Presentation is loading. Please wait.

Presentation is loading. Please wait.

Properties of SPT schedules Eric Angel, Evripidis Bampis, Fanny Pascual LaMI, university of Evry, France MISTA 2005.

Similar presentations


Presentation on theme: "Properties of SPT schedules Eric Angel, Evripidis Bampis, Fanny Pascual LaMI, university of Evry, France MISTA 2005."— Presentation transcript:

1 Properties of SPT schedules Eric Angel, Evripidis Bampis, Fanny Pascual LaMI, university of Evry, France MISTA 2005

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. C 3 =4) Main quality criteria: Makespan -> (P||C max ) Sum of completion times ( ∑ Cj ) -> (P|| ∑ Cj ) P1 P2 P3 n tasks m machines time

4 SPT schedules SPT= Shortest Processing Time first Smith’s rule: SPT greedy –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 A schedule minimizes ∑Cj iff it is an SPT schedule SPT schedules [Bruno et al]: notion of rank Rank 1: Rank 2: Rank 3: The tasks of rank i are counted i times in the ∑Cj : ∑Cj= C 1 + C 4 + C 7 + C 2 + C 5 + C 8 + … = l(1) + ( l(1) + l(4) ) + ( l(1) + l(4) + l(7) ) + … = 3 l(1) + 2 l(4) + l(7) + … machine 1

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 SPT greedy 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. i  {1,…,m}j on Pi P1 P P1

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: C={ x1, x2,..., xn } –Min. Max ∑Cj: k= ½ Min ∑Cj ;2 machines;2n tasks Example : C={ x1, x2, x3 } Tasks = P1 P2 3 x1 3 x1 3 x1 + 2 x2 3 x1 + 2 x3 + 3 x1 + 2 ≠ce contrib ∑Cj = 3 x1 + 2 x2 3 x1 3 x1 3 x1 + 2,,,,, 3 x1 + 2 x3 + x1 x2x3 + x1+ x2 + x3  x1 + x2 = x3 Claim: Solution of (Min. Max ∑Cj)  ∑Cj = ∑Cj = k = ½ Min ∑Cj P1P2

10 Min Max ∑Cj is NP-complete tasks ≠ce length rank ≠ce contrib ∑Cj transformation: –Partition: C={ x1, x2,..., xn }. –Min. Max ∑Cj: k= ½ Min ∑Cj ; 2 machines; 2n tasks n - 2 n n - 1

11 Min. Max ∑Cj : analysis of SPT greedy Theorem 1 : –The approx. ratio of SPT greedy is ≤ 3 – 3/m + 1/m 2. Theorem 2 : –The approx. ratio of SPT greedy is ≥ 2 – 2/(m 2 + m).

12 Min. Max ∑Cj : analysis of SPT greedy Theorem 2 : –The approx. ratio of SPT greedy is ≥ 2 – 2/(m 2 + m). ( example: for m=3, ratio ≥ 11/6 ) Proof: –m(m-1) tasks of length 1 –A task of length B= m(m+1)/2 –Example for m=3: Max ∑Cj = 11 Max ∑Cj = 6

13  Definition of an SPT schedule  Quality of SPT schedules on these criteria: Min. Max ∑Cj. Fairness measure.  Conclusion Outline

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 i th task of S, and the min. completion time of the i th 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) 1 4 Vector X = (1, 3, 4) 2

15 Fairness measure Example: Vector X = (1, 2, 4) 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) C (X) = 4/3 C* (I) = 4/3 I={,, }

16 Fairness measure Theorem 1: –C(X SPTgreedy ) ≤ 2 – 1/m. ( example: for m=2, C(X SPTgreedy ) ≤ 3 / 2 ) Theorem 2: –C* = 3/2 when m=2. Proof of theorem 2: Vector X = (1, 1, 3) C (I) = C* = 3/2

17 Conclusion – Future work Conclusion –Minimization ofMax ∑Cj = NP-complete pb. –SPT greedy between 2 – 2/(m 2 + m) and 3 – 3/m + 1/m 2 for Min. Max ∑Cj. –Good fairness measure for SPT greedy. Future work –A better bound for SPT greedy for Min. Max ∑Cj. –Study of fairness measure on other problems.


Download ppt "Properties of SPT schedules Eric Angel, Evripidis Bampis, Fanny Pascual LaMI, university of Evry, France MISTA 2005."

Similar presentations


Ads by Google