Bicriteria Scheduling for Parallel Jobs

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Bicriteria Scheduling for Parallel Jobs

• Dror Feitelson and Ahuva Mualem
• Hebrew University

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Talk Structure

• The Model
• Reordering and Reductions to Single Machine
• List Algorithm for Parallel Jobs
• Applications

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Parallel Job and Machine (def.)

• Parallel machine has
• m identical processors
• Parallel job j has
• Processing time pj ?? 0
• Number of processors for execution mj ?1, , m
• Release time rj ?? 0
• Preemptions We also assume that a job may be
  preempted and later continued to run on a
  possibly different set of mj processors without
  affecting its processing time.

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Makespan (Cmax) and Sum of Completion Times (?Cj
)

• Goal to find a schedule that simultaneously
  minimizes both objectives P mj , rj , pmtn
  ? wj Cj , Cmax

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Makespan (Cmax) and Sum of Completion Times (?Cj
)

• Goal to find a schedule that simultaneously
  minimizes both objectives P mj , rj , pmtn
  ? wj Cj , Cmax
• But
• Conflicting objectives

time
proc
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Makespan (Cmax) and Sum of Completion Times (?Cj
)

• Goal to find a schedule that simultaneously
  minimizes both objectives. P mj , rj , pmtn
  ? wj Cj , Cmax
• But
• Conflicting objectives.
• Optimization of each objective alone is NP-hard
• LLLR84 1 rj , pmtn ? wj Cj
• Droz 94 P mj , pmtn Cmax
• Goal to find an efficient schedule that
  approximately meets both objectives.

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Bicriteria Results
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Reorderings and Reductions to Single Machine
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Reordering Technique

• Framework Reorder Phillips, Stein, Wein 95
• Simulate a schedule for relaxed version of the
  problem.
• Reorder the jobs by their relaxed completion
  times, and use List schedule.

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Reordering Technique

• Framework Reorder Phillips, Stein, Wein 95
• Simulate a schedule for relaxed version of the
  problem.
• Reorder the jobs by their relaxed completion
  times, and use List schedule.

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Reordering Technique

• Framework Reorder Phillips, Stein, Wein 95
• Simulate a schedule for relaxed version of the
  problem.
• Reorder the jobs by their relaxed completion
  times, and use List schedule.
• Thm PSW95 Simulating SRPT, algorithm Reorder
  achieves (2, 2)-approximation
  for
• 1 rj ? Cj , Cmax .
• On-line result.

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Reduction to Single Machine

• Framework Resize Chekuri, Motwani, Natarajan,
  Stein 97
• Resize the jobs. Simulate a relaxed schedule on
  a single machine.
• Reorder the jobs by their simulated completion
  times, and use List schedule.

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Reduction to Single Machine

• Framework Resize Chekuri, Motwani, Natarajan,
  Stein 97
• Resize the jobs. Simulate a relaxed schedule on
  a single machine.
• Reorder the jobs by their simulated completion
  times, and use List schedule.

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Reduction to Single Machine

• Thm CMNS97 Simulating SRPT on the instance
• rj rj , pj pj / m, algorithm Resize is
  a
• (3, 3) -approximation for P rj ? Cj ,
  Cmax .
• Remark It is not clear how to plug in the List
  algorithm for parallel jobs. The analysis of the
  former algorithm relies on the prefix property
  Cj essentially depends only on the preceding
  jobs 1,..,j.
• (Cj depends on the reordered prefix, not on
  the suffix).

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List Algorithmfor Parallel Jobs
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List Algorithm for Parallel Jobs

• Algorithm Parallel List Garey and Graham 75
• Scan the input list and run any job whose
  processors demand is not greater than current
  number of free processors. Repeat until the input
  list is empty.

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List Algorithm for Parallel Jobs

• Algorithm Parallel List Garey and Graham 75
• Scan the input list and run any job whose
  processors demand is not greater than current
  number of free processors. Repeat until the input
  list is empty.

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List Algorithm for Parallel Jobs

• Thm GG75 Parallel List is a 2-approximation
• for P mj Cmax.
• The makespan achieved by parallel list in
  presence of release times is
• ?

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Preemptions provide the prefix property

• Algorithm Preemptive Parallel List
• Scan the input list and run any job whose
  processor demand is not greater than current
  number of free processors. Whenever a job
  finishes, preempt all running jobs. Repeat until
  the input list is empty.

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Preemptions provide the prefix property

• Algorithm Preemptive Parallel List
• Scan the input list and run any job whose
  processor demand is not greater than current
  number of free processors. Whenever a job
  finishes, preempt all running jobs. Repeat until
  the input list is empty.

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Applications
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SRA (Smallest Remaining Area)

• On-line Algorithm SRA
• Resize Define the instance I
• rj max rj , pj , pj mj pj / m,
  mj 1.
• Simulate SRPT on I on a single machine.
• Reorder the jobs by their simulated completion
  times, and use Preemptive Parallel List.

Remark to ensure makespan factor of 3 we can do
the simulation step and meanwhile apply the
Preemptive Parallel List. Then we avoid the idle
times in this example.
time
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Bounds

• Thm (A slightly modified) SRA is a
• (6, 3)-approximation for P mj , rj , pmtn ?
  Cj , Cmax.
• Proof
• Show that ? CjOPT ( I) ? 2 ? CjOPT ( I ).
• Then, CjSRA ?
• ? 3 CjOPT ( I).

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