CN111459655B - Deterministic method for solving optimal solution of multiprocessor overload scheduling problem - Google Patents

Deterministic method for solving optimal solution of multiprocessor overload scheduling problem Download PDF

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CN111459655B
CN111459655B CN202010141625.5A CN202010141625A CN111459655B CN 111459655 B CN111459655 B CN 111459655B CN 202010141625 A CN202010141625 A CN 202010141625A CN 111459655 B CN111459655 B CN 111459655B
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廖晓鹃
张辉
黄荣
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Chengdu Univeristy of Technology
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Abstract

The invention discloses a deterministic method for solving the optimal solution of multiprocessor overload scheduling problem, the algorithm comprises the steps of (S1) determining the scheduling problem, and coding the regular attribute of the scheduling problem into a MaxSAT hard clause; (S2) encoding the scheduling target as a MaxSAT soft clause; (S3) connecting the hard clause obtained in the step (S1) and the soft clause obtained in the step (S2) by a conjunction operator, thereby obtaining a MaxSAT problem; (S4) calculating an optimal solution of the MaxSAT problem by the MaxSAT solver. Through the scheme, the invention achieves the purpose of obtaining the optimal scheduling scheme of the multiprocessor scheduling system, and has strong expandability, practical value and popularization value.

Description

Deterministic method for solving optimal solution of multiprocessor overload scheduling problem
Technical Field
The invention belongs to the technical field of computer application, and particularly relates to a deterministic algorithm for solving an optimal solution of a multiprocessor overload scheduling problem.
Background
The scheduling problem on multiprocessors has long been studied extensively. Since Graham et al expressed the scheduling problem in the form of α | β | γ for processor attributes, task attributes, scheduling objectives in 1979, much research has been devoted to the complexity analysis of the scheduling problem, i.e., the complexity of the problem is analyzed by adjusting the α, β, γ attributes of the scheduling problem. For example: p | pmtn | Sigma Uj(preemptible scheduling on parallel processors, all tasks ready at the same time, with the goal of maximizing the number of tasks completed by the deadline) proved to be an NP-hard problem in 1983. If the tasks have different ready times, the problem is denoted as P | pmtn, rj|∑UjDu et al demonstrate that when the number of processors is 2, the problem is already an NP-hard problem. Assigning different weights to each task further increases the difficulty of the problem, for example: p | pmtn, Pj＝p|∑Uj(preemptible scheduling on parallel processors, all tasks being ready at the same time and task processing being equal in duration, with the goal of maximizing the number of tasks completed before the deadline) is a polynomial solvable problem, and P | pmtn, Pj＝p|∑wjUj(preemptible scheduling on parallel processors, all tasks being ready at the same time and task processing being equal in duration, with the goal of maximizing the sum of the task weights completed by the deadline) is an NP-hard problem. Due to the difficulty of the multiprocessor scheduling problem, there are currently many approaches to P | pmtn, rj|∑wjUj(preemptible scheduling on parallel processors, all tasks having different ready times, with the goal of maximizing the sum of the weights of the tasks completed before the deadline) problem, the solving algorithm only works onApproximate solutions to the problem are found which, while giving a more efficient scheduling scheme than the classical scheduling algorithm, do not ensure that the output approximate solution is stable close to the optimal solution.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a deterministic algorithm for solving the optimal solution of the multiprocessor overload scheduling problem, the scheduling is carried out in a preemptible mode according to different task ready times, and the aim is to maximize the sum of task weights completed before the deadline so as to determine the optimal solution of the scheduling problem.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a deterministic algorithm for solving an optimal solution of a multiprocessor overload scheduling problem comprises the following steps:
(S1) determining a scheduling problem, encoding a rule attribute of the scheduling problem into a MaxSAT hard clause;
(S2) encoding the scheduling target as a MaxSAT soft clause;
(S3) connecting the hard clause obtained in the step (S1) and the soft clause obtained in the step (S2) by a conjunction operator, thereby obtaining a MaxSAT problem;
(S4) calculating an optimal solution of the MaxSAT problem by the MaxSAT solver.
Further, the MaxSAT hard clause corresponding to the rule attribute coded in the step (S1) should satisfy the following rule at the same time;
rule one is as follows:
any task slicing must be performed by some single processor, encoding this rule attribute as a hard clause as shown in equation (1):
