CS293S Redundancy Removal: SVN & DVN & GCSE Yufei Ding - - PowerPoint PPT Presentation

CS293S Redundancy Removal: SVN & DVN & GCSE

Yufei Ding

2

Review of Last Class

Removing redundant expressions DAG: version tracking Linear representation: value numbering

3

Missed opportunities

(need stronger methods) m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

Local Value Numbering <-> Linear IR

Local Value Numbering

• 1 block at a time
• Strong local results
• No cross-block effects

*

4

Missed opportunities

(need stronger methods)

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

Can we find set of blocks that also ensures the sequential execution

• rder in the basic block?

Local Value Numbering <-> Linear IR

Local Value Numbering

• 1 block at a time
• Strong local results
• No cross-block effects

5

Topics of This Class

Scope of optimization Basic block -> Local value numbering Extended basic block -> Superlocal value numbering (SVN) Dominator -> Dominator-based value numbering (DVN) Global Common Subexpression Elimination (GCSE) More close to DAG-based methods Work on lexical notation instead of expression values.

6

Basic blocks

A basic block is a maximal-length segment of straight-line,

unpredicated code. In another word, it has one entry point (i.e., no code within it is the destination of a jump instruction), one exit point and no jump instructions contained within it.

Example

L2: L1:

m = 2; c = m + n; if(c>0) goto L1; d = 4; goto L2; c = 5;

7

CFG

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

Control-flow graph (CFG)

• Nodes for basic blocks
• Edges for branches
• Basis for many program

analysis & transformation 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

Extended basic block (EBB)

An EBB is a set of blocks B1,

B2, ..., Bn, where Bi, 2<= i <= n has a unique predecessor, which is in the EBB.

May have multiple exits A tree structure If a block is added to the EBB,

all of its predecessors must be

• included. Bi is the one with on

predecessor, i.e., the root of the 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 r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

Can you find the maximum EBB

9

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 r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

• 1. First find the maximum EBB:

ABCDE, F, G

• 2. Apply local method to EBBs’ paths
• Do {A,B}, {A,C,D}, {A,C,E}, {F}, {G}

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Implementation

Reuse the value numbering results of some common blocks for

efficiency

Which necessitates the undoing of a block’s effect After {A,C,D}, it must recreate the state of {A,C} before

processing E.

Options:

1. Record the state of the tables at each block boundary, and restore the state when needed 2. Walking backward and undo the effect. Need record the “lost” information. 3. Scoped hash tables (Lowest cost) keep the table produced at the current block

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Scoped Value Table

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

r ¬ c + d q ¬ a + b

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

t ¬ c + d u ¬ a + b

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

a->1 b->2 1+2->3 m->3 n->3 c->4 d->5 4+5->6 r->6 q->3 t->6 u->3 c->4 d->5 4+5->6 p->6 r->6

12

Rewritten

a ¬ b + c e ¬ b - c

b -> 1 c -> 2 1 + 2 ->3 a -> 3 1->b 2->c 3->a

d ¬ b - c f ¬ b - c

1-2 -> 4 e -> 4 4 -> e 4 -> d 1-2 -> 4 d-> 4 f-> 4

d ¬ b - c f ¬ d

13

Rewritten

a ¬ b + c a ¬ 17 e ¬ b + c d ¬ b + c

Renaming is still needed. But does it work in all scenarios?

a1 ¬ b1 + c1 a2 ¬ 17 e1 ¬ b1 + c1 d1 ¬ b1 + c1

Extra Complexity

14

a1 ¬ b + c a3 ¬ 17 a2 ¬ a1 + c d ¬ a + c ?

15

SSA (Single Static Assignment) Name Space

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 f-functions at merge points to reconcile name space

x ¬ ... x ¬ ... ... ¬ x + ... x0 ¬ ... x1 ¬ ... x2 ¬f(x0,x1) ¬ x2 + ... becomes

Another SSA Example

16

x ¬ ... x ¬ ... ... ¬ x + ... x3 ¬ ... x4 ¬ ... x5 ¬f(x3,x4) ¬ x5 + ... becomes x ¬ x + ... x1 ¬f(x0,x5) x2 ¬ x1 + ...

Detail: CT-2ndEd: Section 5.4.2; CT-1stEd: Section 5.5.

