implicit-deadline task set is schedulable on 2 processors or whether 3 processors are necessary. Assuming , this rules out the possibility of any efficient algorithm with approximation factor better than , as shown in [11]. (This lower bound is further lifted to for sporadic tasks in Section 5.) The problem with this example is that it does not rule out the possibility of an algorithm that only needs processors. Clearly, such an algorithm is almost as good as optimum when is large and would be very desirable.111Indeed, there are (very ingenious) algorithms known for the implicit-deadline partitioning problem that use only processors [25], based on the connection to the bin-packing problem. To get around this issue, a more refined measure is the so-called asymptotic approximation factor. An algorithm has an asymptotic approximation factor if we can find a schedule using at most processors, where is a constant that does not depend on . An algorithm is called an asymptotic polynomial-time approximation scheme (APTAS) if, given an arbitrary accuracy parameter as input, it finds a schedule using processors and its running time is polynomial assuming is a fixed constant.

For implicit-deadline task sets, the multiprocessor partitioned scheduling problem, by speeding up, is equivalent to the Makespan problem [21], and the multiprocessor partitioned packing problem, by allocating more processors, is equivalent to the bin packing problem [20]. The Makespan problem admits polynomial-time approximation schemes (PTASes), by Hochbaum and Shmoys [22], and the bin packing problem admits asymptotic polynomial-time approximation schemes (APTASes), by de la Vega and Lueker [16, 25].

deals with the multiprocessor partitioned scheduling problem as a vector scheduling problem

[7] by constructing (roughly) dimensions and then applies the PTAS of the vector scheduling problem developed by Chekuri and Khanna [7] in a black-box manner. Bansal et al. [1] exploit the special structure of the vectors and give a faster vector scheduling algorithm that is a quasi-polynomial-time approximation scheme (qPTAS) even if is polynomially bounded.

For the results, from the literature and also this paper, related to the multiprocessor partitioned scheduling and packing problems, Table 1 provides a short summary.

Our Contributions This paper studies the multiprocessor partitioned packing problem in much more detail. On the positive side, when the ratio of the period of a constrained-deadline task to the relative deadline of the task is at most , in Section 3, we provide a simple polynomial-time algorithm with a -approximation factor. In Section 4, we show that the deadline-monotonic partitioning algorithm in [3, 4] has an asymptotic -approximation factor for the packing problem, where . In particular, when and are not constant, adopting the worst-fit or best-fit strategy in the deadline-monotonic partitioning algorithm is shown to have an approximation factor, where is the number of tasks. In contrast, from [10], it is known that both strategies have a speed-up factor , when the resource augmentation is to speed up processors. We also show that speeding up processors can be much more powerful than allocating more processors. Specifically, in Section 5, we provide input instances, in which the only feasible schedule is to run each task on an individual processor but the system requires only one processor with a speed-up factor of , where .

On the negative side, in Section 6, we show that there does not exist any asymptotic polynomial-time approximation scheme (APTAS) for the multiprocessor partitioned packing problem for task sets with constrained deadlines, unless . As there is already an APTAS for the implicit deadline case, this together with the result in [11] gives a complete picture of the approximability of multiprocessor partitioned packing for different types of task sets, as shown in Table 1.

## 2 System Model

### 2.1 Task and Platform Model

In this paper we focus on partitioned scheduling, i.e., each task is statically assigned to a fixed processor and all jobs of the task is executed on the assigned processor. On each processor, the jobs related to the tasks allocated to that processor are scheduled using preemptive earliest deadline first (EDF) scheduling. This means that at each point the job with the shortest absolute deadline is executed, and if a new job with a shorter absolute deadline arrives the currently executed job is preempted and the new arriving job starts executing. A task set can be feasibly scheduled by EDF (or EDF is a feasible schedule) on a processor if the timing constraints can be fulfilled by using EDF.

### 2.2 Problem Definition

Given a task set , a feasible task partition on identical processors is a collection of subsets, denoted , such that

• for all ,

• is equal to the input task set , and

• set can meet the timing constraints by EDF scheduling on a processor .

The multiprocessor partitioned packing problem: The objective is to find a feasible task partition on identical processors with the minimum .

We assume that and for any task since otherwise there cannot be a feasible partition.

