Factored Use-Def Chains and Static Single Assignment Forms

1

Factored Use-Def Chains and Static Single Assignment Forms

Addresses use-def difficulty with optimization: Multiple defs of the same

register Which def reaches use? Is RHS expression available?

FUD/SSA forms: Each def of a variable is given

a unique name All uses reached by that

assignment are renamed DU chains become obvious

FUD chains and SSA forms are similar

r2 = r2 + r3r6 = r4 - r5

r6 = r2 + r6r7 = r4

r4 = 4r6 = 8

2

Reaching Defs

Keeping all reaching defs for each use is not space efficient

FUD properties: Each use of a var is reached

by a single def Insert pseudo assignments

with nodes where multiple defs meet

r23 = r21?+r32?

r67 = r45?-r56?

r612 = r2103+r6117,9

r714 = r413?,8

r48 = 4r69 = 8

r6117,9

use defs

3

Converting to FUD Form

r23 = r21?+r32?

r67 = r45?-r56?

r612 = r2103+r6117,9

r714 = r413?,8

r48 = 4r69 = 8

r23 = r21?+r32?

r67 = r45?-r56?

(r4)10?,8

(r6)117,9

r614 = r2123+r61311

r716 = r41510

r48 = 4r69 = 8

4

Converting to SSA Form

r2 = r2 + r3r6 = r4 - r5

r6 = r2 + r6r7 = r4

r4 = 4r6 = 8

r22 = r21 + r31

r61 = r41 - r51

r43 = (r41,r42)r63 = (r61,r62)r64 = r22 + r63

r71 = r43

r42 = 4r62 = 8

5

Converting Loops to SSA Form

r1 = r1 + 1

…

r1 = 0

r12 = (r11,r13)r13 = r12 + 1…

r11 = 0

6

FUD/SSA Pros and Cons

Pro Trivial to find what defs reach a use: each use has

exactly one def Explicit merging of values at nodes Simplifies optimizations that need use-def info

Cons When transforming code, must either recompute (slow)

or incrementally update (tedious)

7

Phi Nodes

A node merges two reaching definitions for a variable (or register) V

A node for V is required when two non-empty paths XZ and YZ converge at node Z, and nodes X and Y contain assignments to V

More precisely: Z is a node with two distinct predecessors Y1 and Y2

p1 : X1* Y1 and p2 : X2* Y2 are two paths in the CFG p1 and p2 have no nodes in common And there are no defs of V along either p1 or p2 (except at X1 and

X2)

8

FUD/SSA Construction

Naïve algorithm Insert nodes at every join for every variable Solve reaching definitions Rename each use to the def that reaches it

Problem: too many nodes Precision Space Time

9

FUD Construction

Step 1: -term placement using an algorithm that similar to SSA insertion, but for FUD we also add a slicing edge

Step 2: FUD chaining connects each variable use with its unique reaching definition, similar to SSA variable renaming

10

SSA Construction

Step 1: Insert nodes

Step 2: Rename variables

11

Phi Node Insertion Algorithm

Compute dominance frontiers Find global names:

Global if name is live on entry to some block For each name build a list of blocks that define it

Insert nodes (-term placement): For each global name N

For each basic block B in which N is defined For each basic block D in B’s dominance frontier insert a node for N in D add N to the list of defining basic blocks

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Dominance Frontiers

The dominance frontier of a node X is the set of nodesZ = DF(X) such that:

If there are two predecessors Y1 and Y2 of Z and X dominates Y1 but not Y2

A node dominates itself A node X dominates a node Y if every path from entry to

Y goes through X

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Computing Dominance Frontiers

Algorithm1. Compute dominator tree2. For each join point X in the CFG

For each predecessor of X in the CFG go up to the immediate dominator D of X in the dominator tree, adding X to DF(Y) for each Y up to D (excluding X and D)

14

Computing Dominance Frontiers (Example Dom. Tree)

