Energy-efficient scheduling on volatile platforms Guillaume Aupy - - PowerPoint PPT Presentation

Energy-efficient scheduling

• n volatile platforms

Guillaume Aupy

based on a work with Anne Benoit

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

1.0

Energy: a crucial issue

• Data centers
• 330, 000, 000, 000 Watts hour in 2007: more than France
• 533, 000, 000 tons of CO2: in the top ten countries
• Exascale computers (1018 floating operations per second)
• Need effort for feasibility
• 1% of power saved 1 million dollar per year
• Lambda user
• 1 billion personal computers
• 500, 000, 000, 000, 000 Watts hour per year
• crucial for both environmental and economical reasons

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

1.0

Energy: a crucial issue

• Data centers
• 330, 000, 000, 000 Watts hour in 2007: more than France
• 533, 000, 000 tons of CO2: in the top ten countries
• Exascale computers (1018 floating operations per second)
• Need effort for feasibility
• 1% of power saved 1 million dollar per year
• Lambda user
• 1 billion personal computers
• 500, 000, 000, 000, 000 Watts hour per year
• crucial for both environmental and economical reasons

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

2.0

1 Model

Reliability Makespan Energy

2 Approximation for linear chains

Intractability FPTAS

3 Approximation for independent tasks

Inapproximability Approximation

4 Conclusion

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

3.0

1 Model

Reliability Makespan Energy

2 Approximation for linear chains

Intractability FPTAS

3 Approximation for independent tasks

Inapproximability Approximation

4 Conclusion

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

4.0

Architecture

p identical processors. Speed Scaling: one can modify the execution speed f of any task, f ∈ [fmin, fmax].

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

5.0

Into details

Let Ti of weight wi executed on processor pj: time

pj

· · · · · · Exe(wi, fi) fi

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

5.0

Into details

Let Ti of weight wi executed on processor pj: time

pj

· · · · · · Exe(wi, fi) fi

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

6.0

Hypothesis

In this model, we further suppose that there is a speed frel, such that, for any Ti executed at speed fi: If fi < frel then we need to execute Ti a second time.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

6.0

Hypothesis

In this model, we further suppose that there is a speed frel, such that, for any Ti executed at speed fi: If fi < frel then we need to execute Ti a second time. How come?

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

Transient failure = local failure (no impact on the system, the processor impacted can restart to work immediately after the failure) The rate of transient failures follow a Poisson Law of parameter: λ(f ) = λ0ed

fmax−f fmax−fmin

The reliability of Ti executed at speed fi: Ri(fi) = e−λ(fi)Exe(wi,fi) ≈ 1 − λ(fi)Exe(wi, fi)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

fi Ri(fi)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

fi Ri(fi) frel Ri(frel)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

fi Ri(fi) frel Ri(frel)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

The reliability of Ti executed at speed fi: Ri(fi) = e−λ(fi)Exe(wi,fi) ≈ 1 − λ(fi)Exe(wi, fi)

Reliability for two executions

Ri = 1 − (1 − Ri(f (1)

i

))(1 − Ri(f (2)

i

))

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

7.0

Reliability

The reliability of Ti executed at speed fi: Ri(fi) = e−λ(fi)Exe(wi,fi) ≈ 1 − λ(fi)Exe(wi, fi)

Reliability for two executions

Ri = 1 − (1 − Ri(f (1)

i

))(1 − Ri(f (2)

i

)) 1 − (1 − Ri(fmin))2 ≥ Ri(frel) ֒ → Let’s suppose that two execution are enough to match the reliability constraint.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

8.0

Makespan

The execution time for Ti at speed fi is: Exe(wi, fi) = wi fi If we call ti the end of the last execution of Ti: time

p1 p2 T (1)

i

f (1)

i T (2)

i

f (2)

i

ti

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

8.0

Makespan

The execution time for Ti at speed fi is: Exe(wi, fi) = wi fi If we call ti the end of the last execution of Ti: time

p1 p2 T (1)

i

f (1)

i T (2)

i

f (2)

i

ti

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

8.0

Makespan

Constraint on the makespan:

We ask ∀i, ti ≤ D (the deadline D is fixed by the user)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

9.0

Why should we decrease the speed then?