### 2.3 Demand Bound Function

This paper focuses on the case where the arrival times of the sporadic tasks are not specified, i.e., they arrive according to their interarrival constraint and not according to a pre-defined pattern. Baruah et al. [5] have shown that in this case the worst-case pattern is to release the first job of tasks synchronously (say, at time for notational brevity), and all subsequent jobs as early as possible. Therefore, as shown in [5], the demand bound function of a task that specifies the maximum demand of task to be released and finished within any time interval with length is defined as

 (1)

The exact schedulability test of EDF, to verify whether EDF can feasibly schedule the given task set on a processor, is to check whether the summation of the demand bound functions of all the tasks is always less than for all [5].

## 3 Reduction to Bin Packing

When considering tasks with implicit deadlines, the multiprocessor partitioned packing problem is equivalent to the bin packing problem [20]. Therefore, even though the packing becomes more complicated when considering tasks with arbitrary or constrained deadlines, it is pretty straightforward to handle the problem by using existing algorithms for the bin packing problem if the maximum ratio of the period to the relative deadline among the tasks, i.e., , is not too large.

For a given task set , we can basically transform the input instance to a related task instance by creating task based on task in such that

• is , is , and is when for , and

• is , is and is when for .

Now, we can adopt any greedy fitting algorithms (i.e., a task is assigned to “one” allocated processor that is feasible; otherwise, a new processor is allocated and the task is assigned to the newly allocated processor) for the bin packing problem by considering only the utilization of transformed tasks in for the multiprocessor partitioned packing problem, as presented in [30, Chapter 8]. The construction of has a time complexity of , and the greedy fitting algorithm has a time complexity of .

Any greedy fitting algorithm by considering for task assignment is a -approximation algorithm for the multiprocessor partitioned packing problem.

###### Proof.

Clearly, as we only reduce the relative deadline and the periods, the timing parameters in are more stringent than in . Hence, a feasible task partition for on processors also yields a corresponding feasible task partition for on processors. As has implicit deadlines, we know that any task subset in with total utilization no more than can be feasibly scheduled by EDF on a processor, and therefore the original tasks in that subset as well. For any greedy fitting algorithms that use processors, using the same proof as in [30, Chapter 8], we get .

By definition, we know that . Therefore, any feasible solution for uses at least processors and the approximation factor is hence proved. ∎

## 4 Deadline-Monotonic Partitioning under EDF Scheduling

This section presents the worst-case analysis of the deadline-monotonic partitioning strategy, proposed by Baruah and Fisher [4, 3], for the multiprocessor partitioned packing problem. Note that the underlying scheduling algorithm is EDF but the tasks are considered in the deadline-monotonic (DM) order. Hence, in this section, we index the tasks accordingly from the shortest relative deadline to the longest, i.e., if . Specifically, in the DM partitioning, the approximate demand bound function is used to approximate Eq. (1), where

 \scdbf∗(τi,t)={0if t

Even though the DM partitioning algorithm in [4, 3] is designed for the multiprocessor partitioned scheduling problem, it can be easily adapted to deal with the multiprocessor partitioned packing problem. For completeness, we revise the algorithm in [4, 3] for the multiprocessor partitioned packing problem and present the pseudo-code in Algorithm 1. As discussed in [4, 3], when a task is considered, a processor among the allocated processors where both the following conditions hold

 Ci+∑τj∈Tm\scdbf∗(τj,Di)≤Di (3) ui+∑τj∈Tmuj≤1 (4)

is selected to assign task , where is the set of the tasks (as a subset of ), which have been assigned to processor before considering . If there is no where both Eq. (3) and Eq. (4) hold, a new processor is allocated and task is assigned to the new processor. The order in which the already allocated processors are considered depends on the fitting strategy:

• first-fit (FF) strategy: choosing the feasible with the minimum index;

• best-fit (BF) strategy: choosing, among the feasible processors, with the maximum approximate demand bound at time ;

• worst-fit (WF) strategy: choosing with the minimum approximate demand bound at time .

For a given number of processors, it has been proved in [10] that the speed-up factor of the DM partitioning is at most , independent from the fitting strategy. However, if the objective is to minimize the number of allocated processors, we will show that DM partitioning has an approximation factor of at least (in the worst case) when the best-fit or worst-fit strategy is adopted. We will prove this by explicitly constructing two concrete task sets with this property. Afterwards, we show that the asymptotic approximation factor of DM partitioning is at most for packing, where .

