1. Efficiently Computing SSA

2. Static Single Assignment form
Each name is defined exactly once
Each use refers to exactly one name
Why use SSA form?

3. Joins
What to do when two different values meet on the Control Flow Graph?
Φ-Functions
A Φ-function is a special kind of copy that selects one of its parameters

4. Sparse Representation
Only want to propagate facts where relevant
Don’t care about the rest
Use the SSA Graph

5. SSA Graph
Add edges from definitions to uses

6. Example i = j = k = l = 1 repeat if (p) then begin j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T

7. Example Number existing defns i = j = k = l = 1 repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T
i1 = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i2 = i + 6
until T

8. Example Add ϕ-functions where needed i1 = j = k = l = 1
repeat
i3 = ϕ()
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i2 = i + 6
until T
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T

9. Example Fill in the use numbers --- Then repeat for other variables
i1 = j = k = l = 1
repeat
i3 = ϕ(i1,i2)
if (p) then begin
j = i3
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i3,j,k,l)
if R then l = l + 4
until S
i2 = i3 + 6
until T
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T

10. Example i = j = k = l = 1 repeat if (p) then begin j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T
i1 = j1 = k1 = l1 = 1
repeat
i3 = f(i1,i2) j2 = f(j1,j4)
k2 = f(k5,k1) l2 = f(l9,l1)
if (p) then begin
j3 = i2
if Q then l3 = 2
else l4 = 3
l5 = f(l3,l4)
k3 = k2 + 1
end else k4 = k2 + 2
j4 = f(j3,j2) k5 = f(k3,k4)
l6 = f(l2,l5)
print (i3,j4,k5,l6)
l7 = f(l9,l6)
if R then l8 = l7 + 4
l9 = f(l7,l8)
until S
i2 = i3 + 6
until T

11. Example - CFG Entry 1 2 3 4 5 7 6 8 9 10 11 Exit 12 i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 3 4 5 7 6 8 9 10 11
Exit 12

12. Constructing SSA (Naively)
Insert Φ-functions at every join for every name
Solve Reaching Definitions
Rename each use to the definition that reaches it

13. Constructing SSA (Naively)
Why is the previous bad?
Too many ϕ-functions!
Can we do better?

14. Efficiently Constructing SSA
Perform Control-flow analysis
Insert ϕ-Functions
Rename Values
However – where do we put the
ϕ-Functions?

15. Formalizing ϕ placement
We need a Φ function at node Z if
Two non-null CFG paths that both define v
Such that both paths start at two distinct nodes and end at Z

16. Dominance Frontiers Illustration
Dominated by X
Dominance Frontier of X
(Not Dominated by X)

17. Dominance Frontiers
If z is the first node we encounter on the path from x which x does not strictly dominate, z is in the dominance frontier of x
For this to happen, there is some path from node x to z, x  …  y  z
where (x SDOM y) but not (x SDOM z).

18. Example {1} {7} {6} {6} {1, 7} {7} ∅ DF(1) = DF(2) = DF(3) = DF(4) =

19. Computing Dominance Frontiers
Two components to DF(X):
DFlocal(X) = {Y∊ succ(X) | X> Y}
Any child of X not (strictly) dominated by X is in DF(X)
Let Z be such that idom(Z) = X
idom(Z) is the parent of Z in the dominator tree
DFup(Z) = {Y∊ DF(Z) | X>Y}
Nodes from DF(Z) that are not strictly dominated by X are also in DF(X)

20. Algorithm Let SDOM(X) = {Y | X>Y}
For each Z such that idom(Z) = X do
DF(X) = DFlocal(X) ∪ (DF(Z) - SDOM(X))
I.e., DF(X) = DFlocal(X) ∪ DFup(Z)

21. SSA f-placement For each variable M in the program
Set Worklist = DM, the set of CFG nodes that contain assignments to variable M.
While items left in the Worklist
remove some node X from the Worklist
for all W in DF(X), if W has never been in M's Worklist, add a f-term at W. Place W into the Worklist.