BB0

BB1

BB3BB2

BB4 BB5

BB6

BB7

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

Dominator tree

BB Dominatedby

0 0

1 0,1

2 0,1,2

3 0,1,3

4 0,1,3,4

5 0,1,3,5

6 0,1,3,6

7 0,1,7

15

Computing Dominance Frontiers (Example 1)

BB0

BB1

BB3BB2

BB4 BB5

BB6

BB7

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

Dominatorfrontiers

For each join point X in the CFG For each predecessor of X in the CFG go up to the immediate dominator D of X in the dominator tree, adding X to DF(Y) for each Y up to D (excluding X and D)

16


BB0

D = BB1

BB3Y = BB2

BB4 BB5

BB6

X = BB7

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

Dominatorfrontiers


17


BB0

D = BB1

Y = BB3

BB4 BB5

Y = BB6

X = BB7

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

Dominatorfrontiers


BB2

18


BB0

D = BB3

Y = BB4 BB5

X = BB6

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

Dominatorfrontiers


BB1

BB2

BB7

19


D = BB0

BB5

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

Dominatorfrontiers


X = BB1

BB2

Y = BB7

BB3

BB4

BB6

20

Phi Node Insertion Algorithm

Compute dominance frontiers Find global names:

Global if name is live on entry to some block For each name build a list of blocks that define it

Insert nodes: For each global name N

For each basic block B in which N is defined For each basic block D in B’s dominance frontier insert a node for N in D add N to the list of defining basic blocks

21

Phi Node Insertion

a = b =c =i =

For each global name N For each basic block B in which N is defined For each basic block D in B’s dominance frontier insert a node for N in D add N to the list of defining basic blocks

BB DF

0 -

1 -

2 7

3 7

4 6

5 6

6 7

7 1

BB0

a = c = BB1

a = d =

b =c = d =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7

a is defined in 0, 1, 3:add in 7

then a is defined in 7:add in 1

b is defined in 0, 2, 6:add in 7

then b is defined in 7:add in 1

c is defined in 0, 1, 2, 5:add in 6, 7

then c is defined in 6, 7:add in 1

d is defined in 2, 3, 4:add in 6, 7

then d is defined in 6, 7:add in 1

i is defined in 7:add in 1

a = (a,a)b = (b,b)c = (c,c)d = (d,d)i = (i,i)

c = (c,c)d = (d,d)

a = (a,a)b = (b,b)c = (c,c)d = (d,d)

22

SSA Construction

Step 1: Insert nodes

Step 2: Rename variables

23

Rename Variables

Algorithm (outline) Use an array of stacks, one per global variable V For each basic block B in a preorder traversal of the dominator

tree: Generate unique names for each node Rewrite each operation in the basic block:

for uses of V: get current name from stackfor defs of V: create and push new name

Fill in node parameters of successor blocks Recurse on B’s children in the dominator tree On exit from B: pop names generated in B from stacks

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Renaming Variables:Initial State

a = b =c =i =

BB0

a = c = BB1

a = d =

b =c = d =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7

a = (a,a)b = (b,b)c = (c,c)d = (d,d)i = (i,i)

c = (c,c)d = (d,d)

a = (a,a)b = (b,b)c = (c,c)d = (d,d)

Var: a b c d i

Ctr: 1 1 1 1 1

Stk: a0 b0 c0 d0 i0

BB0

BB1

BB2 BB7

BB4 BB6BB5

BB3

Preorder traversalof dominator tree

25

Renaming Variables:After BB0

a0 = b0 =c0 =i0 =

BB0

a = c = BB1

a = d =

b =c = d =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7

a = (a0,a)b = (b0,b)c = (c0,c)d = (d0,d)i = (i0,i)

c = (c,c)d = (d,d)

a = (a,a)b = (b,b)c = (c,c)d = (d,d)

Var: a b c d i

Ctr: 1 1 1 1 1

Stk: a0 b0 c0 d0 i0

26


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a = d =

b =c = d =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7

a1 = (a0,a)b1 = (b0,b)c1 = (c0,c)d1 = (d0,d)i1 = (i0,i)

c = (c,c)d = (d,d)

a = (a,a)b = (b,b)c = (c,c)d = (d,d)