The approximation factor of the deadline-monotonic partitioning algorithm with the best-fit strategy is at least when and the schedulability test is based on Eq. (3) and Eq. (4).

###### Proof.

The theorem is proven by providing a task set that can be scheduled on two processors but where Algorithm 1 when applying the best-fit strategy uses processors. Under the assumption that is an integer, is , and is sufficiently large, i.e., , such a task set can be constructed as:

• Let , , and .

• For , let , , and .

• For , let , , and .

The task set can be scheduled on two processors under EDF if all tasks with an odd index are assigned to processor 1 and all tasks with an even index are assigned to processor 2. On the other hand, the best-fit strategy assigns

to processor . The resulting solution uses processors. Details are in the Appendix. ∎

The approximation factor of the deadline-monotonic partitioning algorithm with the worst-fit strategy is at least when the schedulability test is based on Eq. (3) and Eq. (4).

###### Proof.

The proof is very similar to the proof of Theorem 4, considering the task set:

• Let , , and .

• For , let , , and .

• For , let , , and .

Odd tasks are assigned to processor 1 and even tasks to processor 2 the task set is schedulable while is assigned to processor using the worst-fit strategy. Details are in the Appendix. ∎

The DM partitioning algorithm is an asymptotic -approximation algorithm for the multiprocessor partitioned packing problem, when and .

###### Proof.

We consider the task which is the task that is responsible for the last processor that is allocated by Algorithm 1. The other processors are categorized into two disjoint sets and , depending on whether Eq. (3) or Eq. (4) is violated when Algorithm 1 tries to assign (if both conditions are violated, the processor is in ). The two sets are considered individually and the maximum number of processors in both sets is determined based on the minimum utilization for each of the processors. Afterwards, a necessary condition for the amount of processors that is at least needed for a feasible solution is provided and the relation between the two values proves the theorem. Details can be found in the Appendix. ∎

## 5 Hardness of Approximations

It has been shown in [11, 2] that a PTAS exists for augmenting the resources by speeding up. A straightforward question is to see whether such PTASes will be helpful for bounding the lower or upper bounds for multiprocessor partitioned packing. Unfortunately, the following theorem shows that using speeding up to get a lower bound for the number of required processors is not useful.

There exists a set of input instances, in which the number of allocated processors is up to , while the task set can be feasibly scheduled by EDF with a speed-up factor on a processor, where .

###### Proof.

We provide a set of input instances, with the property described in the statement:

• Let , , and .

• For any , let , , and .

Since for any task , assigning any two tasks on the same processor is infeasible without speeding up. Therefore, the only feasible processor allocation is processors and to assign each task individually on one processor. However, by speeding up the system by a factor , the tasks can be feasibly scheduled on one processor due to for any . A proof is in the Appendix. Hence, the gap between these two types of resource augmentation is up to . ∎

Moreover, the following theorem shows the inapproximability for a factor without adopting asymptotic approximation.

For any , there is no polynomial-time approximation algorithm with an approximation factor of for the multiprocessor partitioned packing problem, unless .

###### Proof.

Suppose that there exists such a polynomial-time algorithm with approximation factor . can be used to decide if a task set is schedulable on a uniprocessor, which would contradict the -hardness [17] of this problem. Indeed, we simply run on the input instance. If returns a feasible schedule using one processor, we already have a uniprocessor schedule. On the other hand, if requires at least two processors, then we know that any optimum solution needs processors, implying that the task set is not schedulable on a uniprocessor. ∎

## 6 Non-Existence of APTAS for Constrained Deadlines

We now show that there is no APTAS when considering constrained-deadline task sets, unless . The proof is based on an L-reduction (informally an approximation preserving reduction) from a special case of the vector packing problem, i.e., the 2D dominated vector packing problem.

### 6.1 The 2D Dominated Vector Packing Problem

The vector packing problem is defined as follows: The vector packing problem: Given a set of vectors with dimensions, in which is the value for vector in the -th dimension, the problem is to partition into parts such that is minimized and each part is feasible in the sense that for all . That is, for each dimension , the sum of the -th coordinates of the vectors in is at most .

We say that a subset of can be feasibly packed in a bin if for all -th dimensions. Note that for this is precisely the bin-packing problem. The vector packing problem does not admit any polynomial-time asymptotic approximation scheme even in the case of dimensions, unless [31].

