on the value of preemption in schedulinggilsh/presentations/short... · 2021. 4. 20. ·...

OutlineOverview + Uses

The ModelAlgorithm - Upper Bounds

Lower BoundsSummary

Open Problems

On the Value of Preemption in Scheduling

Yair Bartal1 Stefano Leonardi2 Gil Shallom1

Rene Sitters3

1Department of Computer Science, Hebrew University, Jerusalem, Israel

2Dipartimento di Informatica e Sistemistica,Universit di Roma La Sapienza, Rome, Italy

3Max-Planck-Insitut fur Informatik, Saarbrucken, Germany

TAU Algorithms Seminar, 2007

Bartal,Leonardi,Shallom,Sitters On the Value of Preemption in Scheduling



Lower BoundsSummary

Open Problems

Outline of Topics

Overview + Uses

The Model

Algorithm - Upper Bounds

Lower Bounds

Summary

Open Problems




Lower BoundsSummary

Open Problems

Overview

I Preemptive scheduling algorithms can achieve efficientperformance.

I Example: SRPT is optimal for minimizing total flow timeI Assumption:

I preemption is costlessI unbounded number of preemptions per job

I Real life: Preemption incurs a significant overhead.




Lower BoundsSummary

Open Problems

Focus

I A new model where preemption is costlyI New preemptive scheduling algorithm

I take the cost of preemption into account




Lower BoundsSummary

Open Problems

Competitive analysis

A common technique for measuring effectiveness of onlinealgorithms, by comparing the performance of an online algorithm,for any input sequence, to that of the optimal offline algorithm

ALG (I ) ≤ C · OPT (I ) + α




Lower BoundsSummary

Open Problems

Standard Scheduling Problems

I DefinitionsI Flowtime of Job j: Fj = Cj − rjI Total Flowtime:

∑j∈J Fj

I Goal Function: min∑

j∈J Fj




Lower BoundsSummary

Open Problems

SRPT

SRPT Algorithm (Shortest Remaining Processing Time)

I Optimal with respect to total flowtime in the single machinecase (in the standard model)




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Open Problems

The Standard Model Assumptions

I No info is available regarding future jobsI Online Algorithms (hard to plan)

I All job info is available on its arrivalI Relevant job info (job processing time)

I Preemption is costlessI Model does not take into account resources wasted due to

preemption




Lower BoundsSummary

Open Problems

The Model

I Standard ModelI Minimizing total flowtime∑

j∈J

Fj

I New ModelI Preemption cost is KI Minimize the sum of total flowtime and the total cost of

preemptions ∑j∈J

Fj + K ·M




Lower BoundsSummary

Open Problems

Model Boundaries

I Cost of preemption is K∑j∈J

Fj + K ·M

I K = 0 =⇒ standard model

I K is large =⇒ non-preemptive model




Lower BoundsSummary

Open Problems

WTP Algorithm (Wait to Preempt)

I When starting a new job, choose the job with the shortestremaining processing time. (Which job?)

I If there is an idle machine and a job is available, then startprocessing the shortest of the available jobs. (When toexecute?)

I Denote ε as the processing time of the shortest job

I Preempt a job as soon as it has been processed withoutinterruption for exactly

√Kε time units. That is, every job is

partitioned in parts of length√

Kε. (When to preempt?)




Lower BoundsSummary

Open Problems

WTP Algorithm (Wait to Preempt)

I One proof for single and parallel machines

I WTP competitive ratio:

(C (α + 1) + α)

I SRPT competitive ratio:where C = 1 or C = O(log(min n

m , ρ))and α = minρ,

√κ , κ = K

ε

I ρ is the ratio between the longest and shortest job




Lower BoundsSummary

Open Problems

WTP Analysis

1. Total Flowtime ∑j∈J

Fj

2. Total cost of preemption

K ·M

∑j∈J

Fj + K ·M




Lower BoundsSummary

Open Problems

Total cost of preemption

I At mostpj√Kε

preemptions for job j

I Total cost WTPP(I ) ≤ K (P

pj )√K ε

=√

κ∑

pj

OPT(I ) ≥∑

pj

I WTPP(I ) ≤√

κ OPT(I ) ≤ α ·OPT(I )

