Diapositiva 1

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Kernel Partitioning of Streaming Applications A
Statistical Approach to an NP-complete Problem
Petar Radojkovic, Paul M. Carpenter, Miquel
Moretó, Alex Ramirez, Francisco J. Cazorla
Vancouver, BC, Canada. December 5, 2012.
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Stream programming languages

• Programming of multithreaded applications is
  difficult
• A possible solution Expose the parallelism to
  the compiler

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A lot of pressure on the compiler
Source code Explicit dependencies
(Optimal) Multithreaded executable
Compiler
Complex codeanalysis and optimizations
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One of the problems

• How to (optimally) partition kernels into the
  software threads

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The importance of a good kernel partitioning

• StreamIt 2.1.1 benchmark suite
• Exactly four software threads
• Observe the performance of good and bad kernel
  partitions

Performance difference ranges
from
to
2.4x
3.9x
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Kernel partitioning is an intractable problem

• NP-complete Garey and Johnson, 1979
• Vast exploration space
• channelvocoder benchmark (StreamIt 2.1.1)
  contains 54 kernels

1034
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The performance of the optimal kernel partition?

• How good is Kernel Partitioning (KP) method X?

• Problem for the industry
• What is a possible performance improvement?

Optimal KP?
Performance
State of the art
KP method X
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We cannot compute the performance of the optimal
KP! Can we estimate it?
Performance (cost) of a kernel partition in a
random sample
15728
12659
14124
13564
15684
16428
10627
14551
ranges from X to Y confidence level 0.9 0.95
0.99
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Extreme Value Theory application

• Example How high should be a river embankment?

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Extreme Value Theory

• 1928 R.A. Fisher and L.H.C. Tippett _at_
  Proceedings of the Cambridge Philosophical
  Society, vol. 24.
• 1943 B. Gnedenko _at_ Annals of Mathematics,
  vol 44.
• 1974 A. A. Balkema and L. de Haan _at_ Annals
  of Probability, vol. 2.
• 1975 J. Pickands _at_ Annals of Statistics,
  vol. 3.

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Estimate the performance of the optimal
KP Extreme Value Theory Peak Over Threshold
analysis
Cost
Step 1 Execute random (i.i.d.) KPs Measure
performance (cost) of each of them
Sample of KPs
1
Step 2 Plot cumulative distribution function
F(x)
0
Min cost
Max cost
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PickandsBalkemade Haan theorem Generalized
Pareto Distribution (GPD)
Step 3 Fit Fu(y) to GPD GPD is fully described
with s and ?
1
F(x)
Fu(y)
0
x
u
Min cost
Max cost
Step 4 Estimate the cost of the optimal
KP Point estimation Confidence bounds for a
given confidence interval
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Can I apply this analysis to my problem?

• Theorem For a large class of underlying
  distribution functions
• Is my problem in this class?

Generate a random (i.i.d.) sample
Statistical analysis
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Evaluation

• Experimental
• StreamIt benchmarks
• Compiled into exactly four software threads
• The cost is each KP is based on the StreamIt
  compiler estimate

• Results
• Q1 Can we apply the presented statistical
  analysis to the kernel partitioning problem?
• Q2 Is the estimation precise?
• Q3 Is the estimation accurate?

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Q1 Can we apply the presented statistical
analysis to the kernel partitioning problem?

• How to generate random (i.i.d.) kernel partitions?

• The samples should be selected following the
  uniform distribution
• i.i.d. may not be sufficient

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Q2 Is the estimation precise?

• The precision of the method depends on the number
  of the KPs in the sample
• Larger sample ? Higher precision

serpent_full (slowest convergence)
channelvocoder (fastest convergence)

• StreamIt 2.1.1 benchmarks
• Few 1000 KPs are sufficient for a precise
  estimation

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Q3 Is the estimation accurate?

• In general, finding the optimal KP is impractical
• However, for some benchmarks, it is feasible

serpent_full benchmark

• The same trend for bitonic-sort, des, fft,
  mpeg2-subset, and tde_pp benchmarks

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Can random sampling find a good KP?
We do not need the best KP We need a good one!

• Total number of KPs 1020
• Random sample of 1000 KPs
• Prob(capture the best KP) 0

• Prob(capture one out of the best-performing 1 of
  KPs) 99.99

• Radojkovic et al. _at_ ASPLOS 2012.

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Random sampling vs. heuristics based approach

• Random sampling on its own can provide very good
  results

• Carpenter et al. _at_ CASES 2009.

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Summary
Problem
Our approach
Kernel partitioningof streaming applications
Intractable problem Unknown optimal performance
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Extreme Value Theory is also used in
Civil engineering
Finance