Specifically, the proof in [11] for the non-existence of APTAS for task sets with arbitrary deadlines comes from an L-reduction from the -dimensional vector packing problem as follows: For a vector in , a task is created with , , and . However, a trivial extension from [11] to constrained deadlines does not work, since for the transformation of the task set we need to assume that for any so that for every reduced task . This becomes problematic, as one dimension in the vectors in such input instances for the two-dimensional vector packing problem can be totally ignored, and the input instance becomes a special case equivalent to the traditional bin-packing problem, which admits an APTAS. We will show that the hardness is equivalent to a special case of the two-dimensional vector packing problem, called the two-dimensional dominated vector packing (2D-DVP) problem, in Section 6.2. The two-dimensional dominated vector packing (2D-DVP) problem is a special case of the two-dimensional vector packing problem with following conditions for each vector :

• , and

• if , then is dominated by , i.e., .

Moreover, we further assume that and are rational numbers for every .

Here, some tasks are created with implicit deadlines (based on vector if is ) and some tasks with strictly constrained deadlines (based on vector if is not ). However, the 2D-DVP problem is a special case of the two-dimensional vector packing problem, and the implication for when does not hold in the proof in [31]. We note, that the proof for the non-existence of an APTAS for the two-dimensional vector packing problem in [31] is erroneous. However, the result still holds. Details are in the Appendix. Therefore, we will provide a proper -reduction in Section 6.3 to show the non-existence of APTAS for the multiprocessor partitioned packing problem for tasks with constrained deadlines.

### 6.2 2D-DVP Problem and Packing Problem

We now show that the packing problem is at least as hard as the 2D-DVP problem from a complexity point of view. For vector with , we create a corresponding task with

 Di=1,  Ci=vi,2,  Ti=vi,2vi,1.

Clearly, for such tasks. Let be a common multiple, not necessary the least, of the periods of the tasks constructed above. By the assumption that all the values in the 2D-DVP problem are rational numbers and for every vector , we know that exists and can be calculated in . For vector with , we create a corresponding implicit-deadline task with

 Ti=Di=H,  Ci=vi,1Ti.

The following lemma shows the related schedulability condition.

Suppose that the set of tasks assigned on a processor consists of (1) strictly constrained-deadline tasks, denoted by , with a common relative deadline and (2) implicit-deadline tasks, i.e., , in which the period is a common integer multiple of the periods of the strictly constrained-deadline tasks. EDF schedule is feasible for the set of tasks on a processor if and only if

 ∑τi∈T
###### Proof.

Only if: This is straightforward as the task set cannot meet the timing constraint when or .

If: If and , we know that when , then . When , we have

 ∑τi∈Tm\scdbf(τi,t)=∑τi∈T

Moreover, with , we know that when

 ∑τi∈Tm\scdbf(τi,H)= ∑τi∈T

where comes from the fact that is an integer for any in and so that is equal to .

For any value , the value of is equal to
. Therefore, we know that if and , the task set can be feasibly scheduled by EDF. ∎

If there does not exist any APTAS for the 2D-DVP problem, unless , there also does not exist any APTAS for the multiprocessor partitioned packing problem with constrained-deadline task sets.

###### Proof.

Clearly, the reduction in this section from the 2D-DVP problem to the multiprocessor partitioned packing problem with constrained deadlines is in polynomial time.

For a task subset of , suppose that is the set of the corresponding vectors that are used to create the task subset . By Lemma 6.2, the subset of the constructed tasks can be feasibly scheduled by EDF on a processor if and only if and .

Therefore, it is clear that the above reduction is a perfect approximation preserving reduction. That is, an algorithm with a (asymptotic) approximation factor for the multiprocessor partitioned packing problem can easily lead to a (asymptotic) approximation factor for the 2D-DVP problem. ∎

### 6.3 Hardness of the 2D-DVP problem

Based on Theorem 6.2, we are going to show that there does not exist APTAS for the 2D-DVP problem, which also proves the non-existence of APTAS for the multiprocessor partitioned packing problem with constrained deadlines.

There does not exist any APTAS for the 2D-DVP problem, unless .

###### Proof.

This is proved by an L-reduction, following a similar strategy in [31] by constructing an L-reduction from the Maximum Bounded 3-Dimensional Matching (MAX-3-DM), which is MAX SNP-complete [24]. Details are in the Appendix, where a short comment regarding an erroneous observation in [31] is also provided. ∎

The following theorem results from Theorems 6.2 and 6.3.