Figure GDA0003480248160000021
wherein the content of the first and second substances,
Figure GDA0003480248160000022
is a Boolean variable representing the ith slice f of task ji jIs arranged in the processor MuExecuting;
rule two:
when the k-th slice of task i
Figure GDA0003480248160000023
And the ith slice f of task jl jWhen there is an overlap in the execution time periods of the same processor, there is a possibility that
Figure GDA0003480248160000024
Before fl jCarry out either ofl jPrior to the generation of
Figure GDA0003480248160000025
Executing, encoding such rule attribute into a hard clause shown in formula (2):
Figure GDA0003480248160000026
wherein the content of the first and second substances,
Figure GDA0003480248160000027
is a Boolean variable representing the kth slice of task i
Figure GDA0003480248160000028
Is arranged in the processor MuThe upper side is executed in the upper part,
Figure GDA0003480248160000029
is a Boolean variable representing the kth slice of task i
Figure GDA00034802481600000210
The ith slice f preceding task jl jThe execution is carried out in such a way that,
Figure GDA00034802481600000211
is a boolean variable indicating that the ith slice of task j is executed before the kth slice of task i;
rule three:
the start time of the ith slice of task j is no earlier than the completion time of all task slices before it, and this rule attribute is encoded as a hard clause as shown in equation (3):
Figure GDA00034802481600000212
wherein the content of the first and second substances,
Figure GDA0003480248160000031
is a boolean variable indicating that the ith slice of task j starts executing after time t or t, pjIs the execution duration of the task j, and since any unit time gap of the task in the execution process can be preempted, pjAlso indicates the number of the fragments of the task j and the sequence number of the last fragment of the task, rjIs the ready time of task j, n is the total number of system tasks;
rule four:
the ith fragment of task j is executed before the (i + 1) th fragment of task j, and the rule attribute is encoded into a hard clause shown in formula (4):
Figure GDA0003480248160000032
wherein the content of the first and second substances,
Figure GDA0003480248160000033
is a boolean variable indicating that the ith slice of task j is executed before the (i + 1) th slice of task j;
rule five:
when task slicing fi jStarting at or after time t, it must start after t-1 or t-1, and encodes this rule attribute as a hard clause as shown in equation (5):
Figure GDA0003480248160000034
wherein the content of the first and second substances,
Figure GDA0003480248160000035
is a boolean variable indicating that the ith slice of task j begins execution after time t or t,
Figure GDA0003480248160000036
is a Boolean variable which indicates that the ith slice of the task j starts to be executed after the time t-1 or t-1;
rule six:
when a task ends at t, its last slice must start at or before time t-1, and this rule attribute is encoded as a hard clause as shown in equation (6):
Figure GDA0003480248160000037
wherein, ebjIs a boolean variable that indicates that task j completes before its deadline,
Figure GDA0003480248160000038
is a Boolean variable representing the pth of task jjThe slicing starts to be executed after time t or t.
Further, for the Boolean variables that occurred in rule two and rule four
Figure GDA0003480248160000039
It is used to generate a hard clause represented by equation (7):
Figure GDA0003480248160000041
where t and t' should satisfy, ri+k(k-1)/2≤t≤di-(k+pi)(pi-k+1)/2+1，
Figure GDA0003480248160000042
Wherein r isjIs the ready time of task j, djIs the deadline for task j.
Specifically, in the step (S2), the scheduling objective maximizes the sum of the weights of tasks completed by the deadline, and is encoded as a MaxSAT weighted soft clause expressed by equation (8):
(ebj，wj)(1≤j≤n) (8)
wherein, wjIndicating the revenue obtained from completing task j.
Compared with the prior art, the invention has the following beneficial effects:
(1) the invention obtains the MaxSAT problem by encoding the task attribute and the scheduling target into the MaxSAT clause and connecting the maxSAT clause and the maxSAT clause by using the conjunction operator, and obtains the optimal scheduling scheme by using the MaxSAT solver. Compared with the prior art, the method can obtain the optimal scheduling scheme of the multiprocessor scheduling system, is suitable for occasions where the same scheduling problem repeatedly occurs, and has strong expandability, practical value and popularization value.
(2) The invention has strong expandability and execution efficiency, and can be used for solving the situation that the task attribute changes, such as: and the task ready time is changed, and the preemptive/non-preemptive scheduling method is switched.
Drawings
FIG. 1 is a flow chart of the operation of the present invention.
Detailed Description
The present invention is further illustrated by the following figures and examples, which include, but are not limited to, the following examples.
Examples