17 This is in SSA Form

Superlocal Value Numbering

m0 ¬ a + b n0 ¬ a + b

A

p0 ¬ c + d r0 ¬ c + d

B

r2 ¬ f(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 ¬ f(e0,e1) u2 ¬ f(u0,u1) v0 ¬ a + b w0 ¬ c + d x0 ¬ e + f

F

1.Build SSA form 2.Find EBBs 3.Apply value numbering to each path in each EBB using scoped hash tables

18 This is in SSA Form

Superlocal Value Numbering

m0 ¬ a + b n0 ¬ a + b

A

p0 ¬ c + d r0 ¬ c + d

B

r2 ¬ f(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 ¬ f(e0,e1) u2 ¬ f(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

Dominator-Based Value Numbering

19

20

Regional (Dominator-based) Methods

Dominators of b: all blocks that dominate b if every path from the entry of the graph to b goes through a,

then a is one of b’s dominator.

The full set of dominators for b is denoted by DOM(b). Strict Dominators: If a dominators b and a ≠ b, then we say a strictly dominates b. Immediate Dominator: The immediate dominator of b is the strict dominator of b that

is closest to b. It is denoted IDOM(b).

Example

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

BLOCK A B C D E F G DOM IDOM

22

Dominator-Based Value Numbering

Basic strategy: use table from IDom(x ) to

start value numbering x

Use C for F and A for G Imposes a Dom-based application

• rder

m0 ¬ a + b n0 ¬ a + b

A

p0 ¬ c + d r0 ¬ c + d

B

r2 ¬ f(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 ¬ f(e0,e1) u2 ¬ f(u0,u1) v0 ¬ a + b w0 ¬ c + d x0 ¬ e + f

F

SSA Resolves Name Conflicts

23

a ¬ b + c b ¬ 17 d ¬ b - c e ¬ b + c a ¬ b0 + c b1 ¬ 17 d ¬ b0 - c b2 ¬f(b0,b1) e ¬ b2 + c

Summary

Two methods in a scope beyond a basic block Superlocal value numbering (SVN)

Value numbering across basic blocks

Dominator-based value numbering (DVN)

Uses dominance information to handle join points in CFG

They can be used together First Build SSA Do SVN Do DVN with the value tables built in SVN reused

24

Build SSA form is the prerequisite for both!

Examples

25

e = c + d; f = c + d; g = c + d; x = a + b; c = a - b;

The first data-flow problem A global method

26

Global Common Subexpression Elimination (GCSE)

27

Some Expression Sets

For each block b Let AVAIL(b) be the set of expressions available on entry to b. Let EXPRKILL(b) be the set of expressions killed in b. i.e. one or more operands of the expression are redefined in b. !!!! Must consider all expressions in the whole graph. Let DEEXPR(b) include the downward exposed expressions in b. i.e. expressions defined in b and not subsequently killed in b

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Formula to Compute AVAIL

Now, AVAIL(b) can be defined as:

AVAIL(b) = ÇxÎpred(b) (DEEXPR(x) È (AVAIL(x) Ç EXPRKILL(x) ))

• preds(b) is the set of b’s predecessors in the control-flow graph.

(Again, a predecessor is an immediate parent, not including other ancestors.)

29

Computing Available Expressions

The Big Picture

• 1. Build a control-flow graph
• 2. Gather the initial data: DEEXPR(b) & EXPRKILL(b)
• 3. Propagate information around the graph, evaluating the equation

Works for loops through an iterative algorithm: finding the fixed- point. All data-flow problems are solved, essentially, this way.

30

Making Theory Concrete

Computing AVAIL for the example

AVAIL(A) = Ø AVAIL(B) = {a+b} È (Ø Ç all) = {a+b} AVAIL(C) = {a+b} AVAIL(D) = {a+b,c+d} È ({a+b} Ç all) = {a+b,c+d} AVAIL(E) = {a+b,c+d} AVAIL(F) = [{b+18,a+b,e+f} È ({a+b,c+d} Ç {all - e+f})] Ç [{a+17,c+d,e+f} È ({a+b,c+d} Ç {all - e+f})] = {a+b,c+d,e+f} AVAIL(G) = [ {c+d} È ({a+b} Ç all)] Ç [{a+b,c+d,e+f} È ({a+b,c+d,e+f} Ç all)] = {a+b,c+d}

