List Scheduling on Related Machines

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List Scheduling on Related Machines
Related machines machines may have different
speeds
1
1
1
? 0.25
? 1
? 0.5
1
jobs
processors 1 2 3
2
Hey, the opt makespan is at most L
6
3
5
4
7
1
1
4
5
2
6
? 0.25
? 1
? 0.5
3
2
7
jobs
processors 1 2 3
3
Lemma If the little birdie is right (opt
makespan L) then 2PACK will succeed.
Proof Suppose 2PACK fails on job h

• hs length on processor 1 L , so
• load of processor 1 gt L

• r first processor with load L
• (or m1, if no such processor)

• Claim if opt executes k on
• a machine in r,r1,,m then so
• does 2PACK

optimum 2PACK
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(No Transcript)
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Lemma If the little birdie is right (opt
makespan L) then 2PACK will succeed.
Proof Suppose 2PACK fails on job h

• hs length on processor 1 L , so
• load of processor 1 gt L

• r first processor with load L
• (or m1, if no such processor)

• In other words if 2PACK executes
• k on a machine in 1,2,,r-1 then so
• does opt

• So opts (speed-weighted) total load
• on processors 1,2,,r-1 is gt (r-1)L

r
r

• So some opts processor has load
• gt L -- contradiction

optimum 2PACK
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• Algorithm
• 1. Let Bj 2( 1 2 4 2j ) 2(2j1-1)
• bucket j time interval Bj-1 , Bj (of
  length 22j)
• 2. j 0
• while there are unassigned jobs
• apply 2PACK with L 2j in bucket j
• if 2PACK fails on job k
• let j j1 and continue (starting with job
  k)

7
k
bucket j
k
k
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Analysis

• Suppose the optimal makespan is u
• Choose j such that 2j-1 lt u 2j
• Then 2PACK will succeed in j th bucket (L 2j
  )
• so algorithms makespan 2(2021 2j)
• and

We get competitive ratio 8
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List Scheduling on Related Machines

• Theorem
• There is an 8-competitive online algorithm for
  list scheduling on related machines (to minimize
  makespan).
• With randomization the ratio can be improved to
  2e.
• Aspnes, Azar, Fiat, Plotkin, Waarts 06
• World records
• upper bound 5.828 (4.311 randomized)
• lower bound 2.438 (2 randomized)
• Berman, Charikar, Karpinski 97
• Epstein, Sgall 00