Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Integer Linear Programming formulations for Optimal Task Scheduling with Communication Delays on Parallel Systems | Sarad Venugopalan and Oliver Sinnen Department of Electrical and Computer Engineering University of Auckland, New Zealand
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Problem Scheduling task graphs with communication delays on homogeneous processors
P |prec , cij |Cmax
Strong NP-hard
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Optimal Schedules
Schedule is optimal when the overall nish time is minimised.
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Motivation
Here: Finding optimal solutions for small to mid sized instances Important for time critical systems Evaluation of heuristics
When same schedule is repeated many times
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Scheduling Model
Tasks are assigned in order of their dependence. The parallel processors are considered to be fully interconnected. Each task has its processing time ( weight of the task). Tasks communicating across processors incur a communication cost (weight of the edge).
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Constraints(Tasks on Same processor)
Given by task graph
G = (V , E )
Li : execution time of task i weight of node
p p p p
Processor constraint ( i = j ) i= j⇒ February 2014
or
ti + Li ≤ tj tj + Lj ≤ ti OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Constraints(Tasks on Same processor)
Given by task graph
G = (V , E )
Li : execution time of task i weight of node
p p p p
Processor constraint ( i = j ) i= j⇒ February 2014
or
ti + Li ≤ tj tj + Lj ≤ ti OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Constraints(Tasks on Same processor)
Given by task graph
G = (V , E )
Li : execution time of task i weight of node
p p p p
Processor constraint ( i = j ) i= j⇒ February 2014
or
ti + Li ≤ tj tj + Lj ≤ ti OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Constraints(Tasks on dierent processor)
p p pi 6= pj ⇒ or ttij ≤≤ ttji
Processor constraint ( i 6= j )
p p i j tj ≥ ti + Li + γij
Processor constraint ( i 6= j and edge → )
i
γij : remote communication cost between tasks and weight of edgeFebruary 2014
OptALI014
j
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Constraints(Tasks on dierent processor)
p p pi 6= pj ⇒ or ttij ≤≤ ttji
Processor constraint ( i 6= j )
p p i j tj ≥ ti + Li + γij
Processor constraint ( i 6= j and edge → )
i
γij : remote communication cost between tasks and weight of edgeFebruary 2014
OptALI014
j
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
Communication costs arises between tasks with an edge running on dierent processors They give rise to bilinear forms that needs to be linearised February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
How do they look like?
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
i
h
( 1 task runs on processsor , ih = 0 otherwise.
x Let Let Let Let
ti be the start time of task i . tj be the start time of task j . Li be the execution time of task i .
i
j
γij be the communication cost between tasks and .
Constraints for remote communication is bi-linear
j V : i ∈ δ − (j )
∀ ∈
ti + Li +
February 2014
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
where
j V : i ∈ δ −(j ), h, k ∈ P (z hkij = xih .xjk ).
∀ ∈
x x
The boolean multiplication of ih . jk needs to be linearised How is the linearisation done?
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
where
j V : i ∈ δ −(j ), h, k ∈ P (z hkij = xih .xjk ).
∀ ∈
x x
The boolean multiplication of ih . jk needs to be linearised How is the linearisation done?
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
where
j V : i ∈ δ −(j ), h, k ∈ P (z hkij = xih .xjk ).
∀ ∈
x x
The boolean multiplication of ih . jk needs to be linearised How is the linearisation done?