22. CFG for example – Variable k
Entry 1 2
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 3 2 4 5 7 3 7 8 6 4 5 6 9 8 10 11 9 12 10 11
Exit 12

23. Worklist = {1,6,7} – location of k’s assignments
1 dominates the
entire graph
Worklist = {1,6,7} – location of k’s assignments
Compute DF(1) = {2,3,4,5,6,7,8,9,10,11,12,Exit } – {2,3,4,5,6,7,8,9,10,11,12,Exit} = { }
Remember that DF(x) = (SUCC(DOM-1(x)) – SDOM-1(x) where SUCC(x) is set of successors of x in the CFG)
Entry 1 2 3 4 5 7 6 8 9 10 11
Exit 12

24. Compute DF(6) = {8} – {} = {8} put f in node 8
Worklist = {6,7}
Compute DF(6) = {8} – {} = {8}
put f in node 8
Remember that DF(x) = (SUCC(DOM-1(x)) – SDOM-1(x) where SUCC(x) is set of successors of x in the CFG)
Entry 1 2 3 4 5 7
node 6 only
dominates itself 6 8 f 9 10 11
Exit 12

25. Compute DF(7) = {8} – {} = {8} node 8 already in worklist
Remember that DF(x) = (SUCC(DOM-1(x)) – SDOM-1(x) where SUCC(x) is set of successors of x in the CFG)
Entry 1 2 3 4 5 7
node 7 only
dominates itself 6 8 f 9 10 11
Exit 12

26. Compute DF(8) = {2,9,10,11,12, exit} – {9,10,11,12,exit} = {2}
Worklist = {8}
Compute DF(8) = {2,9,10,11,12, exit} – {9,10,11,12,exit} = {2}
put f in node 2
Remember that DF(x) = (SUCC(DOM-1(x)) – SDOM-1(x) where SUCC(x) is set of successors of x in the CFG)
Entry 1 2 f 3 4 5 7
node 8 dominates
{8,9,10,11,12,exit} 6 8 f 9 10 11
Exit 12

27. Compute DF(2) = {2-12, exit} – {3-12,exit} = {2}
Worklist = {2}
Compute DF(2) = {2-12, exit} – {3-12,exit} = {2}
Remember that DF(x) = (SUCC(DOM-1(x)) – SDOM-1(x) where SUCC(x) is set of successors of x in the CFG)
Entry 1 2 f 3 4 5 7
node 2 dominates
{2-12,exit} 6 8 f 9 10 11
Exit 12

28. CFG for example – Variable k
Entry 1
k = 1 2
k = f(k,k) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 4 5 7
k = k + 2 6
k = k + 1 8
k = f(k,k) 9 10 11 12
Exit

29. Linking it all together
Initially, for each variable V
C(V) = /* counter */
S(V) = Empty /* stack for each variable*/
Other important information:
RHS(A) = set of variables used on right side of assignment
LHS(A) = set of variables assigned to on left side
WhichPred(X,Y) = j where Y is the jth predecessor of X in CFG
Children(X) = children in the IDOM tree
‘Search(entry)’ will do renaming of all variables in program.

30. New assignment – update the counter and push
Search (X: CFG node)
for each statement A in X
If A is an ordinary assignment statement
For each v in RHS(A) replace the use of v with vi, where i = top(S(v))
For each v in LHS(A)
i = C(v); replace v by vi
push i on S(v); C(v)++
end
For each Y in Succ(X) do
J = WhichPred(X,Y) /* predecessors have unique # wrt successor */
For each f-function F in Y
Replace j-th operand v in RHS(F) by vi where i = Top(S(v))
End
For each Y in Children() call Search() end
For each assignment A in X
For each v in LHS(A) pop S(v) end
top(S(v)) will hold
the ‘current’ # for
variable v
New assignment – update the counter and push

31. Look at successors of this node in CFG
Search (X: CFG node)
for each statement A in X
If A is an ordinary assignment statement
For each v in RHS(A) replace the use of v with vi, where i = top(S(v))
For each v in LHS(A)
i = C(v); replace v by vi
push i on S(v); C(v)++
end
For each Y in Succ(X) do
j = WhichPred(X,Y) /* predecessors have unique # wrt successor */
For each f-function F in Y
Replace j-th operand v in RHS(F) by vi where i = Top(S(v))
End
For each Y in Children() call Search() end
For each assignment A in X
For each v in LHS(A) pop S(v) end
Look at successors of this node in CFG
Resolve any ϕ functions in successors

