1 Online Scheduling With Precedence Constraints Yumei Huo Department of Computer Science College

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Online Scheduling With Precedence Constraints

Yumei Huohttp://www.cs.csi.cuny.edu/~yumei/

[email protected]

Department of Computer ScienceCollege of Staten Island, CUNY

Feb. 26, 2008

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Outline• Introduction:

– Background and Notations– Three basic scheduling algorithms

• Online Scheduling of Precedence Constrained Tasks– Four nonpreemptive scheduling problems– Three preemptive scheduling problems

• Approximation Algorithms for Online Scheduling of Equal Processing Time Task System

• Future Work

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• Future Work

4

What is Scheduling ?

• Resource: machines, runways, CPUs• Jobs: operations, takeoffs and landings, programs• Constraints: priority, release date, due date, preemption• Objectives:

– minimize total completion time– minimize the maximum completion time– maximize the number of on-time jobs

Scheduling deals with the allocation of scarce resources to jobs over time subject to some constraints. It is a decision-making process with the goal of optimizing one or more objectives. (Pinedo, 2001)

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3-field Notation of Scheduling• We use || to represent a scheduling problem;

: the machine environment : the processing requirement and constraints of the jobs : the performance measure to optimize.

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2

3 4

6 7

8

1

Cmax=5

642

875311 2 3 4 5 6 7 8 9 10 11 120

P1P2

/Ci

Ci= C1+ C2+…+Cn

Ci=21

• Example: P2|pj=1, prec|Cmax– P2: 2 identical machines in parallel– pj=1, prec: the processing time of any job is unit time, the

jobs have arbitrary precedence constraints– Ci is the completion time of job i, – Cmax=max{C1, C2,…Cn} (makespan);

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Precedence constraints

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chains

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intree

4 5 6

7 8 9

10 11 12

14 13

16 15

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22 21

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outtree

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14 13

16 18

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prec

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Preemption• Jobs can be preempted and later resumed possibly

on a different machine.

4144

3313

22211 2 3 4 5 6 7 8 9 10 11 120

S1

333

222

4441111 2 3 4 5 6 7 8 9 10 11 120

S2

Cmax=4Cj=15

Cmax=6Cj=15

P3|pj=3,pmtn|Cmax:

P3|pj=3|Cmax:

• Example: 3 processors, 4 independent jobs, the processing time of each job is 3.

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• Future Work

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Coffman-Graham algorithm(P2|pj=1, prec|Cmax and P2|pj=1, prec| Ci )

Step 1. LabelingStep 2. Scheduling

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Coffman-Graham algorithm (cont.)

• Comparing two decreasing sequences of positive integers: by lexicographical order

• Example: (8, 6, 4, 3) < (8, 6, 5) and (9, 8, 6) < (9, 8, 6, 4, 3).

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Example of Coffman-Graham algorithm1 2

3

4 5

76

8 9

(9) (8)

(7)

(6)(5)

(4)(3)

(1) (2)

8462

9753176543210

(1) (2,1)

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Hu’s algorithm(P|pj=1, intree|Cmax and P|pj=1, outtree|Cmax )

Step 1. LabelingStep 2. Scheduling

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Hu’s algorithm---Level

• Definition: The level of a job i with no immediate successor is its processing time pi. The level of a job with immediate successor(s) is its processing time plus the maximum level of its immediate successor(s).

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Example for Hu’s algorithm

(1)(2)(3)(5)

(6)

(4)

(7)

(1)(2)(3)(5)

(6)

(4)

(7)

(1)(2)(3)(5)

(6)

(4)

(7)

(8)(9)(10)

(11)

(1)(2)(3)(5)

(6)

(4)

(7)

(8)(9)(10)

(11)(12)(13)

(1)(2)(3)(5)

(6)

(4)

(7)

(8)(9)(10)

(11)(12)(13)

(14)(15)

(8)(9)(10)

(11)(12)(13)

(14)(15)

(16)

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Muntz-Coffman algorithm (P2|pmtn, prec |Cmax, P|pmtn, intree|Cmax and P|pmtn, outtree|Cmax )

Assign one processor each to the jobs at the highest level. If there is a tie among y jobs (because they are at the same level) for the last x (x < y) processors, then assign x/y processor to each of these y jobs. Whenever either of the two events below occurs, reassign the processors to the unexecuted portion of the unfinished tasks according to the above rule.