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Method 1: The usual approach
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
j V , i ∈ δ −(j ), h, k ∈ P j V , i ∈ δ −(j ), h, k ∈ P j V , i ∈ δ −(j ), h, k ∈ P
∀ ∈ ∀ ∈ ∀ ∈
February 2014
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
xih ≥ z hkij xjk ≥ z hkij xih + xjk − 1 ≤ z hkij
OptALI014
(0)
(1) (2) (3)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Method 1: The usual approach
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
j V , i ∈ δ −(j ), h, k ∈ P j V , i ∈ δ −(j ), h, k ∈ P j V , i ∈ δ −(j ), h, k ∈ P
∀ ∈ ∀ ∈ ∀ ∈
February 2014
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
xih ≥ z hkij xjk ≥ z hkij xih + xjk − 1 ≤ z hkij
OptALI014
(0)
(1) (2) (3)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
(0)
Method 2: The compact linearisation
i j V ,k ∈ P ∀i 6= j ∈ V , h, k ∈ P ∀ 6= ∈
z
x
∑ hk ij = jk h∈P hk = kh ij ji
z
February 2014
z
OptALI014
(4) (5)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms
j V : i ∈ δ −(j )
∀ ∈
ti + Li +
x x
t
∑ γij ( ih . jk ) ≤ j h,k ∈P
(0)
Method 2: The compact linearisation
i j V ,k ∈ P ∀i 6= j ∈ V , h, k ∈ P ∀ 6= ∈
z
x
∑ hk ij = jk h∈P hk = kh ij ji
z
February 2014
z
OptALI014
(4) (5)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms (Eq. 4) is obtained by multiplying both sides of the equality (Eq. 6) with
x
i j V , k ∈ P.
with jk ; ∀ 6= ∈
i V
x
(6) ∑ ih = 1 h∈P Eq 6. implies that any given task can run on exactly one processor. ∀ ∈
Method 2: The compact linearisation
i j V ,k ∈ P ∀i 6= j ∈ V , h, k ∈ P ∀ 6= ∈
z
x
∑ hk ij = jk h∈P hk = kh ij ji
z
February 2014
z
OptALI014
(4) (5)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Problems?
z Result in a high number of variables, O (|V | |P | ) . Linearisation step uses an additional variable . 2
2
First linearisation method has a constraint complexity of (| || 2 |).
O E P
Second linearisation method has a constraint complexity of (| 2 || |).
O V P
Can we get rid of it?
Yes, we use problem speci c knowledge Why get rid of it?
Eliminate
z variable and the associated constraint complexity. February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Problems?
z Result in a high number of variables, O (|V | |P | ) . Linearisation step uses an additional variable . 2
2
First linearisation method has a constraint complexity of (| || 2 |).
O E P
Second linearisation method has a constraint complexity of (| 2 || |).
O V P
Can we get rid of it?
Yes, we use problem speci c knowledge Why get rid of it?
Eliminate
z variable and the associated constraint complexity. February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Problems?
z Result in a high number of variables, O (|V | |P | ) . Linearisation step uses an additional variable . 2
2
First linearisation method has a constraint complexity of (| || 2 |).
O E P
Second linearisation method has a constraint complexity of (| 2 || |).
O V P
Can we get rid of it?
Yes, we use problem speci c knowledge Why get rid of it?
Eliminate
z variable and the associated constraint complexity. February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Problems?
z Result in a high number of variables, O (|V | |P | ) . Linearisation step uses an additional variable . 2
2
First linearisation method has a constraint complexity of (| || 2 |).
O E P
Second linearisation method has a constraint complexity of (| 2 || |).
O V P
Can we get rid of it?
Yes, we use problem speci c knowledge Why get rid of it?
Eliminate
z variable and the associated constraint complexity. February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Scheduling Model Constraints Bi-linear Forms
Bi-linear Forms Problems?
z Result in a high number of variables, O (|V | |P | ) . Linearisation step uses an additional variable . 2
2
First linearisation method has a constraint complexity of (| || 2 |).
O E P
Second linearisation method has a constraint complexity of (| 2 || |).
O V P
Can we get rid of it?
Yes, we use problem speci c knowledge Why get rid of it?
Eliminate
z variable and the associated constraint complexity. February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
First Contribution
Formulate a simple and eective linearisation for the Bi-Linear forms arising out of communication delays. All linearisation variables | |2 | |2 , variables.
