1. Optimization through Redundancy Elimination: Value Numbering at Different Scopes COMP 512 Rice University Houston, Texas Fall 2003
Copyright 2003, Keith D. Cooper & Linda Torczon, all rights reserved.
Students enrolled in Comp 512 at Rice University have explicit permission to make copies of these materials for their personal use.

2. (need stronger methods)
Value Numbering
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
Missed opportunities
(need stronger methods)
Local Value Numbering
1 block at a time
Strong local results
No cross-block effects
COMP 512, Fall 2003 *

3. An Aside on Terminology
Control-flow graph (CFG)
Nodes for basic blocks
Edges for branches
Basis for much of program analysis & transformation
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
This CFG, G = (N,E)
N = {A,B,C,D,E,F,G}
E = {(A,B),(A,C),(B,G),(C,D), (C,E),(D,F),(E,F),(F,E)}
|N| = 7, |E| = 8
COMP 512, Fall 2003

4. Extended Basic Blocks A B G C D E F An Extended Basic Block (EBB)
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
An Extended Basic Block (EBB)
Set of blocks b1, b2, …, bn
b1 has > 1 predecessor
All other bi have 1 predecessor
EBBs provide more context for
optimization
COMP 512, Fall 2003 *

5. Extended Basic Blocks A B G C D E F An EBB contains 1 or more path
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
An EBB contains 1 or more path
If b1, b2, …, bn is a path
b1 has > 1 predecessor
bi has 1 predecessor, bi-1
{A, B, C, D, E} is an EBB
{A,B}, {A,C,D}, and {A,C,E} are
paths in {A, B, C, D, E}
{F} and {G} are also EBBs
They have only trivial paths
COMP 512, Fall 2003 *

6. Superlocal Value Numbering
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
The Concept
Apply local method to each
path in the EBB
Do {A,B}, {A,C,D}, & {A,C,E}
Obtain reuse from ancestors
Does not help with F or G
Key: avoid re-analyzing A & C
COMP 512, Fall 2003 *

7. Superlocal Value Numbering
Efficiency
Use A’s table to initialize tables for B & C
To avoid duplication, use a scoped hash table
A, AB, A, AC, ACD, AC, ACE, F, G
Need a VN  name mapping to handle kills
Must restore map with scope
Adds complication, not cost
To simplify matters
Unique name for each definition
Makes name  VN
Use the SSA name space
EaC: § & App. B
m  a + b
n  a + b A
p  c + d
r  c + d B
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
v  a + b
w  c + d
x  e + f F
Subscripted names from example in last lecture
COMP 512, Fall 2003

8. SSA Name Space (locally)
Example (from lecture 3)
These are SSA names …
Original Code
a0  x0 + y0
 b0  x0 + y0
a1  17
 c0  x0 + y0
Renaming:
Give each value a
unique name
Makes it clear
With VNs
a03  x01 + y02
 b03  x01 + y02
a14  17
 c03  x01 + y02
Notation:
While complex,
the meaning is
clear
Rewritten
a03  x01 + y02
 b03  a03
a14  17
 c03  a03
Result:
a03 is available
rewriting works
COMP 512, Fall 2003

9. SSA Name Space (in general)
Two principles
Each name is defined by exactly one operation
Each operand refers to exactly one definition
To reconcile these principles with real code
Insert -functions at merge points to reconcile name space
Add subscripts to variable names for uniqueness
We’ll look at how to construct SSA form later in the course
x  ...
...  x + ...
x0  ...
x1  ...
x2 (x0,x1)
 x
becomes
COMP 512, Fall 2003

10. Superlocal Value Numbering
m0  a + b
n0  a + b A
p0  c + d
r0  c + d B
r2  (r0,r1)
y0  a + b
z0  c + d G
q0  a + b
r1  c + d C
e0  b + 18
s0  a + b
u0  e + f D
e1  a + 17
t0  c + d
u1  e + f E
e3  (e0,e1)
u2  (u0,u1)
v0  a + b
w0  c + d
x0  e + f F
With all the bells & whistles
Find more redundancy
Pay little additional cost
Still does nothing for F & G
This is in SSA Form
Superlocal techniques
Some local methods extend cleanly to superlocal scopes
VN does not back up
If C adds to A, it’s a problem
COMP 512, Fall 2003