Event 1: A task is completed. Event 2: We reach a point where, if we were to continue the

present assignment, we would be executing some tasks at a lower level at a faster rate than other tasks at a higher level.

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Example for Muntz-Coffman algorithm

Preemptive schedule

Processor-sharing schedule

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• Future Work

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Online Scheduling• Tasks are released, along with their constraints, at

various different times. The scheduler schedules tasks with no future information.

• We say that an online scheduling algorithm is optimal if it always produces a schedule with the minimum Cmax, i.e., a schedule as good as any schedule produced by any scheduling algorithm with full knowledge of future releases of tasks.

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Classical Scheduling Problems with polynomial optimal algorithms

• P|pj=1, intree|Cmax and P|pj=1, outtree|Cmax --- Hu’s algorithm

• P2|pj=1, prec|Cmax and P2|pj=1, prec| Cj --- Coffman-Graham algorithm

• P2|pmtn, prec |Cmax, P|pmtn, intree|Cmax and P|pmtn, outtree|Cmax --- Muntz-Coffman algorithm

• P|pmtn|Cmax --- McNaughton’s Rule

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Online version of these problems• Nonpreemptive scheduling problems:

– P2|pj=p, chainsi are released at ri|Cmax

– P|pj=1, intreei is released at ri|Cmax

– P|pj=1, outtreei is released at ri|Cmax

– P2|pj=1, preci is released at ri|Cmax

• Preemptive scheduling problems:– P2|pmtn, preci is released at ri|Cmax

– P|pmtn, intreei is released at ri|Cmax

– P|pmtn, outtreei is released at ri|Cmax

– P|pmtn, rj|Cmax (K.S.Hong and J.Y-T.Leung, 1988)

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Our results• Nonpreemptive scheduling problems:

– P2|pj=p, chainsi are released at ri|Cmax-----no optimal algorithm can possibly exist

– P|pj=1, intreei is released at ri|Cmax-----no optimal algorithm can possibly exist

– P2|pj=1, preci is released at ri|Cmax ---- there is optimal algorithm

– P|pj=1, outtreei is released at ri|Cmax----- there is optimal algorithm

• Preemptive scheduling problems:– P|pmtn, intreei is released at ri|Cmax---- no optimal algorithm can

possibly exist

– P2| pmtn, preci is released at ri|Cmax ---- there is optimal algorithm

– P|pmtn, outtreei is released at ri|Cmax---- there is optimal algorithm

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possibly exist



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P2|pj=p, chainsi are released at ri|Cmax (Counterexample)

M=2pj=2

654

11109873212 4 6 8 10 12 14 16 18 20 22 240

S1

S26 21 23 250 195

S254

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11109872 4 7 9 11 13 15 176 21 23 250 195

654

3212 4 6 8 10 12 14 16 18 20 22 240

S1

456

123

456

456

123

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Chains released at r1=0 Chains released at r2=5

7891011

7891011

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possibly exist



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P|pj=1, intreei is released at ri|Cmax (Counterexample)

4 5 67 8 910 11 12

14 1316 1518 1720 1922 21

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Tree1 released at r1=0

M=3 24

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S11296

22201816141185

23211917151310741 2 3 4 5 6 7 8 9 10 11 120

22192017151296

323129271816141185

232130282625241310741 2 3 4 5 6 7 8 9 10 11 120

312129192712171513

32302826162411975

232220182514108641 2 3 4 5 6 7 8 9 10 11 120

S2

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possibly exist



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P2|pj=1, preci is released at ri|Cmax

• Algorithm A:Whenever new tasks arrive, do {

t = the current time;U = the set of tasks active (i.e., not finished) at time t;Call the Coffman-Graham algorithm to reschedule the tasks in U;

}

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Example

1 2

3

4 5

76

8 9

(9) (8)

(7)

(6)(5)

(4)(3)

(1) (2)

8462

9753176543210

(9)(8)

(5)

(1) (2)

Tasks not finished at t=2

(3) (4)

(6) (7)

(10)

10 (11)

Tasks released at t=2

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8 9

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3110 2 9

9151245

814713111086543 7

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Our result

Theorem 2.1.4 Algorithm A is optimal for P2|pj=1, preci is released at ri|Cmax.Moreover, the schedule produced by Algorithm A has the largest number of tasks completed at any time instant t.

