Transcript Kein Folientitel

Dataflow Frequency Analysis
based on Whole Program Paths
Bernhard Scholz
Eduard Mehofer
Institute of Computer Languages
Vienna University of Technology
Institute for Software Science
University of Vienna
[email protected]
www.complang.tuwien.ac.at/scholz
[email protected]
www.par.univie.ac.at/~mehofer
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 2
Outline

Motivation

WPP profiling

Properties of bi-distributive DFAs

Algorithm

Experiments

Conclusion
Page 3
Classical Approach

Classical Program Optimization:
optimizer
data flow analysis
Program
binary
information
Optimized
program
transformation

Drawback:
heavily
rarely
never
Optimizer
Page 4
Profiling Approach

Probabilistic Program Optimization:
Optimizer based on profiling
Program
dataflow freq. analysis
frequency
Profile

information
Optimized
program
transformation
Advantage:
heavily
rarely
never
Optimizer
Page 5
Running Example

– simple code fragment
– 8 times left branch
– terminates via right branch
s
1

2
d1: x:=...
3
d2: x:=...
...x...
Reaching definitions problem
– two definitions: d1, d2
– d1 kills d2 and vice versa
– use of x at the end of loop
4
5
CFG Example

Questions
– How often does d1 hold at node 5?
– How often does d2 hold at node 5?
Page 6
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
Page 7
WPP Example

CFG Example
s

WPP Graph & Grammar
S  a AAA b c
Abb
S
1
2
A
3
4
a
5


Program Run
- 8x left branch
- 1x right branch
b
c
Terminals:
a: [s,1,2,4]
b: [1,2,4]
c: [1,3,4,5]
Page 8
Bi-Distributive Dataflow Problems

Properties
– finite lattice 2D (power set of dataflow facts)
– transition functions are monotone
– transition functions distribute
f ( X  Y )  f ( X )  f (Y )
f ( X  Y )  f ( X )  f (Y )
– representation relation
– covers bit-vector problems

Due to properties
– transition functions represented as 0/1-matrices
– states represented as 0/1-vectors
Page 9
Representation Relation

Transition function f: 2D 2D
– represented by f r : D  2D
f r ( ) 
f r (d )

f (0)  {}
f ({d })  f (0)
– artificial data fact 

Example
1
2
d1: x:=...
D
M(24)r
d1 {}
d2 {}
4

{d1, }
Page 10
Matrix Representation

Matrix representation of function f
1, if d i  f r (d j )
aij  
0, otherwise

Example
M (2  4)d1 , d2   d1
 0 0 1 1  1 

   
A
M (2  4)   d1 , d 2    0 0 0 1   0 
 0 0 1 1  1 

   
Page 11
Dataflow Frequencies

Definition of dataflow frequencies for node v

y r (v) 
–
–
–
–
–

 (state( ))
 Prefix(
s
r ,v )
r whole program path
v
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
Page 12
Frequency Matrix

Definition of frequency matrices
F (v) 
M ( 
[u1, u2 ,, uk ])
A
[u1 ,u2 ,,uk 1 ,v ]Prefix( ,v )
– sum computation due matrix calculus

Frequency matrices for eliminating sum

y r (v) 
 (state( ))
 Prefix(
r ,v )
 F r (v)   (c)

Computation of frequency matrices for grammar symbols
Page 13
Terminals

Transition function
– compose function for acyclic path t:[u1, u2, ..., uk]
– represent transition function as matrix
M (t ) A

 M ([u1 , u2 ,, uk ])A
A
 M (uk 1  uk )   M (u1  u2 )
Frequency matrix
M ([u1 , u2 ,, v]) A if v  u1 , u2 ,, uk 
Ft (v)  
otherwise
0
Page 14
Nonterminals

Transition function
– compose transition function for ntX1, X2, ..., Xk
– represent transition function as matrix
M (nt) A

 M ( X 1 X 2  X k ]) A
 M ( X k ) A   M ( X 2 ) A  M ( X 1 ) A
Frequency matrix
Fnt (v)  FX1 (v)  FX 2 (v) M ( X 1 ) A  
 FX k (v) M ( X 1 X 2  X k ) A
Page 15
Example


 0 0 1


A
Fb (4)  M ([1,2,4])   0 0 0 
 0 0 1


Terminal b: [1,2,4]
1
2
d1: x:=...
4
Nonterminal Abb
M ( A) A  M (bb) A  M (b) A  M (b) A  M (b) A
 0 0 2


A
FA (4)  Fb (4)  Fb (4) M (b)   0 0 0 
 0 0 2


Page 16
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

y(v)  FS (v) (c)
endfor
Page 17
Example

Transition matrices and frequency matrices for terminals
S
S
a
A
a
b
a
c
b
A
b
c
S
S
c
A
A
a
b
c
a
b
c
Page 18
Example

Transition matrices and frequency matrices for nonterminals
S
S
A
A
a
b
c
S
A
a
b
c
S
Frequency matrix of start symbol S
contains the dataflow frequency information!
A
a
b
c
Page 19
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 20
Node Statistics
12000
Nodes
10000
not executed
executed w/o DFA
analyzed
8000
6000
4000
2000
0
0
.g
99
1


o
8
.m
4
2
s
8k
1
im
.c
29
res
p
om
s
13
0.l
i
13
jp
2.i
eg
13
e
4.p
rl
about 40% of nodes are executed
no computations for 60% of nodes required
Page 21
WPP Size & Overhead

WPP Size in Kbytes
Compile Overhead in %

35
20000
30
15000
25
20
10000
15
10
5000
5
0
9
09
0
.go
12
k
88
4.m
sim
1
.c
29
pre
om
ss
13
i
0.l
13
jp
2.i
eg
1
.p
34
erl
9
09
.go
12
k
88
4.m
sim
1
.c
29
pre
om
ss
13
0.l
i
13
jp
2.i
eg
1
.p
34
erl
- Compile time overhead almost proportional to WPP size
Page 22
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 23
Stop!
Page 24