11. What About Larger Scopes?
We have not helped with F or G
Multiple predecessors
Must decide what facts hold in F and in G
For G, combine B & F?
Merging state is expensive
Fall back on what’s known G
m0  a + b
n0  a + b A
p0  c + d
r0  c + d B
r2  (r0,r1)
y0  a + b
z0  c + d
q0  a + b
r1  c + d C
e0  b + 18
s0  a + b
u0  e + f D
e1  a + 17
t0  c + d
u1  e + f E
e3  (e0,e1)
u2  (u0,u1)
v0  a + b
w0  c + d
x0  e + f F
COMP 512, Fall 2003

12. Dominators Definitions
x dominates y if and only if every path from the entry of the control-flow graph to the node for y includes x
By definition, x dominates x
We associate a Dom set with each node
|Dom(x )| ≥ 1
Immediate dominators
For any node x, there must be a y in Dom(x ) closest to x
We call this y the immediate dominator of x
As a matter of notation, we write this as IDom(x )
COMP 512, Fall 2003

13. Dominators Dominators have many uses in analysis & transformation
Finding loops
Building SSA form
Making code motion decisions
We’ll look at how to compute dominators later
m0  a + b
n0  a + b A
p0  c + d
r0  c + d B
r2  (r0,r1)
y0  a + b
z0  c + d G
q0  a + b
r1  c + d C
e0  b + 18
s0  a + b
u0  e + f D
e1  a + 17
t0  c + d
u1  e + f E
e3  (e0,e1)
u2  (u0,u1)
v0  a + b
w0  c + d
x0  e + f F
Dominator sets A B C G F E D
Dominator tree
Back to the discussion of value numbering over larger scopes ...
COMP 512, Fall 2003 *

14. What About Larger Scopes?
We have not helped with F or G
Multiple predecessors
Must decide what facts hold in F and in G
For G, combine B & F?
Merging state is expensive
Fall back on what’s known
Can use table from IDom(x ) to start x
Use C for F and A for G
Imposes a Dom-based application order
Leads to Dominator VN Technique (DVNT)
m0  a + b
n0  a + b A
p0  c + d
r0  c + d B
r2  (r0,r1)
y0  a + b
z0  c + d G
q0  a + b
r1  c + d C
e0  b + 18
s0  a + b
u0  e + f D
e1  a + 17
t0  c + d
u1  e + f E
e3  (e0,e1)
u2  (u0,u1)
v0  a + b
w0  c + d
x0  e + f F
COMP 512, Fall 2003 *

15. Dominator Value Numbering
The DVNT Algorithm
Use superlocal algorithm on extended basic blocks
Retain use of scoped hash tables & SSA name space
Start each node with table from its IDom
DVNT generalizes the superlocal algorithm
No values flow along back edges (i.e., around loops)
Constant folding, algebraic identities as before
Larger scope leads to (potentially) better results
Local + Superlocal + good start for new EBBs
COMP 512, Fall 2003

16. Dominator Value Numbering
DVNT advantages
Find more redundancy
Little additional cost
Retains online character
m  a + b
n  a + b A
p  c + d
r  c + d B
r2  (r0,r1)
y  a + b
z  c + d G
q  a + b C
e  b + 18
s  a + b
u  e + f D
e  a + 17
t  c + d E
e3  (e1,e2)
u2  (u0,u1)
v  a + b
w  c + d
x  e + f F
DVNT shortcomings
Misses some opportunities
No loop-carried CSEs or constants
COMP 512, Fall 2003

17. End of Lecture …
COMP 512, Fall 2003

18. The Story So Far, … Local algorithm (Balke, 1967)
Superlocal extension of Balke (many)
Dominator VN technique (Simpson, 1996)
All these propagate along forward edges
None are global methods
Global Methods
Classic CSE (Cocke 1970)
Partitioning algorithms (Alpern et al. 1988, Click 1995)
Partial Redundancy Elimination (Morel & Renvoise 1979)
SCC/VDCM (Simpson 1996)
We will look at several global methods
Future lectures
Next lecture
COMP 512, Fall 2003 *