(Note: Algorithm A is optimal for P2|pj=1, preci is released at ri| Cj)

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possibly exist



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P|pj=1, outtreei is released at ri|Cmax

• Algorithm B:Whenever new tasks arrive, do {

t =the current time;U= the set of tasks active (i.e., not finished) at time t;Call Hu's algorithm to reschedule the tasks in U;

}

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Example

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Our result

Theorem 2.1.5 Algorithm B is optimal for P|pj=1, outtreei is released at ri|Cmax.Moreover, the schedule produced by Algorithm B has the largest number of tasks completed at any time instant t.

(Note: Algorithm B is optimal for P|pj=1, outtreei is released at ri| Cj)

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possibly exist



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P|pj=1, pmtn, intreei is released at ri|Cmax (Counterexample)

4 5 67 8 910 11 12

14 1316 1518 1720 1922 21

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M=3 24

25

26 27

28 29

30 31

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S11296

22201816141185

23211917151310741 2 3 4 5 6 7 8 9 10 11 120

12171513

211911975

232220181614108641 2 3 4 5 6 7 8 9 10 11 120

S2

22192017151296

323129271816141185

232130282625241310741 2 3 4 5 6 7 8 9 10 11 120

312129192712171513

32302826162411975

232220182514108641 2 3 4 5 6 7 8 9 10 11 120

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• Preemptive scheduling problems:– P|pmtn, pj=1, intreei is released at ri|Cmax---- no optimal algorithm can

possibly exist



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P2| pmtn,preci is released at ri|Cmax

• Algorithm C:Whenever new tasks arrive, do {

t =the current time;U= the set of tasks active (i.e., not finished) at time t;Call Muntz-Coffman algorithm to reschedule the tasks in U;

}

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Example

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Our resultTheorem 2.2.4 Algorithm C is an optimal

online algorithm for P2| pmtn, preci is released at ri|Cmax.

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possibly exist



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P|pmtn, outtreei is released at ri|Cmax

Theorem 2.2.6 Algorithm C is an optimal online algorithm for P|pmtn, outtreei is released at ri|Cmax.

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• Future Work

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known results

• P| pj=1, outtreei released at ri | Cmax

– Online version of Hu’s algorithm is known to be optimal (Huo & Leung)

• P2| pj=1, preci released at ri | Cmax

– Online version of Coffman-Graham algorithm is known to be optimal (Huo & Leung)

• P| prec, online | Cmax

– Online version of List scheduling has competitive ratio 2-1/m (Sgall)

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Problems

• P | pj=1, intreei released at ri | Cmax

• P | pj=p, intreei released at ri | Cmax

• P | pj=p, outtreei released at ri | Cmax

• P2 | pj=p, preci released at ri | Cmax

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Problems





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On-line version of Hu’s Algorithm(revisited)

• Hu’s Algorithm:– Whenever a machine is idle, pick a job from the highest level

and schedule it.– Hu’s algorithm is optimal for the off-line problem of p|pj=1, intree|cmax

• On-line version:Whenever new jobs arrive, do{ T=the current time U=the set of jobs not finished at time T reschedule the tasks in U by Hu’s algorithm.}

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An Example of On-line Hu’s Algorithm

• A set of jobs released at time 0

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An Example of On-line Hu’s Algorithm (Algorithm B)

• A new set of jobs released at time 1.

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New Release:

Jobs has been scheduled:

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An Example

• Final Schedule3

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612

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106

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Result 1

• For p | intree,pj=1,online | cmax problem, let S be the cmax produced by the on-line version of Hu’s algorithm, and S* be the cmax of the optimal schedule, then S/S*≤ 1.5

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Problems





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Meta-Algorithm Delay-X (from pj=1 to pj=p)

• Suppose X is an algorithm which solves online problem for pj=1.

• For any job released at time r such that kp<r<(k+1)p, delay it from consideration until the time instant (k+1)p.

