Analysis of cooperation in multiorganization Scheduling

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Analysis of cooperation in multi-organization
Scheduling

• Pierre-François Dutot (Grenoble University)
• Krzysztof Rzadca (Polish-Japanese school, Warsaw)
• Fanny Pascual (LIP6, Paris)
• Denis Trystram (Grenoble University and INRIA)
• NCST, CIRM, may 13, 2008

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Goal
The evolution of high-performance execution
platforms leads to distributed entities
(organizations) which have their own  local 
rules. Our goal is to investigate the
possibility of cooperation for a better use of a
global computing system made from a collection of
autonomous clusters. Work partially supported
by the Coregrid Network of Excellence of the EC.
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Main result
We show in this work that it is always possible
to produce an efficient collaborative solution
(i.e. with a theoretical performance guarantee)
that respects the organizations selfish
objectives. A new algorithm with theoretical
worst case analysis plus experiments for a case
study (each cluster has the same objective
makespan).
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Target platforms






Our view of computational grids is a collection
of independent clusters belonging to distinct
organizations.
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Computational model(one cluster)
Independent applications are submitted locally on
a cluster. The are represented by a precedence
task graph. An application is a parallel rigid
job. Let us remind briefly the model (close to
what presented Andrei Tchernykh this morning)
See Feitelson for more details and classification.
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Local queue of submitted jobs
J1
J2
J3

• Cluster

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Job
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overhead
Computational area
Rigid jobs the number of processors is fixed.
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Runtime pi
of required processors qi
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Runtime pi
of required processors qi
high jobs (those which require more than m/2
processors) and low jobs (the others).
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Scheduling rigid jobsPacking algorithms
Scheduling independent rigid jobs may be solved
as a 2D packing Problem (strip packing). List
algorithm (off-line).
m
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n organizations.
J1
J2
J3

• Cluster

Organization k
m processors (identical for the sake of
presentation)
k
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n organizations (here, n3, red 1, green 2 and
blue 3)
Cmax(Ok) organization
Cmax(Mk) cluster
Each organization k aims at minimizing  its 
own makespan max(Ci,k) We want also that the
global makespan is minimized
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n organizations (here, n3, red 1, green 2 and
blue 3)
Cmax(O3) Cmax(M1)
Cmax(Ok) organization
Cmax(Mk) cluster
Cmax(O2)
Each organization k aims at minimizing  its 
own makespan max(Ci,k) We want also that the
global makespan (max over all local ones) is
minimized
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Problem statement
MOSP minimization of the  global  makespan
under the constraint that no local schedule is
increased. Consequence taking the restricted
instance n1 (one organization) and m2 with
sequential jobs, the problem is the classical 2
machines problem which is NP-hard. Thus, MOSP is
NP-hard.
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Multi-organizations
Motivation A non-cooperative solution is that
all the organizations compute their local jobs
( my job first  policy). However, such a
solution is arbitrarly far from the global
optimal (it grows to infinity with the number of
organizations n). See next example with n3 for
jobs of unit length.
O1
O1
O2
O2
O3
O3
with cooperation (optimal)
no cooperation
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Preliminary results
Single organization scheduling (packing).
Resource constraint list algorithm (Garey-Graham
1975).

• List-scheduling (2-1/m) approximation ratio
• HF Highest First schedules (sort the jobs by
  decreasing number of required processors). Same
  theoretical guaranty but better from the
  practical point of view.

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Analysis of HF (single cluster)
Proposition. All HF schedules have the same
structure which consists in two consecutive zones
of high (I) and low (II) utilization. Proof. (2
steps) By contracdiction, no high job appears
after zone (II) starts
low utilization zone (II)
high utilization zone
(I) (more than 50 of processors are busy)
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Collaboration between clusters can substantially
improve the results.
Other more sophisticated algorithms than the
simple load balancing are possible matching
certain types of jobs may lead to bilaterally
profitable solutions.
no cooperation
with cooperation
O1
O1
O2
O2
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If we can not worsen any local makespan, the
global optimum can not be reached.
local
globally optimal
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1
O1
O1
1
1
1
2
2
2
O2
O2
2
2
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If we can not worsen any local makespan, the
global optimum can not be reached.
local
globally optimal
2
1
O1
O1
1
1
1
2
2
2
O2
O2
2
2
1
2
O1
1
best solution that does not increase Cmax(O1)
O2
2
2
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If we can not worsen any local makespan, the
global optimum can not be reached.

• Lower bound on approximation ratio greater than
  3/2.

2
1
O1
O1
1
1
1
2
2
2
O2
O2
2
2
1
2
O1
1
best solution that does not increase Cmax(O1)
O2
2
2
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Multi-Organization Load-Balancing
1 Each cluster is running local jobs with Highest
First LB max (pmax,W/nm) 2.
Unschedule all jobs that finish after 3LB. 3.
Divide them into 2 sets (Ljobs and Hjobs) 4. Sort
each set according to the Highest first order 5.
Schedule the jobs of Hjobs backwards from 3LBon
all possible clusters 6. Then, fill the gaps with
Ljobs in a greedy manner
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Hjob
Ljob
let consider a cluster whose last job finishes
before 3LB
3LB
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Hjob
Ljob
3LB
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Hjob
Ljob
3LB
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Hjob
Ljob
3LB
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Ljob
3LB
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Ljob
3LB
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Notice that the centralized scheduling
mechanism isnot work stealing, moreover, the
idea is to changeas minimum as we can the local
schedules.
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Feasibility (insight)
Zone (I)
Zone (II)

• Zone (I)

3LB
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Sketch of analysis
Proof by contradiction let us assume that it is
not feasible, and call x the first job that does
not fit in a cluster.
Case 1 x is a small job. Global surface
argument Case 2 x is a high job. Much more
complicated, see the paper for technical details
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3-approximation (by construction)This bound is
tight
Local HF schedules
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3-approximation (by construction)This bound is
tight
Optimal (global) schedule
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3-approximation (by construction)This bound is
tight
Multi-organization load-balancing
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Improvement
We add an extra load-balancing procedure
O1
O2
Local schedules
O3
O4
O5
O1
O2
Multi-org LB
O3
O4
O5
O1
O2
O3
Compact
O4
O5
O1
O2
O3
load balance
O4
O5
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Some experiments
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Conclusion

• We proved in this work that cooperation may help
  for a better global performance (for the
  makespan).
• We designed a 3-approximation algorithm.
• It can be extended to any size of organizations
  (with an extra hypothesis).

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Conclusion

• We proved in this work that cooperation may help
  for a better global performance (for the
  makespan).
• We designed a 3-approximation algorithm.
• It can be extended to any size of organizations
  (with an extra hypothesis).
• Based on this, it remains a lot of interesting
  open problems, including the study of the problem
  for different local policies or objectives
  (Daniel Cordeiro)

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Thanks for attentionDo you have any questions?
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Using Game Theory?
We propose here a standard approach using
Combinatorial Optimization. Cooperative Game
Theory may also be usefull, but it assumes that
players (organizations) can communicate and form
coalitions. The members of the coalitions split
the sum of their playoff after the end of the
game. We assume here a centralized mechanism and
no communication between organizations.