Online Scheduling With Precedence Constraints

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Online Scheduling With Precedence Constraints

• Yumei Huo
• http//www.cs.csi.cuny.edu/yumei/
• huo_at_mail.csi.cuny.edu
• Department of Computer Science
• College 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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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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What is Scheduling ?
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)

• 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

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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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Coffman-Graham algorithm(P2pj1, precCmax and
P2pj1, prec ?Ci )?

• Step 1. Labeling
• Step 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) lt (8, 6, 5) and (9, 8, 6) lt (9, 8,
  6, 4, 3).

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Hus algorithm(Ppj1, intreeCmax and Ppj1,
outtreeCmax )?

• Step 1. Labeling
• Step 2. Scheduling

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Hus 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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Muntz-Coffman algorithm (P2pmtn, prec Cmax,
Ppmtn, intreeCmax and Ppmtn, outtreeCmax )?

• 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 lt 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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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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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

• Ppj1, intreeCmax and Ppj1, outtreeCmax ---
  Hus algorithm
• P2pj1, precCmax and P2pj1, prec ?Cj ---
  Coffman-Graham algorithm
• P2pmtn, prec Cmax, Ppmtn, intreeCmax and
  Ppmtn, outtreeCmax --- Muntz-Coffman algorithm
• PpmtnCmax --- McNaughtons Rule

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Online version of these problems

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax
• Ppj1, intreei is released at riCmax
• Ppj1, outtreei is released at riCmax
• P2pj1, preci is released at riCmax
• Preemptive scheduling problems
• P2pmtn, preci is released at riCmax
• Ppmtn, intreei is released at riCmax
• Ppmtn, outtreei is released at riCmax
• Ppmtn, rjCmax (K.S.Hong and J.Y-T.Leung, 1988)?

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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P2pj1, preci is released at riCmax

• 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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Our result

• Theorem 2.1.4 Algorithm A is optimal for
  P2pj1, preci is released at riCmax.
• 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 P2pj1, preci
  is released at ri ?Cj)?

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppj1, outtreei is released at riCmax

• 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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Our result

• Theorem 2.1.5 Algorithm B is optimal for Ppj1,
  outtreei is released at riCmax.
• 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 Ppj1,
  outtreei is released at ri ?Cj)?

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, pj1, intreei is released at riCmax----
  no optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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P2 pmtn,preci is released at riCmax

• 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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Our result

• Theorem 2.2.4 Algorithm C is an optimal online
  algorithm for P2 pmtn, preci is released at
  riCmax.

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppmtn, outtreei is released at riCmax

• Theorem 2.2.6 Algorithm C is an optimal online
  algorithm for Ppmtn, outtreei is released at
  riCmax.

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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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known results

• P pj1, outtreei released at ri Cmax
• Online version of Hus algorithm is known to be
  optimal (Huo Leung)?
• P2 pj1, 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 pj1, intreei released at ri Cmax
• P pjp, intreei released at ri Cmax
• P pjp, outtreei released at ri Cmax
• P2 pjp, preci released at ri Cmax

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Problems

• P pj1, intreei released at ri Cmax
• P pjp, intreei released at ri Cmax
• P pjp, outtreei released at ri Cmax
• P2 pjp, preci released at ri Cmax

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On-line version of Hus Algorithm(revisited)?

• Hus Algorithm
• Whenever a machine is idle, pick a job from the
  highest level and schedule it.
• Hus algorithm is optimal for the off-line
  problem of
• ppj1, intreecmax
• On-line version
• Whenever new jobs arrive, do
• Tthe current time
• Uthe set of jobs not finished at time T
• reschedule the tasks in U by Hus algorithm.

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An Example of On-line Hus Algorithm

• A set of jobs released at time 0

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An Example of On-line Hus Algorithm (Algorithm
B)?

• A new set of jobs released at time 1.

New Release
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Jobs has been scheduled
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An Example

• Final Schedule

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Result 1

• For p intree,pj1,online cmax problem, let S
  be the cmax produced by the on-line version of
  Hus algorithm, and S be the cmax of the optimal
  schedule, then S/S 1.5

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Problems

• P pj1, intreei released at ri Cmax
• P pjp, intreei released at ri Cmax
• P pjp, outtreei released at ri Cmax
• P2 pjp, preci released at ri Cmax

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Meta-Algorithm Delay-X (from pj1 to pjp)?

• Suppose X is an algorithm which solves online
  problem for pj1.
• For any job released at time r such that
  kpltrlt(k1)p, delay it from consideration until
  the time instant (k1)p.
• Call algorithm X at time instant (k1)p

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Result 2

• Suppose algorithm X gives competitive ratio of ?
  for apj1,onlinecmax problem, then,
• For the problemapjp,onlinecmax, let s be the
  cmax produced by Delay-X and s be the cmax of
  the optimal schedule, then s?s ? p

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Result 3

• P pjp, intreei released at ri cmax
• Let S be the cmax produced by Delay-Hus
  algorithm and S be the cmax of optimal schedule,
  then S1.5(Sp). The asymptotic competitive
  ratio is 1.5 since Sgtgtp.
• P pjp, outtreei released at ri cmax
• Let S be the cmax produced by Delay-Hus
  algorithm and S be the cmax of optimal schedule,
  then S (Sp). The asymptotic competitive ratio
  is 1 since Sgtgtp.
• P2 pjp, 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 (Sp). The asymptotic
  competitive ratio is 1 since Sgtgtp.

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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 (Continued Work)?

• For p pj1, intreei released at ri cmax
  problem
• Better approximation algorithms? Tight
  Competitive Ratio?
• For p pjp, preci released at ri cmax
• Asymptotic competitive ratio? (gt2-2/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,
  35211-220, 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,177116-134, 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 C2Cn
• 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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Complexity results for scheduling problems

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