• Loss in reliability
• Loss on the makespan

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

The execution of task Ti at speed fi: Ei(fi) = Exe(wi, fi)f 3

i = wif 2 i

→ (Dynamic part of the classical energy model)

Energy consumption with two executions:

Ei = wi

• f (1)

i

2 + wi

• f (2)

i

2

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

The execution of task Ti at speed fi: Ei(fi) = Exe(wi, fi)f 3

i = wif 2 i

→ (Dynamic part of the classical energy model)

Energy consumption with two executions:

Ei = wi

• f (1)

i

2 + wi

• f (2)

i

2

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

fi Ei(fi) frel Ei(frel)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

fi Ei(fi)

wif 2

i + wif 2 i = 2Ei(fi)

frel Ei(frel)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

fi Ei(fi)

wif 2

i + wif 2 i = 2Ei(fi)

frel Ei(frel)

frel √ 2

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

10.0

Energy

fi Ei(fi)

wif 2

i + wif 2 i = 2Ei(fi)

frel Ei(frel)

frel √ 2

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

11.0

To sum up

We need to find for each task:

• the number of execution (one or two)
• the speed of those executions
• the mapping (processor) of those executions

In order to minimize the energy consumption under the constraints:

• ∀i, ti ≤ D (bounded makespan)
• ∀i, fi ≥ frel or the task needs to be executed twice

(minimum reliability)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

12.0

1 Model

Reliability Makespan Energy

2 Approximation for linear chains

Intractability FPTAS

3 Approximation for independent tasks

Inapproximability Approximation

4 Conclusion

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

13.0

A chain of tasks

Let n tasks T1, . . . , Tn of weight w1, . . . , wn with the constraint that for all i, Ti should be executed before Ti+1 time

p1 T1 T2 T3 T4 T5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

13.0

A chain of tasks

Let n tasks T1, . . . , Tn of weight w1, . . . , wn with the constraint that for all i, Ti should be executed before Ti+1 time

p1 T1 T2 T3 T4 T5

We focus here on the case p = 1

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

14.0

Lemma (Without re-execution (folklore))

Let S = wi. The optimal solution without re-execution is to execute all tasks at speed max(frel, S

D ).

Lemma (Speeds of re-executions)

If a task is executed twice, then both execution are at the same speed.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4 T5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4 T5

frel

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4

frel

T (1)

5

T (2)

5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4

frel

T (1)

5

T (2)

5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4 T5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4 T (1)

5

T (2)

5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4

f > frel

T (1)

5

T (2)

5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

D

T1 T2 T3 T4

f > frel

T (1)

5

T (2)

5

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

15.0

Lemma (Speed of the tasks executed once)

The speed of the tasks executed once should be frel (whenever it is possible).

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

We have seen:

• the tasks executed once, should be at speed frel;
• the tasks executed twice are executed at the same speed.

We show the intractability by reducing this issue to the matter

• f selecting the tasks that should be executed twice.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

⇐ ⇒

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

Actually, the set of tasks that should be executed twice caracterize the solution:

• the set X of tasks executed once are executed at frel;
• the set {T1, . . . , Tn} \ X is executed by taking as much

time as possible.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

We reduce this problem using Subset-Sum Subset-Sum Let {a1, . . . , an}, X. Does there exist a subset I of {1, . . . , n} such that

i∈I ai = X?

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

We reduce this problem using Subset-Sum We simply need to call:                ∀i, wi = ai frel = fmax D0 =

S frel + X c frel

E0 = 2X 2c 1 + c frel 2 + (S − X) f 2

rel

where c = 4

• 2

7 cos 1 3(π − tan−1 1 √ 7) − 1 and the result follows

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

16.0

Intractability

We reduce this problem using Subset-Sum We simply need to call:                ∀i, wi = ai frel = fmax D0 =

S frel + X c frel

E0 = 2X 2c 1 + c frel 2 + (S − X) f 2

rel

where c = 4

• 2

7 cos 1 3(π − tan−1 1 √ 7) − 1 and the result follows

c is the only positive root

• f the polynomial 7y3 + 21y2 − 3y − 1

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

17.0

Quick reminder

Approximation Algorithm: an algorithm such that the result is guaranteed to be “not too far” from the optimal solution. Formally A is a λ-approximation if ∀I, Opt(I) ≤ A(I) ≤ λ · Opt(I) FPTAS: is a family of (1 + ε)-approximation algorithms for any ε > 0, that runs in polynomial time in the size of the problem and in 1

ε.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

17.0

Quick reminder

Approximation Algorithm: an algorithm such that the result is guaranteed to be “not too far” from the optimal solution. Formally A is a λ-approximation if ∀I, Opt(I) ≤ A(I) ≤ λ · Opt(I) FPTAS: is a family of (1 + ε)-approximation algorithms for any ε > 0, that runs in polynomial time in the size of the problem and in 1

ε.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

Reminder:

⇐ ⇒

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

We are looking for the set I of tasks to execute twice. What we know:

• i∈I wi

E

XD ED

We know how to compute ED, a lower bound on the energy, however we do not know where E0 the actual optimal solution is.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

We are looking for the set I of tasks to execute twice. What we know:

• i∈I wi

E

XD ED X0 E0

We know how to compute ED, a lower bound on the energy, however we do not know where E0 the actual optimal solution is.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

We are looking for the set I of tasks to execute twice. What we know:

• i∈I wi

E

XD ED X0 E0

We know how to compute ED, a lower bound on the energy, however we do not know where E0 the actual optimal solution is.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

Using an algorithm very similar to the FPTAS to solve SubsetSum, called on the value XD, we show that we can find a subset of tasks Ia satisfying

• i∈Ia

wi = Xa ≤ X0 ≤ Xa(1 + ε)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

18.0

FPTAS for the chain problem

Then we show that Ea ≤ (1 + 28ε)E0

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

19.0

p ≥ 2

When p ≥ 2, we obtain the same results using a similar technique.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

20.0

1 Model

Reliability Makespan Energy

2 Approximation for linear chains

Intractability FPTAS

3 Approximation for independent tasks

Inapproximability Approximation

4 Conclusion

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

21.0

Independent tasks

Let n tasks T1, . . . , Tn of weight w1, . . . , wn. Let us assume that there are no precedence constraints between the tasks.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

22.0

Intractability

With one processor we can reduce this problem to the chain

• problem. This problem is then NP-complete.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

23.0

Inapproximation

Theorem

There does not exist any approximation algorithm that runs in polynomial time for this problem, unless P=NP.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

23.0

Inapproximation

Theorem

There does not exist any approximation algorithm that runs in polynomial time for this problem, unless P=NP.

• p processors
• fmin = frel = fmax = 1

Then, any solution is the optimal solution! Finding an approximation algorithm for the energy problem is equivalent to solving (P||Cmax)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

24.0

Relaxation

We now assume that we can relax the makespan constraint.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

24.0

Relaxation

We now assume that we can relax the makespan constraint. An (α, β)-approximation is such that, if Eopt is the minimum energy for the problem under a maximal makespan of D,

• Ealgo ≤ α × Eopt (Ealgo is the energy of our algorithm)
• Talgo ≤ β × D (Talgo is the makespan of our algorithm)

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

24.0

Relaxation

We now assume that we can relax the makespan constraint.

Theorem

There exist

• 1 + 1

β2 , β

• approximation algorithms, for all

β ≥ max

• 2 −

3 2p+1, 2 − p+2 4p+2

• .

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 D p2 p3 p4 p5 p6 w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11

• 1. At first we sort the tasks by increasing order.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 D w1 p2 p3 p4 p5 p6 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11

• 2. The “big” tasks are scheduled on one processor each

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 D w1 p2 w2 p3 p4 p5 p6 w3 w4 w5 w6 w7 w8 w9 w10 w11

• 2. The “big” tasks are scheduled on one processor each

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

D 2D 3D 4D w3 w4 w5 w6 w7 w8 w9 w10 w11

• 3. We create a virtual processor, “equivalent” to the

remaining processors.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 D 2D 3D 4D w4 w5 w6 w7 w8 w9 w10 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 D 2D 3D 4D w5 w6 w7 w8 w9 w10 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 D 2D 3D 4D w6 w7 w8 w9 w10 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 D 2D 3D 4D w7 w8 w9 w10 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 w7 D 2D 3D 4D w8 w9 w10 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 w7 w10 D 2D 3D 4D w8 w9 w11

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 w7 w10 w11 D 2D 3D 4D w8 w9

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 w7 w10 w11 w(1)

8

w(2)

8

D 2D 3D 4D w9

• 4. We apply the FPTAS for linear chains on this processor...

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 w1 p2 w2 peq

1

w3 w4 w5 w6 w7 w10 w11 w(1)

8

w(2)

8

w(1)

9

w(2)

9

D 2D 3D 4D

• 4. We apply the FPTAS for linear chains on this processor...

... which gives us a set of “new” tasks.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

25.0

p1 D βD w1 p2 w2 p3 w3 w(2)

9

w10 p4 w4 w(1)

9

w11 p5 w(1)

8

w5 p6 w(2)

8

w6 w7

• 5. Finally, we sort greedily the “new” tasks on the remaining

processors.

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

26.0

1 Model

Reliability Makespan Energy

2 Approximation for linear chains

Intractability FPTAS

3 Approximation for independent tasks

Inapproximability Approximation

4 Conclusion

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

27.0

Conclusion

What we did:

• FPTAS for linear chains;
• Inapproximability for independent tasks;
• With a relaxation on the makespan constraint, we can

approximate this result (we can improve it for large p to a (1 + Θ( 1

p), 2 − Θ( 1 p))-approximation).

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

27.0

Conclusion

Next,

• General graphs (we can start by forks, trees etc);
• Other reliability models (global reliability, checkpoints...).

Energy, Reliability, Makespan

• G. Aupy

Introduction Model

Reliability Makespan Energy

Approximation for linear chains

Intractability FPTAS

Approximation for independent tasks

Inapproximability Approximation

Conclusion

What we had: What we aim at: Energy-efficient scheduling + frequency scaling