V P z
z are eliminated freeing upto
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Outline 1
2
3
Problem and Motivation Scheduling Scheduling Model Constraints Bi-linear Forms Our Contribution Speeding up the Formulation Discussion of Proposed MILP Formulation
4
Experimental Results
5
Summary February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed MILP Formulation: Overlap Variables
Overlap Variables
ij V
σij
ij V
εij
∀, ∈
∀, ∈
( 1 = 0 ( 1 = 0
i
j
task nishes before task starts otherwise
i
the PI∗ of task is strictly less than task otherwise
* PI is the Processor Index
February 2014
OptALI014
j
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed MILP Formulation: Overlap Variables
Overlap Variables
ij V
σij
ij V
εij
∀, ∈
∀, ∈
( 1 = 0 ( 1 = 0
i
j
task nishes before task starts otherwise
i
the PI∗ of task is strictly less than task otherwise
* PI is the Processor Index
February 2014
OptALI014
j
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP Formulation Constraints
min W i V ti + Li ≤ W i j V εij + εji ≤ 1 i j V σij + σji ≤ 1 i j V σij + σji + εij + εji ≥ 1 j V i j σij = 1 i j V pj − pi − 1 − (εij − 1)|P | ≥ 0 i j V tj − ti − Li − (σij − 1)Wmax ≥ 0 j V i j h k P ti + Li + γ hkij (xih + xjk − 1) ≤ tj j V i j ti + Li ≤ tj i V ∑ xik = 1 kεP ∀i ∈ V ∑ kxik = pi
∀ ∈ ∀ 6= ∈ ∀ 6= ∈ ∀ 6= ∈ ∀ ∈ : ∈ δ −( ) ∀ 6= ∈ ∀ 6= ∈ ∀ ∈ : ∈ δ − ( ), ∀ , ∈ ∀ ∈ : ∈ δ −( ) ∀ ∈
kεP
February 2014
OptALI014
MinM
Overla
Edge
Proces
Time
Connect
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Formulation Bounds
W ≥0 ti ≥ 0
i ∈V i ∈ V pi ∈ {1, . . . , |P |} i V , k ∈ P xik ∈ {0, 1} i j ∈ V σij , εij ∈ {0, 1}
∀ ∀ ∀ ∈ ∀,
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP Formulation ( 1 ih = 0 ( 1 jk = 0
x x
task
i
runs on processsor ,
h
j
runs on processsor ,
otherwise. task
k
otherwise.
j V : i ∈ δ −(j ), ∀h, k ∈ P ∀j ∈ V : i ∈ δ − (j )
∀ ∈
ti + Li + γ hkij (xih + xjk − 1) ≤ tj ti + Li ≤ tj
Done only for tasks with edges Also equivalent to a Boolean multiplication.
x
x
The second equation compensates when ih and jk are both 0 February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP Formulation ( 1 ih = 0 ( 1 jk = 0
x x
task
i
runs on processsor ,
h
j
runs on processsor ,
otherwise. task
k
otherwise.
j V : i ∈ δ −(j ), ∀h, k ∈ P ∀j ∈ V : i ∈ δ − (j )
∀ ∈
ti + Li + γ hkij (xih + xjk − 1) ≤ tj ti + Li ≤ tj
Done only for tasks with edges Also equivalent to a Boolean multiplication.
x
x
The second equation compensates when ih and jk are both 0 February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP Formulation ( 1 ih = 0 ( 1 jk = 0
x x
task
i
runs on processsor ,
h
j
runs on processsor ,
otherwise. task
k
otherwise.
j V : i ∈ δ −(j ), ∀h, k ∈ P ∀j ∈ V : i ∈ δ − (j )
∀ ∈
ti + Li + γ hkij (xih + xjk − 1) ≤ tj ti + Li ≤ tj
Done only for tasks with edges Also equivalent to a Boolean multiplication.
