A Study of Spilling and Coalescing in Register Allocation as...

Introduction Difficulty of register allocation The coalescing problem Conclusion

A Study of Spilling and Coalescing in RegisterAllocation as Two Separate Phases

Ph.D. Defense

Florent Bouchezflorent.bouchez@ens-lyon.org

CompsysLIP UMR CNRS — Inria — ENS Lyon — UCBL

France

April 30th 2009

BU

1 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Compilation

foo.c foo.bin

Our interest

Register allocation, one of the latest steps.

ZO

2 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

French folklore: Les Shadoks

MEU

3 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Shadoks have a very small brain

BU GA

4 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The shadoks at the library

BU BU

5 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is register allocation?

Assign variables to memorylocations

Registers: , , , . . .

Memory: infinite

Rules of the game

two interfering variablesà different registers

not enough registersà spill to memory

a← 3425n← 0

a 6= 1 ?

n← n + 1a even ?

a← a/2 a← 3× a + 1

print nBU ZO

6 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is register allocation?

Assign variables to memorylocations

Registers: , , , . . .

Memory: infinite

Rules of the game

two interfering variablesà different registers

not enough registersà spill to memory

Plus constraints:

register constraints

pre-colored variables

register pairing,aliasing,. . .

BU ZO

6 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin et al. model

Live-in: k jg := mem[j+12]h := k-1f := g+he := mem[j+8]m := mem[j+16]b := mem[f]c := e+8d := ck := m+4j := b

Live-out: d k j

Live-rangesg

h

f

e

m

b

c

d

g

h

f

e

m

b

c

d

kj

jk

BU MEU

7 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin et al. model

Interference graph

k

j

g

h

f

e

mb c

d

Live-rangesg

h

f

e

m

b

c

d

g

h

f

e

m

b

c

d

kj

jk

BU MEU

7 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin et al. model

Interference graph

k

j

g

h

f

e

mb c

d

Live-rangesg

h

f

e

m

b

c

d

g

h

f

e

m

b

c

d

kj

jk

BU MEU

7 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Coloring the interference graph: the greedy wayRegister allocation is modeled as coloring the interference graph of theprogram.

Problem

Graph-k-coloring is NP-complete (for k ≥ 3), and any interference graphcan arise in programs. (Chaitin et al.’s proof)

à register allocation is NP-complete in this model.

A greedy coloring heuristic is used: Chaitin et al.’s greedy scheme.Greedy scheme

If coloring fails, usually spill.Disadvantages:

The scheme might fail even when there is a solution.

à need to spill more than necessary

A variable is supposed to be in exactly one register.

à restriction on the coloring

ZO GA

8 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Coloring the interference graph: the greedy wayRegister allocation is modeled as coloring the interference graph of theprogram.

Problem

Graph-k-coloring is NP-complete (for k ≥ 3), and any interference graphcan arise in programs. (Chaitin et al.’s proof)

à register allocation is NP-complete in this model.

A greedy coloring heuristic is used: Chaitin et al.’s greedy scheme.Greedy scheme

If coloring fails, usually spill.

Disadvantages: The scheme might fail even when there is a solution.

à need to spill more than necessary

A variable is supposed to be in exactly one register.

à restriction on the coloring

ZO GA

8 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Coloring the interference graph: the greedy wayRegister allocation is modeled as coloring the interference graph of theprogram.

Problem

Graph-k-coloring is NP-complete (for k ≥ 3), and any interference graphcan arise in programs. (Chaitin et al.’s proof)

à register allocation is NP-complete in this model.

A greedy coloring heuristic is used: Chaitin et al.’s greedy scheme.Greedy scheme

If coloring fails, usually spill.Disadvantages:

The scheme might fail even when there is a solution.

à need to spill more than necessary

A variable is supposed to be in exactly one register.

à restriction on the coloring

ZO GA

8 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Coloring the interference graph: the greedy wayRegister allocation is modeled as coloring the interference graph of theprogram.

Problem

Graph-k-coloring is NP-complete (for k ≥ 3), and any interference graphcan arise in programs. (Chaitin et al.’s proof)

à register allocation is NP-complete in this model.

