Properties%20of%20SPT%20schedules

1
Properties of SPT schedules
MISTA 2005

• Eric Angel, Evripidis Bampis, Fanny Pascual
• LaMI, university of Evry, France

2
Outline

• Definition of an SPT schedule
• Quality of SPT schedules on these criteria
• Min. Max ?Cj minimization of the maximum sum of
  completion times per machine.
• Fairness measure
• Conclusion

3
Model

• Example
• Cj completion time of task j. (e.g. C34)
• Main quality criteria
• Makespan -gt (PCmax)
• Sum of completion times ( ?Cj ) -gt (P ?Cj )

1
5
6
P1
n tasks m machines
2
4
P2
P3
3
time
4
SPT schedules

• SPT Shortest Processing Time first
• Smiths rule SPTgreedy
• Sort tasks in order of increasing lengths.
• Schedule them as soon as a machine is available.
• Algo which minimizes ?Cj .
• Class of the schedules which minimize ?Cj
• Bruno et al Algorithms for minimizing mean
  flow time

5
SPT schedules

• Bruno et al notion of rank.

The tasks of rank i are counted i times in the
?Cj ?Cj C1 C4 C7 C2 C5 C8
l(1) ( l(1) l(4) ) ( l(1) l(4)
l(7) ) 3 l(1) 2 l(4) l(7)
machine 1

• A schedule minimizes ?Cj iff it is an SPT
  schedule.

6
Outline

• Definition of an SPT schedule
• Quality of SPT schedules on these criteria
• Min. Max ?Cj minimization of the maximum sum of
  completion times per machine.
• Example
• NP-complete problem
• Analysis of SPTgreedy
• Fairness measure
• Conclusion

7
Minimization of Max ?Cj

• Pb Minimization of Max ? Cj
• To minimize Max ?Cj ? To minimize ?Cj
• Max ?Cj 7 Max ?Cj 6
• ?Cj 10 ?Cj 11
• NP-complete problem.

8
To minimize Max ?Cj is an NP-complete problem

• We reduce the partition problem into a Min. Max
  ?Cj problem.
• Partition Let C x1, x2, . . . , xn be a set
  of numbers. Does there exist a partition (A,B) of
  C such that ?x?A x ?x?B x ?
• Min. Max ?Cj Let n tasks and m machines, and let
  k be a number. Does there exist a schedule such
  that Max ?Cj k ?

9
Min Max ?Cj is NP-complete

• Transformation
• Partition Cx1, x2, . . . , xn
• Min. Max ?Cj k ½ Min ?Cj 2 machines2n tasks
• Example C x1, x2, x3
• Tasks

Claim Solution of (Min. Max ?Cj) ?
?Cj ?Cj k ½ Min ?Cj
P1
P2
?ce contrib ?Cj
x3
? x1 x2 x3
x1
x2
10
Min Max ?Cj is NP-complete

• transformation
• Partition Cx1, x2, . . . , xn.
• Min. Max ?Cj k ½ Min ?Cj 2 machines 2n
  tasks.

tasks ?ce length rank ?ce contrib ?Cj

n
n - 1
n - 2
...
1
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Min. Max ?Cj analysis of SPTgreedy

• Theorem 1
• The approx. ratio of SPTgreedy is
• 3 3/m 1/m2 .
• Theorem 2
• The approx. ratio of SPTgreedy is
• 2 2/(m2 m).

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Min. Max ?Cj analysis of SPTgreedy

• Theorem 2
• The approx. ratio of SPTgreedy is 2 2/(m2
  m).
• ( example for m3, ratio 11/6 )
• Proof
• m(m-1) tasks of length 1
• A task of length B m(m1)/2
• Example for m3

Max ?Cj 6
Max ?Cj 11
13
Outline

• Definition of an SPT schedule
• Quality of SPT schedules on these criteria
• Min. Max ?Cj.
• Fairness measure.
• Conclusion

14
Fairness measure

• Kumar, Kleinberg Fairness Measures For
  Ressources Allocation (FOCS 2000)
• Definition global approx ratio of a schedule S
• Max. ratio between the completion time of the ith
  task of S, and the min. completion time of the
  ith task of any other schedule.
• I instance X (sorted) vector of completion
  times
• C(X) min ? s.t. X ? ? Y ? Y feasible schedule
  of I
• C(I) min C(X) s.t. X feasible schedule of I
• C max C(I)

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Fairness measure
Possible vectors X (1, 2, 5) (1, 3, 3) (1, 3,
5) (2, 3, 3) (2, 3, 4) (1, 3, 6) (1, 4, 6) (2, 3,
6) (2, 5, 6) (3, 4, 6) (3, 5, 6) Min (1, 2, 3)

• Example

Vector X (1, 2, 4)
C (X) 4/3 C (I) 4/3

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Fairness measure

• Theorem 1
• C(XSPTgreedy) 2 1/m.
• ( example for m2, C(XSPTgreedy) 3/2 )
• Theorem 2
• C 3/2 when m2.
• Proof of theorem 2

C (I) C 3/2
Vector X (1, 1, 3)
17
Conclusion Future work

• Conclusion
• Minimization ofMax ?Cj NP-complete pb.
• SPTgreedy between 2 2/(m2 m) and 3 3/m
  1/m2 for Min. Max ?Cj.
• Good fairness measure for SPTgreedy.
• Future work
• A better bound for SPTgreedy for Min. Max ?Cj.
• Study of fairness measure on other problems.