• Call algorithm X at time instant (k+1)p

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Result 2• Suppose algorithm X gives competitive ratio of

λ for α|pj=1,online|cmax problem, then,– For the problemα|pj=p,online|cmax, let s be the cmax

produced by Delay-X and s* be the cmax of the optimal schedule, then s≤λs*+ λ p

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Result 3• P| pj=p, intreei released at ri | cmax

– Let S be the cmax produced by Delay-Hu’s algorithm and S* be the cmax of optimal schedule, then S≤1.5(S*+p). The asymptotic competitive ratio is 1.5 since S*>>p.

• P| pj=p, outtreei released at ri | cmax

– Let S be the cmax produced by Delay-Hu’s algorithm and S* be the cmax of optimal schedule, then S ≤(S*+p). The asymptotic competitive ratio is 1 since S*>>p.

• P2| pj=p, preci released at ri | cmax

– Let S be the cmax produced by Delay-Coffman-Graham algorithm and S* be the cmax of optimal schedule, then S ≤(S*+p). The asymptotic competitive ratio is 1 since S*>>p.

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• Future Work

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Future Work (Continued Work)• For p | pj=1, intreei released at ri | cmax problem:

– Better approximation algorithms? Tight Competitive Ratio?

• For p | pj=p, preci released at ri | cmax

– Asymptotic competitive ratio? (>2-2/m [Lam and Sethi] )

• For p | pmtn, intreei released at ri | cmax problem:– Approximation Algorithm? Competitive Ratio?

• Other Criteria of these online scheduling problems?– Objective function: mean flow time

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Selected Publications• Huo, Y. and J. Y-T. Leung, "Online Scheduling of Precedence Constrained

Tasks," SIAM J. on Computing, Volume 34, Number 3, pp. 743-762. 2005. • Huo, Y. and J. Y-T. Leung, "Minimizing Total Completion Time for UET Tasks

with Release Time and Outtree Precedence Constraints," Mathematical Methods of Operations Research, Vol. 62, No. 2, pp. 275-278, 2005 .

• Huo, Y. and J. Y-T. Leung, "Minimizing Total Completion Time for UET Tasks ," ACM Transactions on Algorithms, Vol. 2, No. 2, pp. 244-262. April 2006 .

• Huo, Y., J. Y-T. Leung and H. Zhao, "Complexity of Two Dual Criteria Scheduling Problems," submitted to Operations Research Letters, 35:211-220, 2007.

• Huo, Y., J. Y-T. Leung and H. Zhao, "Bi-criteria Scheduling Problems: Number of Tardy Jobs and Maximum Weighted Tardiness," European Journal of Operational Research,177:116-134, 2007.

• Huo, Y., J. Y-T. Leung and X. Wang, "Online Scheduling of Equal-Processing-Time Task Systems," Submitted to Theoretical Computer Science.

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Thank you!

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Makespan

• Total completion time: Ci= C1+ C2+…+Cn

• Makespan = (Ci - ri) /n– Ci is the completion time of job i;

– ri is the release time of job i;– n is the number of jobs

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Application of Scheduling• Scheduling of Flexible Resources in Professional Service

Firms• Novel metaheuristic Approaches to Nurse Rostering

Problems in Belgian Hospitals• University Timetabling• Adapting the GATES Architecture to Scheduling Faculty• Constraint Programming for Scheduling• Batch Production Scheduling in the Process Industries• A Composite Very-Large-Scale Neighborhood Search

Algorithm for the Vehicle Routing Problem• Scheduling Problems in the Airline Industry• Bus and Train Driver Scheduling• Sports Scheduling

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Application of Online Scheduling

• Industrial Applications• Multiuser operating systems such as Unix and

Windows• Web servers• Database servers• Load balancers sitting in front of server farms

(Pruhs K., Sgall J., and Torng E., 2004 )

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M3

M2

M1

t=0 r1 r2 r3

Ji1,Ji1+1, … Ji2,Ji2+1, … Ji3,Ji3+1, …

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Complexity results for scheduling problems

• by Brucker and Knust: http://www.mathematik.uni-osnabrueck.de/research/OR/class/

Documents

1 Online Scheduling With Precedence Constraints Yumei Huo Department of Computer Science College