A greedy coloring heuristic is used: Chaitin et al.’s greedy scheme.Greedy scheme

If coloring fails, usually spill.Disadvantages:

The scheme might fail even when there is a solution.à need to spill more than necessary

A variable is supposed to be in exactly one register.à restriction on the coloring

ZO GA

8 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Outline

1 IntroductionLes ShadoksRegister allocation

2 Difficulty of register allocationStrict SSA graphs are chordalQuestions with register allocation in two phases

3 The coalescing problemIntroduction to coalescingIncremental coalescing

ZO BU

9 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Outline

1 IntroductionLes ShadoksRegister allocation

2 Difficulty of register allocationStrict SSA graphs are chordalQuestions with register allocation in two phases

3 The coalescing problemIntroduction to coalescingIncremental coalescing

ZO ZO

10 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

SSA interference graphs are chordal

In 2002, Darte realized that the interference graph of programs understrict SSA are chordal.(The same discovery was made independently by Brisk and Hack in 2004-2005.)

à Starting point of this thesis

Theorem

The interference graph of a program under strict SSA form is chordal.

ZO MEU

11 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

SSA interference graphs are chordal

In 2002, Darte realized that the interference graph of programs understrict SSA are chordal.(The same discovery was made independently by Brisk and Hack in 2004-2005.)

à Starting point of this thesis

Theorem

The interference graph of a program under strict SSA form is chordal.

ZO MEU

11 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a b c

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs

Definition (Chordal graph)

A graph is chordal iff every cycle of size ≥ 4 has a chord.

a

b

c

d

e

Chordal graphs are perfect and easy to color.

MEU GA

12 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Static Single AssignmentSSA : exactly one textual definition per variable

strictness : SSA where the definition always dominates its uses

MEU BU

13 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Static Single AssignmentSSA : exactly one textual definition per variable

strictness : SSA where the definition always dominates its uses

Example (Straight code converted to SSA form)

a← . . ....

. . .← a...

a← . . ....

. . .← a

a1 ← . . ....

. . .← a1...

a2 ← . . ....

. . .← a2

MEU BU

13 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Static Single AssignmentSSA : exactly one textual definition per variable

strictness : SSA where the definition always dominates its uses

Example (Conditional code converted to SSA form)

if (. . . )

a← 1 a← 2

. . .← a

if (. . . )

a1 ← 1 a2 ← 2

a3 ← φ(a1, a2). . .← a3

MEU BU

13 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Static Single AssignmentSSA : exactly one textual definition per variable

strictness : SSA where the definition always dominates its uses

Example (strict SSA or SSA with dominance property)

if (. . . )

a1 ← 1 . . .

. . .← a1

if (. . . )

a1 ← 1 . . .

a2 ← φ(a1,⊥). . .← a2

MEU BU

13 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Proof that strict SSA interference graphs are chordal

Theorem

The interference graph of a program under strict SSA form is chordal.

Proof.

a

b

c

d

e

dominance property

a and b interfere ⇒ def (a) dominates def (b)(or the converse)

direct each edge with dominance

def (d) is dominated by def (c) and def (e)

c and e are live at def (d)

MEU ZO

14 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Strict SSA programs are easy to colorChordal graphs are perfect graphs, hence easy to color. We proved more:

Theorem

Chordal graphs are colorable using Chaitin et al. greedy scheme.They are greedy-k-colorable.

General program:NP-complete

strict SSA program:greedy-k-colorable

Under strict SSA, Maxlive, the maximum number of simultaneously livevariables, is the coloring indicator:

Maxlive ≤ R

MEU MEU

15 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Strict SSA programs are easy to colorChordal graphs are perfect graphs, hence easy to color. We proved more:

Theorem

Chordal graphs are colorable using Chaitin et al. greedy scheme.They are greedy-k-colorable.

General program:NP-complete

strict SSA program:greedy-k-colorable

Under strict SSA, Maxlive, the maximum number of simultaneously livevariables, is the coloring indicator:

Maxlive ≤ R

MEU MEU

15 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Register allocation in two phases

Using Maxlive, it seems possible to use a very simple register allocationscheme:

1 spill variables until Maxlive ≤ R

2 transform program into strict SSA form

3 allocate variables using R registers

4 go out of colored SSA form

Questions

SSA seems to transform an NP-complete problem into polynomial one. . .Where is the complexity now? What else is simplified?