α = minP,√

κ




Lower BoundsSummary

Open Problems

Total Flowtime - Analysis Steps

I Jobs can arrive at any time, but WTP is not always allowed topreempt a job (unlike SRPT)

I For the analysis, We create a new schedule and shift the jobrelease times to the times when WTP is also allowed topreempt

I Now SRPT(I) and WTP(I) act the same on this new inputsequence

I We finally show that the flowtime incurred due to the shift isalso bounded




Lower BoundsSummary

Open Problems

Shifting jobs

Elaborate on that part at TAU - copy the proof from the paper

1. Create a new schedule I ′ from I

2. Result: WTPF (I ) = SRPT (I )

3. Flowtime cost for waiting is r ′j − rj for job j




Lower BoundsSummary

Open Problems

Total Flowtime - The Analysis

WTPF (I ) = WTPF (I ′) +∑

j

(r ′j − rj) =

= SRPT(I ′) +∑

j

(r ′j − rj) ≤

≤ C ·OPT(I ′) +∑

j

(r ′j − rj)




Lower BoundsSummary

Open Problems

Total Flowtime - The Analysis

OPT(I ′) ≤ OPT(I ) +∑

j

(αε + rj − r ′j )

WTPF (I ) ≤ C (OPT(I ) + nαε) + (C − 1)∑

j

(rj − r ′j )

≤ C (α + 1)OPT(I ).




Lower BoundsSummary

Open Problems

Upper Bound Result

WTP = WTPP(I ) + WTPF (I ) ≤ (C (α + 1) + α)OPT(I )




Lower BoundsSummary

Open Problems

Lower Bounds

What we are going to see:

I Lower bound 1:Idle Time Not Allowed Model

I Lower bound 2: Idle Time Allowed Model

I Inapproximability




Lower BoundsSummary

Open Problems

Idle Time Not Allowed Model

I We show a lower bound on the competitive ratio achievable byany online algorithm due to the online nature of the problem.

I The algorithm has to choose an action with information onlyregarding the past and therefore is bound to make mistakes.

I We construct an example which makes the algorithm performat its worst.

Ω(α), where α = minP,√

κ




Lower BoundsSummary

Open Problems

Idle Time Not Allowed Model

0 x 2x 3x

N processes

4x

A

B

(a) The job release sequence in the first case

N processes

0 x 2x 3x 4x

A

B

(b) The job release sequence in the second case

Figure: Idle Time Not Allowed Release Sequences




Lower BoundsSummary

Open Problems

Idle Time Not Allowed

0 x 2x 3x 4x

N processes

N processes

AlgA

OptAB

B

(a) The algorithm chooses to executethe longer process first

0 x 2x 3x 4x

N processes

Alg

OptA

N processes

B

AB

(b) The algorithm chooses to executethe shorter process first

Figure: Idle Time Not Allowed Execution Sequences




Lower BoundsSummary

Open Problems

Idle Time Not Allowed

Either many machines choose to:

I Execute Job A first and incur a significant cost

I Execute Job B and incur a significant cost

I Preempt and incur a significant cost due to preemption

The lower bound result:


κ




Lower BoundsSummary

Open Problems

Idle Time Allowed Model

I The algorithm is allowed not to execute a job even if an idlemachine is available

I Even in this case the lower bound is the same:

The lower bound result:


κ




Lower BoundsSummary

Open Problems

Inapproximability

I Using Woeginger et al. Ω(√

n) lower bound on theapproximation ratio of polynomial-time approximationalgorithms for the problem of minimizing flow time in anon-preemptive environment

I We get an inapproximability result (in our model) of:

Ω(min

κ

14−δ,P

13−δ

)




Lower BoundsSummary

Open Problems

Summary

I More realistic scheduling model

I Efficient online algorithms for this setting

I Lower bounds for the online settings

I Inapproximability bound




Lower BoundsSummary

Open Problems

Open Problems

I Minimizing completion time, weighted flow time, or stretch

I Cost of preemption to be dependent on the jobs involved, ormore generally on the state of the system

I Semi-clairvoyance with costly preemption




Lower BoundsSummary

Open Problems

The end

Questions?


on the value of preemption in schedulinggilsh/presentations/short... · 2021. 4. 20. ·...

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