As shown in fig. 1, a deterministic algorithm for solving an optimal solution of a multiprocessor overload scheduling problem includes the following specific steps:
(S1) determining a scheduling problem, and encoding the task attribute of the scheduling problem into a MaxSAT hard clause corresponding to the task attribute, wherein the following rules are satisfied simultaneously during encoding;
rule one is as follows: any task slicing must be performed by some single processor, encoding this rule attribute as a hard clause as shown in equation (1):
Figure GDA0003480248160000051
wherein the content of the first and second substances,
Figure GDA0003480248160000052
is a Boolean variable representing the ith slice f of task ji jIs arranged in the processor MuExecuting;
rule two: when the k-th slice of task i
Figure GDA0003480248160000053
And the ith slice f of task jl jWhen there is an overlap in the execution time periods of the same processor, there is a possibility that
Figure GDA0003480248160000054
Before fl jCarry out either ofl jPrior to the generation of
Figure GDA0003480248160000055
Executing, encoding such rule attribute into a hard clause shown in formula (2):
Figure GDA0003480248160000056
wherein the content of the first and second substances,
Figure GDA0003480248160000057
is a Boolean variable representing the kth slice of task i
Figure GDA0003480248160000058
Is arranged in the processor MuThe upper side is executed in the upper part,
Figure GDA0003480248160000059
is a Boolean variable representing the kth slice of task i
Figure GDA00034802481600000510
The ith slice f preceding task jl jThe execution is carried out in such a way that,
Figure GDA00034802481600000511
is a boolean variable indicating that the ith slice of task j is executed before the kth slice of task i;
rule three: the start time of the ith slice of task j is no earlier than the completion time of all task slices before it, and this rule attribute is encoded as a hard clause as shown in equation (3):
Figure GDA00034802481600000512
wherein the content of the first and second substances,
Figure GDA00034802481600000513
is a boolean variable indicating that the ith slice of task j starts executing after time t or t, pjIs the execution duration of the task j, and since any unit time gap of the task in the execution process can be preempted, pjAlso indicates the number of the fragments of the task j and the sequence number of the last fragment of the task, rjIs the ready time of task j, n is the total number of system tasks;
rule four: the ith fragment of task j is executed before the (i + 1) th fragment of task j, and the rule attribute is encoded into a hard clause shown in formula (4):
Figure GDA00034802481600000514
wherein the content of the first and second substances,
Figure GDA00034802481600000515
is a boolean variable indicating that the ith slice of task j is executed before the (i + 1) th slice of task j;
rule five: when task slicing fi jStarting at or after time t, it must start after t-1 or t-1, and encodes this rule attribute as a hard clause as shown in equation (5):
Figure GDA0003480248160000061
wherein the content of the first and second substances,
Figure GDA0003480248160000062
is a boolean variable indicating that the ith slice of task j begins execution after time t or t,
Figure GDA0003480248160000063
is a Boolean variable which indicates that the ith slice of the task j starts to be executed after the time t-1 or t-1;
rule six: when a task ends at t, its last slice must start at or before time t-1, and this rule attribute is encoded as a hard clause as shown in equation (6):
Figure GDA0003480248160000064
wherein, ebjIs a boolean variable that indicates that task j completes before its deadline,
Figure GDA0003480248160000065
is a Boolean variable representing the pth of task jjOne slice (i.e., the last slice) starts to execute after time t or t.
For both the Boolean variables present in rule two and rule four
Figure GDA0003480248160000066
It is used to generate a hard clause represented by equation (7):
Figure GDA0003480248160000067
where t and t' should satisfy, ri+k(k-1)/2≤t≤di-(k+pi)(pi-k+1)/2+1，
Figure GDA0003480248160000068
Wherein r isjIs the ready time of task j, djIs the deadline for task j.
(S2) encoding the scheduling objective as a MaxSAT weighted soft clause implementing the scheduling objective: maximizing the sum of the weights of tasks completed before the cutoff time is shown in equation (8):
(ebj，wj)(1≤j≤n) (8)
wherein, wjIndicating the revenue obtained from completing task j.
(S3) connecting the hard clause obtained in the step (S1) and the soft clause obtained in the step (S2) by a conjunction operator a, thereby obtaining a MaxSAT problem;
(S4) calculating an optimal solution of the MaxSAT problem by the MaxSAT solver, the optimal solution including the start execution time of each task slice, whereby an optimal scheduling scheme can be obtained.
The above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention, but all changes that can be made by applying the principles of the present invention and performing non-inventive work on the basis of the principles shall fall within the scope of the present invention.