x
x
The second equation compensates when ih and jk are both 0 February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP Complexity
Bi-Linear 1
VARIABLES
E P z
| |.| | , variables 2
February 2014
Bi-Linear 2
Improved-1[3
| | .| | , variables
free of
V P z 2
OptALI014
2
z
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Proposed ILP
min W ∀i ∈ V ti + Li ≤ W ∀i 6= j ∈ V εij + εji ≤ 1 ∀i 6= j ∈ V σij + σji ≤ 1 ∀i 6= j ∈ V σij + σji + εij + εji ≥ 1 ∀j ∈ V : i ∈ δ − (j ) σij = 1 ∀i 6= j ∈ V pj − pi − 1 − (εij − 1)|P | ≥ 0 ∀i 6= j ∈ V tj − ti − Li − (σij − 1)Wmax ≥ 0 ∀j ∈ V : i ∈ δ − (j ), ∀h, k ∈ P ti + Li + γ hk ij (xih + xjk − 1) ≤ tj − ∀j ∈ V : i ∈ δ (j ) ti + Li ≤ tj ∀i ∈ V ∑ xik = 1 kεP ∀i ∈ V ∑ kxik = pi kεP
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improve Proposed ILP
If we are able to remove all processor indices running on the subscript, the constraint complexity reduces from (| |2 +| || |2 ) to (| |2 )
O V
E P
O V
We however retain the bounds on the processors for the number of processors to be nitely bounded
i V
∀ ∈
pi ∈ {1, . . ., |P |}
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improve Proposed ILP
If we are able to remove all processor indices running on the subscript, the constraint complexity reduces from (| |2 +| || |2 ) to (| |2 )
O V
E P
O V
We however retain the bounds on the processors for the number of processors to be nitely bounded
i V
∀ ∈
pi ∈ {1, . . ., |P |}
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improved Proposed ILP
min ∀i ∈ V ∀i 6= j ∈ V ∀i 6= j ∈ V ∀i 6= j ∈ V ∀i 6= j ∈ V ∀i 6= j ∈ V ∀i 6= j ∈ V ∀j ∈ V : i ∈ δ − (j ) ∀j ∈ V : i ∈ δ − (j )
W ti + Li ≤ W
(1) (2)
σij + σji ≤ 1 (3) εij + εji ≤ 1 (4) σij + σji + εij + εji ≥ 1 (5) j − i − 1 − (εij − 1)| | ≥ 0 (6) j − i − εij | | ≤ 0 (7)
p p
p p
P P
ti + Li + (σij − 1)Wmax ≤ tj ti + Li + γij (εij + εji ) ≤ tj
(8) (9)
σij = 1 (10)
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improved ILP Bounds
ij V ∀i ∈ V ∀i ∈ V
σij , εij ∈ {0, 1} (11) i ∈ {1, . . ., | |} (12)
∀, ∈
p ∑ Li + ∑
i ∈V
February 2014
i ,j ∈V
OptALI014
γij −
P
ti ≥ 0 W ≥0
Wmax = 0
(13) (14) (15)
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improve Proposed ILP
j V : i ∈ δ −(j ),∀h, k ∈ P ti + Li + γ hkij (xih + xjk − 1) ≤ tj ∀j ∈ V : i ∈ δ − (j ) ti + Li ≤ tj ∀i ∈ V ∑ xik = 1 kεP ∀i ∈ V ∑ kxik = pi kεP ∀i ∈ V , k ∈ P xik ∈ {0, 1}
∀ ∈
Are the constarints that left the ILP
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Speeding up the Formulation Discussion of Proposed MILP Formulation
Improve Proposed ILP
Two more very problem speci c improvmenets were done NOT discussed but the main ideas have been presented
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results The computations are carried out using CPLEX 11.0 on an Intel Core i3 processor 330M, 2.13 GHZ CPU and 2 GB RAM running with no parallel mode and on a single thread on Windows 7. Graph
n p 20
BI-LINEAR
PROPOSED-1
PROPOSED-2
12h 32.09%
12h 26.15%
12h 25.78%
8
35m:56s
6m:11s
6s
16
12h 3.92%
1h:38m:5s
10s
2 4
ogra20_55
8m:49s
February 2014
18s
OptALI014
14s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results The computations are carried out using CPLEX 11.0 on an Intel Core i3 processor 330M, 2.13 GHZ CPU and 2 GB RAM running with no parallel mode and on a single thread on Windows 7. Graph
n p 20
BI-LINEAR
PROPOSED-1
PROPOSED-2
12h 32.09%
12h 26.15%
12h 25.78%
8
35m:56s
6m:11s
6s
16
12h 3.92%
1h:38m:5s
10s
2 4
ogra20_55
8m:49s
February 2014
18s
OptALI014
14s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results The computations are carried out using CPLEX 11.0 on an Intel Core i3 processor 330M, 2.13 GHZ CPU and 2 GB RAM running with no parallel mode and on a single thread on Windows 7. Graph
n p 20
BI-LINEAR
PROPOSED-1
PROPOSED-2
12h 32.09%
12h 26.15%
12h 25.78%
8
35m:56s
6m:11s
6s
16
12h 3.92%
1h:38m:5s
10s
2 4
ogra20_55