32. Look at all nodes dominated by X
Search (X: CFG node)
for each statement A in X
If A is an ordinary assignment statement
For each v in RHS(A) replace the use of v with vi, where i = top(S(v))
For each v in LHS(A)
i = C(v); replace v by vi
push i on S(v); C(v)++
end
For each Y in Succ(X) do
J = WhichPred(X,Y) /* predecessors have unique # wrt successor */
For each f-function F in Y
Replace j-th operand v in RHS(F) by vi where i = Top(S(v))
End
For each D in Children(X) call Search(D) end
For each assignment A in X
For each v in LHS(A) pop S(v) end
Look at all nodes dominated
by X

33. Pop the var # pushed earlier Search (X: CFG node)
for each statement A in X
If A is an ordinary assignment statement
For each v in RHS(A) replace the use of v with vi, where i = top(S(v))
For each v in LHS(A)
i = C(v); replace v by vi
push i on S(v); C(v)++
end
For each Y in Succ(X) do
J = WhichPred(X,Y) /* predecessors have unique # wrt successor */
For each f-function F in Y
Replace j-th operand v in RHS(F) by vi where i = Top(S(v))
End
For each D in Children(X) call Search(D) end
For each assignment A in X
For each v in LHS(A) pop S(v) end
Pop the var #
pushed earlier

34. 1 1 2 2 3 3 7 8 4 5 7 9 6 12 8 9 10 11 12

35. Just deal with k in this example Search(1) C(k) = 1 S(k) = {0}
Children(1) = {2}
k0 = 1 1 2
k = f(k0,k) 3 4 5 7 1
k = k + 2 6 2
k = k + 1 8 3 7 8
k = f(k,k) 9 9 10 12 11
Exit 12

36. Search(2)
C(k) = 2
S(k) = {0,1}
None of the successors
have f functions
Children(2) = {3,7,8}
k0 = 1 1 2
k1 = f(k0,k) 3 4 5 7 1
k = k + 2 6 2
k = k + 1 8 3 7 8
k = f(k,k) 9 9 10 12 11
Exit 12

37. Search(3)
C(k) = 2
S(k) = {0,1}
Children(3) = {4,5,6}
Search(4)
Children(4) = {}
Search(5)
Children(5) = {}
Search(6)
C(k) = 3
S(k) = {0,1,2}
Children(6) = {}
Pop(S(k)) = {0,1} 1
k0 = 1 2
k1 = f(k0,k) 3 4 5 7
k = k + 2 6 1
k2 = k1 + 1 8
k = f(k2,k) 2 9 3 7 8 10 9 11
Exit 12 12

38. Search(7)
C(k) = 4
S(k) = {0,1,3}
Children(7) = { }
Pop(S(k)) = {0,1} 1
k0 = 1 2
k1 = f(k0,k) 3 4 5 7 1
k3 = k1 + 2 6 2
k2 = k1 + 1 8
k = f(k2,k3) 3 7 8 9 9 10 11 12
Exit 12

39. Search(8)
C(k) = 5
S(k) = {0,1,4}
Children(8) = {9}
Search(9)
Children(9) = {10,11}
Search(10)
Search(11) 1
k0 = 1 2
k1 = f(k0,k) 3 4 5 7
k3 = k1 + 2 1 6
k2 = k1 + 1 2 8
k4 = f(k2,k3) 9 3 7 8 9 10 11
Exit 12 12

40. Search(12)
C(k) = 5
S(k) = {0,1,4} 1
k0 = 1 2
k1 = f(k0,k4) 3 4 5 7
k3 = k1 + 2 1 6 2
k2 = k1 + 1 8
k4 = f(k2,k3) 3 7 8 9 9 10 11 12
Exit 12

41. i = j = k1 = l = 1
Repeat
k2 = f(k1,k5)
if (p) then begin
j = i
if Q then l = 2
else l = 3
k3 = k2 + 1
end else k4 = k2 + 2
k5 = f(k3,k4)
print (i,j,k5,l)
repeat
if R then l = l + 4
until S
i = i + 6
until T
i = j = k = l = 1
Repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
repeat
if R then l = l + 4
until S
i = i + 6
until T