There does not exist any APTAS for the multiprocessor partitioned packing problem for constrained-deadline task sets, unless .

## 7 Concluding Remarks

This paper studies the partitioned multiprocessor packing problem to minimize the number of processors needed for multiprocessor partitioned scheduling. Interestingly, there turns out to be a huge difference (technically) in whether one is allowed faster processors or additional processors. Our results are summarized in Table 1. For general cases, the upper bound and lower bound for the first-fit strategy in the deadline-monotonic partitioning algorithm are both open. The focus of this paper is the multiprocessor partitioned packing problem. If global scheduling is allowed, in which a job can be executed on different processors, the problem of minimizing the number of processors has been also recently studied in a more general setting by Chen et al. [14, 13] and Im et al. [23]. They do not explore any periodicity of the job arrival patterns. Among them, the state-of-the-art online competitive algorithm has an approximation factor (more precisely, competitive factor) of by Im et al. [23]. These results are unfortunately not applicable for the multiprocessor partitioned packing problem since the jobs of a sporadic task may be executed on different processors.

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## Appendix

### Proofs related to Section 4

Proof of Theorem 4. We provide a task set that can be scheduled on two processors but where Algorithm 1 when applying the best-fit strategy uses processors. Let be an integer, is , and is sufficiently large, i.e., .

• Let , , and .

• For , let , , and .

• For , let , , and .

Hence, in this input instance, , , , . For the simplicity of presentation, we will omit any term multiplied with by assuming that this is positive and arbitrarily small. When applying DM partitioning, tasks and are both assigned on processor . Then, we know that at time , . Clearly, are not eligible for processor , because for we have

 C2i+1+\scdbf∗(τ1,D2i+1)+\scdbf∗(τ2,D2i+1) ≈ Ki−Ki−1+1K+KiK>Ki=D2i+1. (6)

Therefore, is assigned on processor . When considering , both processors are feasible, and processor has a higher approximate demand at time , i.e., and . Therefore, is assigned on processor under the best-fit strategy. Similarly, are not eligible for processor , because for we have

 C2i+1+\scdbf∗(τ3,D2i+1)+\scdbf∗(τ4,D2i+1) ≈ Ki−Ki−1+C3+KiK>Ki=D2i+1. (7)

When considering , the allocated three processors are all feasible, but processor has a higher approximate demand at time . One can formally prove that task is assigned to processor because for any . Moreover, since for any and due to the assumption , we know that processor has the highest approximate demand at time among the first (allocated) processors. Thus, task is assigned to processor due to the best-fit strategy. Therefore, we conclude that the best-fit strategy assigns to processor . The resulting solution uses processors.

Now, consider the following task assignment, in which is assigned on processor (resp., ) if is an odd (resp. even) number. Let be the set of tasks that are assigned on processor . The assignment is feasible on processor , as all the tasks are with implicit deadlines, and the total utilization is . The assignment is also feasible on processor by verifying the schedulability by using , i.e., the demand bound function without approximation! Since all tasks in have the same period, we only have to verify at time , in which for .

We will now show that when , the of at time will still be no more than , i.e., showing that . Since the tasks in have the same period, for the simplicity of presentation, let be with . We can divide the time interval into . Suppose that is a non-negative integer and is an index , where is in interval . Here, is an auxiliary parameter set to and is an auxiliary parameter set to for brevity.

Then, due to the parameters of task and , for task , we have and . As a result, when and , we have

 ∑τi∈T′1dbf(τi,t)=ℓ∑τi∈T′1Ci+∑τi∈T′1 and i

Moreover, when , we have . When , we have . When , we have . Therefore, we reach the conclusion that .

Hence, there exists a feasible solution by using only processors, but the DM partitioning algorithm under BF uses processors.

Proof of Theorem 4. Suppose that is an integer, is , and is sufficiently large, i.e., . Consider the following input task set:

• Let , , and .

• For , let , , and .

• For , let , , and .

We know that , , , . The proof is very similar to that of Theorem 4. For the simplicity of presentation, we will omit any term multiplied with by assuming that this is positive and arbitrarily small.

When applying DM partitioning, task and are both assigned on processor . One can formally prove that task is assigned to processor because for any . Moreover, since