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Analysis and Approximation of Optimal Co-Scheduling on Chip Multiprocessors Yunlian Jiang Xipeng Shen The College of William & Mary, USA Jie Chen DoE Jefferson Lab , USA Rahul Tripath University of South Florida, USA Cache

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Analysis and Approximation

  • f Optimal Co-Scheduling on

Chip Multiprocessors

Yunlian Jiang Xipeng Shen The College of William & Mary, USA Jie Chen DoE Jefferson Lab , USA Rahul Tripath University of South Florida, USA

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Cache Sharing on CMP

 Shorten inter-thread

communication

 Flexible usage of cache

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CPU

Shared Cache

CPU

 degrade performance  impair fairness  hurt performance isolation

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Degradation is affected by co-runner

Performance degradation range

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20 40 60 80 100 120 140 160 180 200 Degradation min median max

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Job Co-Scheduling

 To assign jobs to chips in a manner to

minimize contention

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Shared cache 1 Shared cahe 2

P1 P3 P4 P2

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Job Co-Scheduling

 To assign jobs to chips in a manner to

minimize contention

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Shared cache 1 Shared cahe 2

P1 P3 P4 P2 Resource Waste Resource Contention

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Job Co-Scheduling

 To assign jobs to chips in a manner to

minimize contention

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Shared cache 2 Shared cache 1

P1 P3 P2 P4

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Job Co-Scheduling

 To assign jobs to chips in a manner to

minimize contention

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Shared cache 2 Shared cache 1

P1 P3 P2 P4

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The Goal of this Work

 Related work

 Snavely etc. [00’ ASPLOS]

 Goal of this work

 Find the optimal schedule on CMP system

 Benefits

 Evaluate current schedule quality  Applied in real system

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Contributions

 Polynomial optimal solution on Dual-core

systems

 NP-Completeness proof on K-core (K>2)

systems

 Polynomial approximation algorithms on K-

core (K>2) systems

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Contributions

 Polynomial optimal solution on Dual-core

systems

 NP-Completeness proof on K-core (K>2)

systems

 Polynomial approximation algorithms on K-

core (K>2) systems

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Problem Formulation

 M jobs  N Core processors The College of William and Mary

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Problem Formulation

 M jobs  N Core processors The College of William and Mary

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Problem Formulation

 Assignment The College of William and Mary

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i i i i

cCPI sCPI Deg sCPI − =

 Goal Minimize ∑Degi

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Dual-Core System

 Polynomial Solution

 Minimum-weight perfect matching

[Edmonds: 1965]  Matching

 A matching M in graph G is a set of edges with no

common vertex.

 perfect matching is a matching which

matches all vertices of the graph

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Dual-Core System

 Minimum-weight perfect

matching

 In edge weighted

graph

 Sum of weight of

edges in the match is minimum

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Dual-Core System

 Job

  • Nodes

 Corun-Degradation

  • Edge Weight

 Optimal Schedule

  • Minimum weight

perfect matching

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Contributions

 Polynomial optimal solution on Dual-core

systems

 NP-Completeness proof on K-core (K>2)

systems

 Polynomial approximation algorithms on K-

core (K>2) systems

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NP-Completeness proof

 NP proof

 Given a schedule, can compute

 Reduction

 NP-Complete problem  Job Co-scheduling

 Multidimensional Assignment Problem (MAP)

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NP-Completeness Proof

 MAP the College of William and Mary

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NP-Completeness Proof

 MAP the College of William and Mary

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NP-Completeness Proof

 MAP the College of William and Mary

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Minimize Total Weight

Weight

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NP-Completeness Proof

 Job Co-Scheduling on CMP the College of William and Mary

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NP-Completeness Proof

 Job Co-Scheduling on CMP the College of William and Mary

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Minimize Total Weight

Weight

Sum of Degradations in the Assignment

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NP-Completeness Proof

 MAP  Job Co-Scheduling the College of William and Mary

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NP-Completeness Proof

 MAP  Job Co-Scheduling the College of William and Mary

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=

Weight

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NP-Completeness Proof

 MAP  Job Co-Scheduling the College of William and Mary

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= =∞

Weight

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Contributions

 Polynomial optimal solution on Dual-core

systems

 NP-Completeness proof on K-core (K>2)

systems

 Polynomial approximation algorithms on K-

core (K>2) systems

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Approximation algorithms

 Hierarchical Perfect Matching  Greedy the College of William and Mary

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Hierarchical Perfect Matching

 Dual-core system optimal solution the College of William and Mary

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N Core N/2 Core Dual Core

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Hierarchical Perfect Matching

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Hierarchical Perfect Matching

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Hierarchical Perfect Matching

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Hierarchical Perfect Matching

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Hierarchical Perfect Matching

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Greedy Algorithm

 Basic idea

 Schedule the least “polite” job first

 “politeness” of a Job

 Sum of degradations of all the assignments

contain this job.

 Impact of a job on others

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Greedy Algorithm

I.

Sort unassigned jobs based on politeness

II.

Pick the least politeness job J to schedule

  • III. Add assignment contains J with least

degradation to schedule

  • IV. Update unassigned job list

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Local Optimization

 Main Scheme the College of William and Mary

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for i  1 to K-1 for j  i+1 to K Local-Optimization( i, j )

K: number of assignments

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Performance Evaluation

 Machine

 AMD Opteron 4 core processors

 Benchmarks

 15 SPEC CPU2000, 1 Stream

 Metrics

 Performance Degradation  Scheduling time  Fairness

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Performance Degradation

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10 20 30 40 50 60 70

  • Perf. Degradation(%)

Benchmarks

OPT Greedy Hierarchical Random

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Performance Degradation

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10 20 30 40 50 60 70 ammp art applu bzip crafty equake facerec gap mcf parser stream swim twolf vpr Average Benchmarks

  • Perf. Degradation(%)

OPT Greedy-Opt Hierarchical-Opt Random

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Scheduling Time

5 10 15 20 16 32 48 64 80 96 112 128 144 Number of Jobs Running Time(s)

Greedy Greedy-opt Hierarchical Hierarchical-opt

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Fairness

0.05 0.1 0.15 0.2 0.25

1

Unfairness Factor

OPT Greedy

  • pt

Greedy Hierarchical

  • pt

Hierarchical Random the College of William and Mary

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 Unfairness Factor

 Coefficient of Variation of normalized degradation

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Conclusion

 Job co-scheduling on CMP is crucial

 Different schedule performance varies

 Dual-core system

 Polynomial solvable

 K-core (K>2) system

 NP-Complete problem  Heuristics

 Hierarchical  Greedy

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Acknowledgement

 Weizhen Mao William and Mary  Cliff Stein Columbia University  William Cook Georgia Tech  National Science Foundation  IBM CAS Fellowship the College of William and Mary

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Thanks!

Questions?

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