19. Roadmap To recap … We have seen value numbering
Local, superlocal, dominator scopes
We may look at global value numbering later
We will look at global redundancy elimination
We have used dominators
We will look at how to calculate Dom later
We have used SSA
We will look at how to build SSA later
Next:
Global redundancy elimination from available expressions
Iterative data-flow analysis
COMP 512, Fall 2003 

20. Using Available Expressions for GCSE
The goal
Find common subexpressions whose range spans basic blocks, and eliminate unnecessary re-evaluations
Safety
Available expressions proves that the replacement value is current
Transformation must ensure right namevalue mapping
Profitability
Don’t add any evaluations
Add some copy operations
Copies are inexpensive
Many copies coalesce away
Copies can shrink or stretch live ranges
COMP 512, Fall 2003 *

21. Using Available Expressions for GCSE
Expressions defined in b
Using Available Expressions for GCSE
The Method
1.  block b, compute DEF(b) and NKILL(b)
2.  block b, compute AVAIL(b)
3.  block b, value number the block starting from AVAIL(b)
4. Replace expressions in AVAIL(b) with references
Two key issues
Computing AVAIL(b)
Managing the replacement process
We’ll look at the replacement issue first
Expressions not killed in b
COMP 512, Fall 2003

22. Computing Available Expressions
For each block b
Let AVAIL(b) be the set of expressions available on entry to b
Let NKILL(b) be the set of expression not killed in b
Let DEF(b) be the set of expressions defined in b and not subsequently killed in b
Now, AVAIL(b) can be defined as:
AVAIL(b) = xpred(b) (DEF(x)  (AVAIL(x)  NKILL(x) ))
preds(b) is the set of b’s predecessors in the control-flow graph
This system of simultaneous equations forms a data-flow problem
Solve it with a data-flow algorithm
COMP 512, Fall 2003

23. Global CSE (replacement step)
Managing the name space
Need a unique name  e  AVAIL(b)
1. Can generate them as replacements are done (Fortran H)
2. Can compute a static mapping
3. Can encode value numbers into names (Briggs 94)
Strategy
1. This works; it is the classic method
2. Fast, but limits replacement to textually identical expressions
3. Requires more analysis (VN), but yields more CSEs
Assume, w.l.o.g., solution 2
COMP 512, Fall 2003

24. Global CSE (replacement step)
Compute a static mapping from expression to name
After analysis & before transformation
 b, e  AVAIL(b), assign e a global name by hashing on e
During transformation step
Evaluation of e  insert copy name(e)  e
Reference to e  replace e with name(e)
The major problem with this approach
Inserts extraneous copies
At all definitions and uses of any e  AVAIL(b),  b
Those extra copies are dead and easy to remove
The useful ones often coalesce away
Common strategy:
Insert copies that might be useful
Let DCE sort them out
Simplifies design & implementation
COMP 512, Fall 2003 *

25. An Aside on Dead Code Elimination
What does “dead” mean?
Useless code — result is never used
Unreachable code — code that cannot execute
Both are lumped together as “dead”
To perform DCE
Must have a global mechanism to recognize usefulness
Must have a global mechanism to eliminate unneeded stores
Must have a global mechanism to simplify control-flow predicates
All of these will come later in the course
COMP 512, Fall 2003

26. Global CSE Now a three step process Compute AVAIL(b),  block b
Assign unique global names to expressions in AVAIL(b)
Perform replacement with local value numbering
Earlier in the lecture, we said
Now, we need to make good on the assumption
Assume, without loss of generality, that we can compute available expressions for a procedure.
This annotates each basic block, b, with a set AVAIL(b) that contains all expressions that are available on entry to b.
COMP 512, Fall 2003

27. Computing Available Expressions
The Big Picture
1. Build a control-flow graph
2. Gather the initial (local) data — DEF(b) & NKILL(b)
3. Propagate information around the graph, evaluating the equation
4. Post-process the information to make it useful (if needed)
All data-flow problems are solved, essentially, this way
Next lecture:
Iterative computation of AVAIL information
From Chapter 8 of EaC
COMP 512, Fall 2003