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Dataflow Frequency Analysis based on Whole Program Paths
Eduard Mehofer
Institute for Software Science
University of Vienna
www.par.univie.ac.at/~mehofer
Bernhard Scholz
Institute of Computer Languages
Vienna University of Technology
www.complang.tuwien.ac.at/scholz
Page 2
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
Page 3
Outline Motivation
WPP profiling
Properties of bi-distributive DFAs
Algorithm
Experiments
Conclusion
Page 4
ProgramProgram
Classical Approach
Drawback:
Classical Program Optimization:
transformation
data flow analysis
optimizer
binary informationOptimizedprogram
Optimizedprogram
OptimizerOptimizer
heavily rarely never
Page 5
ProgramProgram
Profiling Approach
Advantage:
Probabilistic Program Optimization:
transformation
dataflow freq. analysis
Optimizer based on profiling
frequency informationOptimizedprogram
Optimizedprogram
OptimizerOptimizer
heavily rarely never
ProfileProfile
Page 6
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
32
4
5
d1: x:=... d2: x:=...
...x...
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WPP Profiling
Captures the whole program path – Larus at PLDI’99
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
s
1
32
4
5
CFG Example
Program Run- 8x left branch- 1x right branch
A
S
a b c
S a A A A b c
A b b
WPP Graph & Grammar
Terminals:a: [s,1,2,4]
b: [1,2,4]
c: [1,3,4,5]
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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
)()()(
)()()(
YfXfYXf
YfXfYXf
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Representation Relation Transition function f: 2D 2D
– represented by f r : D 2D
– artificial data fact )0(})({)(
}{)0()(
fdfdf
ffr
r
2
4d1: x:=...
1
{d1, }
{}d2
{}d1
M(24)rD
Example
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Matrix Representation
Matrix representation of function f
otherwise,0
)( if,1 jr
iij
dfda
1
0
1
1
1
1
100
000
100
,)42( 21 ddM A
121,)42( dddM
Example
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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
))state(()(),Prefix(
vr
rvy
s
v
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Frequency Matrix Definition of frequency matrices
– sum computation due matrix calculus
Ak
vvuuu
uuuMvFk
]),,,[()( 21),Prefix(],,,,[ 121
)()(
))state(()(),Prefix(
cvF
vy
r
r
rv
Frequency matrices for eliminating sum
Computation of frequency matrices for grammar symbols
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Terminals Transition function
– compose function for acyclic path t:[u1, u2, ..., uk]
– represent transition function as matrix
Akk
Ak
A
uuMuuM
uuuMtM
)()(
]),,,([)(
211
21
otherwise0
,,, if]),,,([)( 2121 k
A
t
uuuvvuuMvF
Frequency matrix
Page 15
Nonterminals Transition function
– compose transition function for ntX1, X2, ..., Xk
– represent transition function as matrix
AAAk
Ak
A
XMXMXM
XXXMntM
)()()(
])()(
12
21
AkX
AXXnt
XXXMvF
XMvFvFvF
k)()(
)()()()(
21
121
Frequency matrix
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Example Terminal b: [1,2,4]
100
000
100
])4,2,1([)4( Ab MF 2
4d1: x:=...
1
AAAAA bMbMbMbbMAM )()()()()(
200
000
200
)()4()4()4( AbbA bMFFF
Nonterminal Abb
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Algorithm
forall vN do
forall tT do
compute terminal t for node v
endfor
forall ntNT in reverse topological order do
compute nonterminal nt for node v
endfor
endfor
)()()( cvFvy S
Pseudo-Code
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Example
A
S
a b c
Transition matrices and frequency matrices for terminals
A
S
a b c
a
A
S
a b c
b
A
S
a b c
c
Page 19
Example
A
S
a b c
Transition matrices and frequency matrices for nonterminals
A
S
a b c
A
A
S
a b c
S
Frequency matrix of start symbol S
contains the dataflow frequency information!
Page 20
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
Page 21
Node Statistics
0
2000
4000
6000
8000
10000
12000
No
des
not executed
executed w/o DFA
analyzed
about 40% of nodes are executed no computations for 60% of nodes required
Page 22
WPP Size & Overhead
0
5000
10000
15000
20000
099.go
124.m88ksim
129.compress130.li
132.ijpeg
134.perl0
5
10
15
20
25
30
35
099.go
124.m88ksim
129.compress130.li
132.ijpeg
134.perl
WPP Size in Kbytes Compile Overhead in %
- Compile time overhead almost proportional to WPP size
Page 23
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
Page 24
Stop!
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