42. Consider L i = j = k = l = 1 repeat if (p) then begin j = i
Entry 1
l = 2 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 4 5 7
l =
l = 6 8
print l 9 10
l = l + 11
Exit 12

43. Computing the DF i = j = k = l = 1 repeat if (p) then begin j = i
Entry 1
l = 2
l = f 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l =
l = 3 7 8 6
l = f
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

44. i = j = k = l = 1
repeat
l = f(l,l)
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T

45. Adding the numbers: search(1)
Entry 1
l1 = 2
l = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l =
l = 3 7 8 6
l = f
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

46. Adding the numbers: search(2)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l =
l = 3 7 8 6
l = f
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

47. Adding the numbers: search(3)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l =
l = 3 7 8 6
l = f
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

48. Adding the numbers: search(4)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l =
l3 = 3 7 8 6
l = f(l3)
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

49. Adding the numbers: search(5)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l = f(l3,l4)
l = f 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

50. Adding the numbers: search(6)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l = f(l5 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

51. Adding the numbers: search(7)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l = f(l5,l2) 4 5 6 9 8
print l 10 11 9
l = f 10 12
l = l +
l = f 11
Exit 12

52. Adding the numbers: search(8)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l6 = f(l5,l2) 4 5 6 9 8
print l6 10 11 9
l = f(l6) 10 12
l = l +
l = f 11
Exit 12

53. Adding the numbers: search(9)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l6 = f(l5,l2) 4 5 6 9 8
print l6 10 11 9
l7 = f(l6) 10 12
l = l +
l = f(l7) 11
Exit 12

54. Adding the numbers: search(10)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l6 = f(l5,l2) 4 5 6 9 8
print l6 10 11 9
l7 = f(l6) 10 12
l8 = l7 +
l = f(l7,l8) 11
Exit 12

55. Adding the numbers: search(11)
Entry 1
l1 = 2
l2 = f(l1) 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l6 = f(l5,l2) 4 5 6 9 8
print l6 10 11 9
l7 = f(l6,l9) 10 12
l8 = l7 +
l9 = f(l7,l8) 11
Exit 12

56. Adding the numbers: search(12)
Entry 1
l1 =
l2 = f(l1,l9) 2 3
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T 1 2 4 5 7
l4 =
l3 = 3 7 8 6
l5 = f(l3,l4)
l6 = f(l5,l2) 4 5 6 9 8
print l6 10 11 9
l7 = f(l6,l9) 10 12
l8 = l7 +
l9 = f(l7,l8) 11
Exit 12

57. i = j = k = l1 = 1
repeat
l2 = f(l1,l9)
if (p) then begin
j = i
if Q then l3 = 2
else l4 = 3
l5 = f(l3,l4)
k = k + 1
end else k = k + 2
l6 = f(l5,l2)
print (i,j,k,l6)
l7 = f(l6,l9)
if R then l8 = l7 + 4
l9 = f(l7,l8)
until S
i = i + 6
until T
i = j = k = l = 1
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k = k + 1
end else k = k + 2
print (i,j,k,l)
if R then l = l + 4
until S
i = i + 6
until T

58. SSA Deconstruction At some point, we need executable code
X17 Ø(x10,x11)
...  x17
...
At some point, we need executable code
Few machines implement ϕ operations
Need to fix up the flow of values
Basic idea
Insert copies ϕ-function pred’s
Simple algorithm
Works in most cases
Adds lots of copies
Most of them coalesce away
...  x17
X17  x10
X17  x11 *

59. i = j = k0 = l = 1
k1 = k0
repeat
if (p) then begin
j = i
if Q then l = 2
else l = 3
k2 = k2 + 1
k4 = k2
end else
k3 = k1 + 2
k4 = k3
print (i,j,k4,l)
if R then l = l + 4
until S
i = i + 6
k1 = k4
until T 1
k0 = 1
k1 = k0 2
k1 = f(k0,k4) 3 4 5 7
k3 = k1 + 2
k4 = k3 6
k2 = k1 + 1
k4 = k2 8
k4 = f(k2,k3) 9 10 11
Exit 12
k1 = k4