Var: a b c d i

Ctr: 3 2 3 2 2

Stk: a0

a1

a2

b0

b1

c0

c1

c2

d0

d1

i0

i1

27


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a = d =

b2 =c3 = d2 =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7


c = (c,c)d = (d,d)

a = (a2,a)b = (b2,b)c = (c3,c)d = (d2,d)

Var: a b c d i

Ctr: 3 3 4 3 2

Stk: a0

a1

a2

b0

b1

b2

c0

c1

c2

c3

d0

d1

d2

i0

i1

28

Renaming Variables:Backtrack (Before BB3)

a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a = d =

b2 =c3 = d2 =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7


c = (c,c)d = (d,d)

a = (a2,a)b = (b2,b)c = (c3,c)d = (d2,d)

Var: a b c d i

Ctr: 3 3 4 3 2

Stk: a0

a1

a2

b0

b1

c0

c1

c2

d0

d1

i0

i1

29


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a3 = d3 =

b2 =c3 = d2 =

d = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7


c = (c,c)d = (d,d)

a = (a2,a)b = (b2,b)c = (c3,c)d = (d2,d)

Var: a b c d i

Ctr: 4 3 4 4 2

Stk: a0

a1

a2

a3

b0

b1

c0

c1

c2

d0

d1

d3

i0

i1

30


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a3 = d3 =

b2 =c3 = d2 =

d4 = c =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7


c = (c2,c)d = (d4,d)

a = (a2,a)b = (b2,b)c = (c3,c)d = (d2,d)

Var: a b c d i

Ctr: 4 3 4 5 2

Stk: a0

a1

a2

a3

b0

b1

c0

c1

c2

d0

d1

d3

d4

i0

i1

31

Renaming Variables:Backtrack to BB3 & After BB5

a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a3 = d3 =

b2 =c3 = d2 =

d4 = c4 =

b =

i =

BB2 BB3

BB5BB4

BB6

BB7


c = (c2,c4)d = (d4,d3)

a = (a2,a)b = (b2,b)c = (c3,c)d = (d2,d)

Var: a b c d i

Ctr: 4 3 5 5 2

Stk: a0

a1

a2

a3

b0

b1

c0

c1

c2

c4

d0

d1

d3

i0

i1

32


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a3 = d3 =

b2 =c3 = d2 =

d4 = c4 =

b3 =

i =

BB2 BB3

BB5BB4

BB6

BB7


c5 = (c2,c4)d5 = (d4,d3)

a = (a2,a3)b = (b2,b3)c = (c3,c5)d = (d2,d5)

Var: a b c d i

Ctr: 4 4 6 6 2

Stk: a0

a1

a2

b0

b1

b3

c0

c1

c2

c5

d0

d1

d5

i0

i1

33


a0 = b0 =c0 =i0 =

BB0

a2 = c2 =

BB1

a3 = d3 =

b2 =c3 = d2 =

d4 = c4 =

b3 =

i2 =

BB2 BB3

BB5BB4

BB6

BB7

a1 = (a0,a2)b1 = (b0,b4)c1 = (c0,c6)d1 = (d0,d6)i1 = (i0,i2)

c5 = (c2,c4)d5 = (d4,d3)

a4 = (a2,a3)b4 = (b2,b3)c6 = (c3,c5)d6 = (d2,d5)

Var: a b c d i

Ctr: 5 5 7 7 3

Stk: a0

a1

a2

a4

b0

b1

b4

c0

c1

c2

c6

d0

d1

d6

i0

i1

i2

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Homework:Convert to SSA Form

a = b =

BB1

b = a =

c =

b =

a = c =

BB0

BB2 BB3

BB4

BB5

Tasks:1. Compute dominator tree2. Compute dominance

frontiers3. Find global names4. Insert nodes5. Rename variables

Date post:	19-Jan-2016
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Factored Use-Def Chains and Static Single Assignment Forms

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