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

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First step is to compute DEEXPR & EXPRKILL

Computing Available Expressions

assume a block b with operations o1, o2, …, ok VARKILL ¬ Ø DEEXPR(b) ¬ Ø for i = k to 1 assume oi is “x ¬ y + z” add x to VARKILL if (y Ï VARKILL) and (z Ï VARKILL) then add “y + z” to DEEXPR(b) EXPRKILL(b) ¬ Ø For each expression e for each variable v Î e if v Î VARKILL(b) then EXPRKILL(b) ¬ EXPRKILL(b) È {e}

Many data-flow problems have initial information that costs less to compute

O(k) steps O(N) steps

N is # operations Backward through block

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Computing Available Expressions

The worklist iterative algorithm

Worklist ¬ { all blocks, bi } while (Worklist ¹ Ø) remove a block b from Worklist recompute AVAIL(b ) as AVAIL(b) = ÇxÎpred(b) (DEEXPR(x) È (AVAIL(x) Ç EXPRKILL(x) )) if ??? then Worklist ¬ ???

33

Computing Available Expressions

The worklist iterative algorithm

Worklist ¬ { all blocks, bi } while (Worklist ¹ Ø) remove a block b from Worklist recompute AVAIL(b ) as AVAIL(b) = ÇxÎpred(b) (DEEXPR(x) È (AVAIL(x) Ç EXPRKILL(x) )) if AVAIL(b ) changed then Worklist ¬ Worklist È successors(b )

• Finds fixed point solution to equation for AVAIL
• That solution is unique

34

Data-flow Analysis

Data-flow analysis is a collection of techniques for compile-time reasoning about run-time flow of values

Almost always involves building a graph Problems are trivial on a basic block Global problems -> control-flow graph (or derivative) Whole program problems -> call graph (or derivative) Usually formulated as a set of simultaneous equations

35

Replacement step in GCSE

Limit to textually identical expressions

(like DAG, unlike value numbering)

e <- d + c

a <- b + c d <- b

e <- b + c

a <- b + c

f <- b + c AVAIL(B) ={b+c}

B2 B1 B

AVAIL(B) ={b+c} Cannot find or remove the redundancy! Should replace b+c with ?

36

GCSE (replacement step)

Compute a static mapping from expression to name After analysis & before transformation " block b, " expression eÎAVAIL(b), assign e a global name by hashing

• n e

During transformation step Evaluation of e Þ insert copy name(e) ¬ e (e is not available and needs to be evaluated) Reference to e Þ replace e with name(e) (e is available and should be replaced)

Example

m=a+b; n=c+d; c = 17; q=c+d; p=c+d; r=c+d; name expression t1 a+b t2 c+d B1 B2 B3 B4 t1 = a+b; m=t1; t2=c+d; n=t2; c = 17; t2=c+d; q=t2; t2=c+d; p=t2; r=t2; B1 B2 B3 B4

AVAIL(B4) ={c+d; a+b}

38

GCSE (replacement step)

The major problem with this approach Inserts extraneous copies At all definitions and uses of any eÎAVAIL(b), " b Not a big issue Those extra copies are dead and easy to remove The useful ones often coalesce away

39

Comparison

m ¬ a + b n ¬ a + b

A

p ¬ c + d r ¬ c + d

B

y ¬ a + b z ¬ c + d

G

q ¬ a + b r ¬ c + d

C

e ¬ b + 18 s ¬ a + b u ¬ e + f

D

e ¬ a + 17 t ¬ c + d u ¬ e + f

E

v ¬ a + b w ¬ c + d x ¬ e + f

F

LVN LVN SVN SVN SVN DVN DVN GCSE DVN GCSE

The VN methods are ordered

• LVN ≤ SVN ≤ DVN
• GCSE is different
• Based on names, not value
• But for this particular

example: DVN ≤ GCSE

• Not always!!!!

40

Redundancy Elimination Wrap-up

Conclusions

Redundancy elimination has some depth & subtlety Variations on names, algorithms & analysis

DVN is probably the method of choice

Results quite close to the global methods (± 1%) Cost is low