BU GA GA

16 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Register allocation in two phases

Using Maxlive, it seems possible to use a very simple register allocationscheme:

1 spill variables until Maxlive ≤ R

2 transform program into strict SSA form

3 allocate variables using R registers

4 go out of colored SSA form

Questions

SSA seems to transform an NP-complete problem into polynomial one. . .Where is the complexity now? What else is simplified?

BU GA GA

16 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

BU GA BU

17 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

BU GA BU

17 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin’s proof revisited

No Yes

* on basic block borders only

Proofcriti

cal edges

variable

splitting

complexity

Chaitin et al.NP-complete

Thm. 3.1NP-complete

Sec. 3.2.1 (SSA) *polynomial

Thm. 3.2 *NP-complete

Thm. 3.4 (Col. prop.)polynomial

Thm. 3.3NP-complete

BU GA ZO

18 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin’s proof revisited

No Yes

* on basic block borders only

Proofcriti

cal edges

variable

splitting

complexity

Chaitin et al.NP-complete

Thm. 3.1NP-complete

Sec. 3.2.1 (SSA) *polynomial

Thm. 3.2 *NP-complete

Thm. 3.4 (Col. prop.)polynomial

Thm. 3.3NP-complete

BU GA ZO

18 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin’s proof revisited

No Yes* on basic block borders only

Proofcriti

cal edges

variable

splitting

Swap instructio

n

complexity

Chaitin et al.NP-complete

Thm. 3.1NP-complete

Sec. 3.2.1 (SSA) *polynomial

Thm. 3.2 *NP-complete

Thm. 3.4 (Col. prop.)polynomial

Thm. 3.3NP-complete

BU GA ZO

18 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin’s proof revisited

No Yes* on basic block borders only

Proofcriti

cal edges

variable

splitting

Swap instructio

n

complexity

Chaitin et al.NP-complete

Thm. 3.1NP-complete

Sec. 3.2.1 (SSA) *polynomial

Thm. 3.2 *NP-complete

Thm. 3.4 (Col. prop.)polynomial

Thm. 3.3NP-complete

BU GA ZO

18 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chaitin’s proof revisited

No Yes* on basic block borders only

Proofcriti

cal edges

variable

splitting

Swap instructio

n

2 simult.

def.

complexity

Chaitin et al.NP-complete

Thm. 3.1NP-complete

Sec. 3.2.1 (SSA) *polynomial

Thm. 3.2 *NP-complete

Thm. 3.4 (Col. prop.)polynomial

Thm. 3.3NP-complete

BU GA ZO

18 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

BU GA MEU

19 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

criticalflow-edgesplitting

π-motion

BU GA MEU

19 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not.

Short answer: No

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not. Short answer: No

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not. Short answer: No

Longer answer:

Chordal graph= SSA case

Interval graph= basic block

weighted

no

yes

no

yes

Ω′ ≤ C

dyn. prog.

Ω′ ≤ R

furthest use

ILP

Ω′ ≤ Ω− 1

X3C

dyn. prog.

polynomial NP-complete new result

Previous studies by Belady, Yannakakis&Gavril, and Liberatore.

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not. Short answer: No

Longer answer:

Chordal graph= SSA case

Interval graph= basic block

weighted

no

yes

no

yes

Ω′ ≤ C

dyn. prog.

Ω′ ≤ R

furthest use

ILP

Ω′ ≤ Ω− 1

X3C

dyn. prog.

polynomial NP-complete new result

Previous studies by Belady, Yannakakis&Gavril, and Liberatore.

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not. Short answer: No

The problem of “holes.”

On RISC architectures, spilled variables leave small intervals behind.

a← . . ..... . .← a.... . .← a

spill a

a1 ← . . .store a1...a2 ← load. . .← a2...a3 ← load. . .← a3

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Spilling under SSA is still a difficult problemIn this thesis, we studied the complexity of spill everywhere under SSA toknow whether it would simplify the problem or not. Short answer: No

With “holes,” it becomes harder even on a basic block:

h = 1

h ≥ 2

h notbounded

weight

no

yes

no

yes

no

yes

Ω′ ≤ C

dyn. prog.

Ω′ ≤ R

?

stable set

stable set

Ω′ ≤ Ω− C

dyn. prog.