8m:49s
February 2014
18s
OptALI014
14s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results Graph
n p 30
t30_56_1
BI-LINEAR
PROPOSED-1
PROPOSED-2
2
9s
2s
16s
4
7m:32s
17s
32s
8
7h:22m:19s
16
12h 8.94%
12h 0.21% 12h 4.07%
h: Hours m:minutes s:seconds February 2014
OptALI014
48s 15s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results Graph
n p 30
t30_56_1
BI-LINEAR
PROPOSED-1
PROPOSED-2
2
9s
2s
16s
4
7m:32s
17s
32s
8
7h:22m:19s
16
12h 8.94%
12h 0.21% 12h 4.07%
h: Hours m:minutes s:seconds February 2014
OptALI014
48s 15s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results Graph
n p 40
2 4
t40_30_1
8 16
BI-LINEAR
PROPOSED-1
PROPOSED-2
6m
46s
3m:29s
29m:33s
4m:25s
12h 7.18% 12h 9.83% 12h 23.66%
12h 5.07% 12h 9.43%
h: Hours m:minutes s:seconds February 2014
OptALI014
3m:34s 4m:3s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results Graph
n p 40
2 4
t40_30_1
8 16
BI-LINEAR
PROPOSED-1
PROPOSED-2
6m
46s
3m:29s
29m:33s
4m:25s
12h 7.18% 12h 9.83% 12h 23.66%
12h 5.07% 12h 9.43%
h: Hours m:minutes s:seconds February 2014
OptALI014
3m:34s 4m:3s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Experimental Results Graph
n p 50
2 4
Ogra50_53
8 16
BI-LINEAR
PROPOSED-1
PROPOSED-2
12h 46.33% 12h 19.78% 12h inf 12h inf
12h 46.15% 12h 5.26%
12h 48.50% 12h 12.94%
February 2014
12h 2.46% 12h 5.58%
OptALI014
17m:17s 58m:2s
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Summary A MILP formulation to solve the Multi-Processor Scheduling with Communication Delays was proposed[1]. The proposed formulation is free of linearisation variables required to linearise bi-linear forms arising out of comunication delays[2]. All variable subscripts are independent of the number of processors.
V
As a result the constraint complexity reduces to O(| |)2 .
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Summary A MILP formulation to solve the Multi-Processor Scheduling with Communication Delays was proposed[1]. The proposed formulation is free of linearisation variables required to linearise bi-linear forms arising out of comunication delays[2]. All variable subscripts are independent of the number of processors.
V
As a result the constraint complexity reduces to O(| |)2 .
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Summary A MILP formulation to solve the Multi-Processor Scheduling with Communication Delays was proposed[1]. The proposed formulation is free of linearisation variables required to linearise bi-linear forms arising out of comunication delays[2]. All variable subscripts are independent of the number of processors.
V
As a result the constraint complexity reduces to O(| |)2 .
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
Summary A MILP formulation to solve the Multi-Processor Scheduling with Communication Delays was proposed[1]. The proposed formulation is free of linearisation variables required to linearise bi-linear forms arising out of comunication delays[2]. All variable subscripts are independent of the number of processors.
V
As a result the constraint complexity reduces to O(| |)2 .
February 2014
OptALI014
Problem and Motivation Scheduling Our Contribution Experimental Results Summary
References
Sarad Venugopalan and Oliver Sinnen, ILP formulations for Optimal Task Scheduling with Communication Delays on Parallel Systems, IEEE Transactions on Parallel and Distributed Systems.DOI: 10.1109/TPDS.2014.2308175, 2014. T. Davidovi¢, L. Liberti, N. Maculan, and N. Mladenovic. Towards the optimal solution of the multiprocessor scheduling problem with communication delays. In , pages 128135, 2007.
3rd Multidisciplinary International Conference on Scheduling: Theory and Application
Sarad Venugopalan and Oliver Sinnen. Optimal linear programming solutions for multiprocessor scheduling with communication delays. In , pages 129138, 2012.
ICA3PP (1)
February 2014
OptALI014
