Rounding Technique

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Rounding Technique

• We can simplify scheduling problem structure by
  rounding job sizes, release dates

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Rounding Scheme 1

• Make all processing times and release dates
  integral powers of 1e
• Rx (1e)x
• Ix Rx, Rx1)
• Length of Ix e Rx

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Effect of rounding on optimal average completion
time

• How much does this rounding increase optimal
  average completion time? Give argument

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Enforce rj gt epj

• Make each job j have a release time of at least
  epj
• Argue this causes at most (1e) effect on optimal
  average completion time

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PTAS for 1rjSCj

• Run SPT until at most 3/e7 jobs are left
• Each job released at maxrj,pj/e2
• Enumerate last jobs optimally

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Key idea 1 Time stretching

• A job is small if pj lt eIx where Ix is interval
  that job j is started
• No small job crosses into interval Ix1
• Stretch interval Ix by (1e) to fit last crossing
  job in

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Observation

• If all jobs are small in their intervals, with
  previous time-stretching idea, we achieve
  optimality as we approximate SRPT which is
  optimal for the preemptive version of this
  problem.

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Key idea 2 Moving large jobs

• We can move large jobs that appear before time t
  e7OPT back in the schedule by some amount to
  where they are small
• Key arguments
• Space to fit these jobs (time stretching)
• Delay caused by stretching is not too large

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Rounding Scheme 2

• Round release dates and processing times to
  integral powers of eP/n2 where P is the largest
  job size
• How much does this rounding affect average flow
  time?
• Key observation all processing times in interval
  0, n2/e and all events occur in interval
  0,2n3/e
• Leads to a polynomial number of decision times to
  be represented in a dynamic programming solution