Claims (1)

1. A deterministic method for solving an optimal solution of a multiprocessor overload scheduling problem is characterized by comprising the following steps:
(S1) determining a scheduling problem, encoding a rule attribute of the scheduling problem into a MaxSAT hard clause; wherein, the rule attribute is coded into the MaxSAT hard clause corresponding to the rule attribute, and the MaxSAT hard clause simultaneously satisfies the following rules;
rule one is as follows:
any task slicing must be performed by some single processor, encoding this rule attribute as a hard clause as shown in equation (1):
Figure FDA0003480248150000011
wherein the content of the first and second substances,
Figure FDA0003480248150000012
is a Boolean variable representing the ith slice f of task ji jIs arranged in the processor MuExecuting;
rule two:
when the k-th slice of task i
Figure FDA0003480248150000013
And the ith slice f of task jl jWhen there is an overlap in the execution time periods of the same processor, there is a possibility that
Figure FDA0003480248150000014
Before fl jCarry out either ofl jPrior to the generation of
Figure FDA0003480248150000015
Executing, encoding such rule attribute into a hard clause shown in formula (2):
Figure FDA0003480248150000016
wherein the content of the first and second substances,
Figure FDA0003480248150000017
is a Boolean variable representing the kth slice of task i
Figure FDA0003480248150000018
Is arranged in the processor MuThe upper side is executed in the upper part,
Figure FDA0003480248150000019
is a Boolean variable representing the kth slice of task i
Figure FDA00034802481500000110
The ith slice f preceding task jl jThe execution is carried out in such a way that,
Figure FDA00034802481500000111
is a boolean variable indicating that the ith slice of task j is executed before the kth slice of task i;
rule three:
the start time of the ith slice of task j is no earlier than the completion time of all task slices before it, and this rule attribute is encoded as a hard clause as shown in equation (3):
Figure FDA00034802481500000112
wherein the content of the first and second substances,
Figure FDA00034802481500000113
is a boolean variable indicating that the ith slice of task j starts executing after time t or t, pjIs the execution duration of the task j, and since any unit time gap of the task in the execution process can be preempted, pjAlso indicates the number of the fragments of the task j and the sequence number of the last fragment of the task, rjIs the ready time of task j, n is the total number of system tasks;
rule four:
the ith fragment of task j is executed before the (i + 1) th fragment of task j, and the rule attribute is encoded into a hard clause shown in formula (4):
Figure FDA0003480248150000021
wherein the content of the first and second substances,
Figure FDA0003480248150000022
is a boolean variable indicating that the ith slice of task j is executed before the (i + 1) th slice of task j;
rule five:
when task slicing fi jStarting at or after time t, it must start after t-1 or t-1, and encodes this rule attribute as a hard clause as shown in equation (5):
Figure FDA0003480248150000023
wherein the content of the first and second substances,
Figure FDA0003480248150000024
is a boolean variable indicating that the ith slice of task j begins execution after time t or t,
Figure FDA0003480248150000025
is a Boolean variable which indicates that the ith slice of the task j starts to be executed after the time t-1 or t-1;
rule six:
when a task ends at t, its last slice must start at or before time t-1, and this rule attribute is encoded as a hard clause as shown in equation (6):
Figure FDA0003480248150000026
wherein, ebjIs a boolean variable that indicates that task j completes before its deadline,
Figure FDA0003480248150000027
is a Boolean variable representing the pth of task jjThe slicing is started to be executed after t or t time;
rule seven:
for the saidBoolean variables present in rule two and rule four
Figure FDA0003480248150000028
It is used to generate a hard clause represented by equation (7):
Figure FDA0003480248150000029
where t and t' should satisfy, ri+k(k-1)/2≤t≤di-(k+pi)(pi-k+1)/2+1，
Figure FDA0003480248150000031
Wherein r isjIs the ready time of task j, djIs the deadline for task j;
(S2) encoding the scheduling target as a MaxSAT soft clause; wherein, the scheduling objective maximizes the sum of the task weights completed before the deadline, and is encoded into a MaxSAT weighted soft clause shown in formula (8):
(ebj，Wj)(1≤j≤n) (8)
wherein, WjRepresenting the revenue gained by completing task j;
(S3) connecting the hard clause obtained in the step (S1) and the soft clause obtained in the step (S2) by a conjunction operator, thereby obtaining a MaxSAT problem;
(S4) calculating an optimal solution of the MaxSAT problem by the MaxSAT solver.
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