Dataflow Frequency Analysis based on Whole Program Paths

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Dataflow Frequency Analysis based on Whole
Program Paths
Eduard Mehofer Institute for Software
Science University of Vienna mehofer_at_par.univie.a
c.at www.par.univie.ac.at/mehofer
Bernhard Scholz Institute of Computer
Languages Vienna University of Technology scholz_at_
complang.tuwien.ac.at www.complang.tuwien.ac.at/sc
holz
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Dataflow Frequency Analysis

• Goal
• accurately computing frequencies of data flow
  facts
• Problem
• high costs for computing accurate frequencies
• requires whole program path
• efficient data structures and algorithm?
• Approach
• exploiting algebraic properties of
  bi-distributive DFA problems
• employing WPPs to capture control flow
• computing frequencies in a bottom-up style on the
  WPP graph

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Outline

• Motivation
• WPP profiling
• Properties of bi-distributive DFAs
• Algorithm
• Experiments
• Conclusion

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Classical Approach

• Classical Program Optimization

optimizer
data flow analysis
Program
Optimized program
binary information
transformation

• Drawback

heavily rarely never
Optimizer
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Profiling Approach

• Probabilistic Program Optimization

Optimizer based on profiling
dataflow freq. analysis
Optimized program
frequency information
transformation

• Advantage

heavily rarely never
Optimizer
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Running Example

• CFG Example
• simple code fragment
• 8 times left branch
• terminates via right branch
• Reaching definitions problem
• two definitions d1, d2
• d1 kills d2 and vice versa
• use of x at the end of loop
• Questions
• How often does d1 hold at node 5?
• How often does d2 hold at node 5?

s
1
3
2
d1 x...
d2 x...
4
...x...
5
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WPP Profiling

• Captures the whole program path
• Larus at PLDI99
• Path profiling techniques for acyclic paths
• minimal insertion of instrumentation code
• keeps executable fast
• Sequitur for compression
• builds a grammar
• terminals are acyclic paths
• nonterminals have only one production
• graph representation of grammar
• grammar has only sentence
• best case logarithmic size reduction

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WPP Example

• CFG Example

• Program Run
• 8x left branch
• 1x right branch

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Bi-Distributive Dataflow Problems

• Properties
• finite lattice 2D (power set of dataflow facts)
• transition functions are monotone
• transition functions distribute
• representation relation
• covers bit-vector problems
• Due to properties
• transition functions represented as 0/1-matrices
• states represented as 0/1-vectors

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Representation Relation

• Transition function f 2D? 2D
• represented by f r D ? 2D
• artificial data fact ?

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Matrix Representation

• Matrix representation of function f

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Dataflow Frequencies

• Definition of dataflow frequencies for node v
• ?r whole program path
• prefix set of all sub-paths from start node to
  node v
• ? converts data flow facts to 0/1-vector
• state(?) data flow facts which hold along path ?
• sums up the occurrences of data flow facts which
  hold in v
• Approach for fast computation
• adopt definition for grammar symbols of SEQUITUR

s
v
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Frequency Matrix

• Definition of frequency matrices
• sum computation due matrix calculus

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Terminals

• Transition function
• compose function for acyclic path tu1, u2,
  ..., uk
• represent transition function as matrix

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Nonterminals

• Transition function
• compose transition function for nt?X1, X2, ...,
  Xk
• represent transition function as matrix

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Example

• Terminal b 1,2,4

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Algorithm

• Pseudo-Code

• forall v?N do
• forall t?T do
• compute terminal t for node v
• endfor
• forall nt?NT in reverse topological order do
• compute nonterminal nt for node v
• endfor
• endfor

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Example

• Transition matrices and frequency matrices for
  terminals

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Example

• Transition matrices and frequency matrices for
  nonterminals

S
A
a
b
c
Frequency matrix of start symbol S contains the
dataflow frequency information!
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Experiments

• Gcc-Compiler 2.95.2
• data flow frequency analysis written in C/C
• implementation of WPP (runtime compiletime)
• Benchmark
• some programs of SpecInt95
• reaching definitions problem
• Environment
• Sun Ultra Enterprise 450 (4 x 296 MhZ) with 2.5 GB

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Node Statistics

• about 40 of nodes are executed
• no computations for 60 of nodes required

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WPP Size Overhead

• WPP Size in Kbytes

• Compile Overhead in

• Compile time overhead almost proportional to WPP
  size

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Conclusion

• Novel dataflow frequency analysis
• designed for bi-distributive dataflow analysis
  problems
• matrix representation of transition functions
• employs SEQUITUR Grammars
• Accurate and efficient algorithm
• Experiments
• platform gcc for Ultra 450
• benchmark reaching definitions problem for
  SpecInt95
• overhead is proportional to the size of WPP

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Stop!