Ω′ ≤ Ω− 1

set cover

polynomial NP-complete new result

BU BU GA

20 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

criticalflow-edgesplitting

π-motion

spilling NOTsimplifiedby SSA

BU BU BU

21 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

criticalflow-edgesplitting

π-motion

spilling NOTsimplifiedby SSA

Coalescing(out-of-SSA)

edgesplitting

π-motion

heuristicscomplexity

BU BU BU

21 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Outline

1 IntroductionLes ShadoksRegister allocation

2 Difficulty of register allocationStrict SSA graphs are chordalQuestions with register allocation in two phases

3 The coalescing problemIntroduction to coalescingIncremental coalescing

BU BU ZO

22 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is coalescing?

Goal of coalescing

Removing the register-to-register copies [move a← b]

Numerous move due to:

live-range splitting toavoid spilling

register constraints

SSA destruction

Not a new problem, butnever studied not intermixedwith spilling.

BU BU MEU

23 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is coalescing?

Goal of coalescing

Removing the register-to-register copies [move a← b]

Numerous move due to:

live-range splitting toavoid spilling

register constraints

SSA destruction

Not a new problem, butnever studied not intermixedwith spilling.

a← . . .b← . . .c← f(a, b)

a← . . .b← . . .move R0,amove R1,bcall fmove c,R0

BU BU MEU

23 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is coalescing?

Goal of coalescing

Removing the register-to-register copies [move a← b]

Numerous move due to:

live-range splitting toavoid spilling

register constraints

SSA destruction

Not a new problem, butnever studied not intermixedwith spilling.

if (. . . )

a1 ← 1 a2 ← 2

a3 ← φ(a1, a2)· · · ← a3

BU BU MEU

23 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is coalescing?

Goal of coalescing

Removing the register-to-register copies [move a← b]

Numerous move due to:

live-range splitting toavoid spilling

register constraints

SSA destruction

Not a new problem, butnever studied not intermixedwith spilling.

if (. . . )

a1 ← 1 a2 ← 2

a3 ← φ(a1, a2)· · · ← a3

move a3 ← a1 move a3 ← a2

BU BU MEU

23 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

What is coalescing?

Goal of coalescing

Removing the register-to-register copies [move a← b]

Numerous move due to:

live-range splitting toavoid spilling

register constraints

SSA destruction

Not a new problem, butnever studied not intermixedwith spilling.

if (. . . )

a1 ← 1 a2 ← 2

a3 ← φ(a1, a2)· · · ← a3

move a3 ← a1 move a3 ← a2

BU BU MEU

23 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Traditional modeling of the coalescing problem

Given an instruction [move a← b]

Fact I Giving the same color to both a and b saves the instruction.Fact II Merging nodes a and b forces them to have the same color.

Idea Express this as an “affinity” between a and b in theinterference graph to drive the algorithm.

a

b c

d

Merge a and bab

c

d

We work on graphs instead of programs.

BU ZO GA

24 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Traditional modeling of the coalescing problem

Given an instruction [move a← b]

Fact I Giving the same color to both a and b saves the instruction.Fact II Merging nodes a and b forces them to have the same color.

Idea Express this as an “affinity” between a and b in theinterference graph to drive the algorithm.

a

b c

d

Merge a and bab

c

d

We work on graphs instead of programs.

BU ZO GA

24 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Traditional modeling of the coalescing problem

Given an instruction [move a← b]

Fact I Giving the same color to both a and b saves the instruction.Fact II Merging nodes a and b forces them to have the same color.

Idea Express this as an “affinity” between a and b in theinterference graph to drive the algorithm.

a

b c

d

Merge a and bab

c

d

We work on graphs instead of programs.

BU ZO GA

24 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Traditional modeling of the coalescing problem

Given an instruction [move a← b]

Fact I Giving the same color to both a and b saves the instruction.Fact II Merging nodes a and b forces them to have the same color.

Idea Express this as an “affinity” between a and b in theinterference graph to drive the algorithm.

a

b c

d

Merge a and bab

c

d

We work on graphs instead of programs.

BU ZO GA

24 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

Incremental

Conservative

Optimistic

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

Coalesce as many affinitiesas possible.

Incremental

Conservative

Optimistic

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Conservative

Optimistic

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Conservative

Coalesce as many affinities aspossible but remains k-colorable.

Optimistic

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

Perform aggressive coalescing, thende-coalescing to get k-colorable.

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

cv

v1

v2

v3cv

c′v

v1

v2

v3

cv

v1

v2

v3

3-vertex-cover

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

Coalesce one affinity while stayingk-colorable.

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

cv

v1

v2

v3cv

c′v

v1

v2

v3

cv

v1

v2

v3

3-vertex-cover

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

T

F

R

ci,1bi,1

ai,1yi,1

ai,2yi,2

ci,2bi,2

ai,3yi,3

ai,4yi,4

3-SAT

NP-complete for general graphs.

Polynomial for chordal graphs.

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

cv

v1

v2

v3cv

c′v

v1

v2

v3

cv

v1

v2

v3

3-vertex-cover

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Different coalescing problems

Aggressive

s1s2

s3

uw

v

edge removeds1

s2

s3

uw

v

affinity notcoalesced

multiway-cut

Incremental

T

F

R

ci,1bi,1

ai,1yi,1

ai,2yi,2

ci,2bi,2

ai,3yi,3

ai,4yi,4

3-SAT

NP-complete for general graphs.

Polynomial for chordal graphs.

Conservative

a

b

d

c

e

a

b

d

c

e

x〈a,c〉

y〈a,c〉x〈b,a〉

y〈b,a〉

x〈d,b〉

y〈d,b〉 x〈c,d〉

y〈c,d〉

x〈c,e〉

y〈c,e〉

graph k-coloring

Optimistic

cv

v1

v2

v3cv

c′v

v1

v2

v3

cv

v1

v2

v3

3-vertex-cover

BU ZO BU

25 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

The problem of incremental coalescing

The goal of incremental is to perform conservative coalescing bycoalescing affinities one by one.

Problem (Incremental coalescing)

Given a k-colorable graph G and two nodes x and y, is it possible to colorG such that x and y have the same color?

Theorem

The incremental coalescing problem is NP-complete.

BU ZO ZO

26 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

y

y

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

y

yzz

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

y

yzz

w

w

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

y

yzz

w

w

v

v

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

First, equivalence between graph-3-coloring and 4-SAT.

3 nodes for True, False, and X

2 nodes for each variable: v and v

. . . and a widget to forbid everyvariable of a clause to be false

X

T F

x

x

y

yzz

w

w

v

v

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

Need a widget that is 3-colorable only if not all 4 variables are false.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

Need a widget that is 3-colorable only if not all 4 variables are false.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

If all 4 variables are false, not 3-colorable.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

If at least one variable is true

, 3-colorable.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

If at least one variable is true

, 3-colorable.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

w

z

y

x

T

F

X

If at least one variable is true, 3-colorable.

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

Now, transform 3-SAT instance into 4-SAT by adding x0 to every clause:

(y ∨ z ∨ w) ∧ · · · ∧ (z ∨ y ∨ u)

becomes

(x0 ∨ y ∨ z ∨ w) ∧ · · · ∧ (x0 ∨ z ∨ y ∨ u)

Clearly, x0 =True satisfies the equation (i.e., the graph is 3-colorable).

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

Now, transform 3-SAT instance into 4-SAT by adding x0 to every clause:

(y ∨ z ∨ w) ∧ · · · ∧ (z ∨ y ∨ u)

becomes

(x0 ∨ y ∨ z ∨ w) ∧ · · · ∧ (x0 ∨ z ∨ y ∨ u)

Clearly, x0 =True satisfies the equation (i.e., the graph is 3-colorable).

Now, ask x0 and False to be coalesced. . .

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing is NP-complete in the general caseReduction from 3-SAT. (Similar to reduction of graph-3-coloring from 3-SAT).

Example on (x ∨ y ∨ z ∨ w) ∧ · · · ∧ (x ∨ z ∨ y ∨ u).

To conclude:

3-SAT is true ⇐⇒ 4-SAT is true with x0 = False

⇐⇒ graph is 3-colorable with x0 in red/False

⇐⇒ incremental coalescing of x0 with False is possible

BU ZO MEU

27 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Incremental coalescing for chordal graphs

Problem (Incremental coalescing for chordal graphs)

Given a k-colorable chordal graph G and two nodes x and y. Is it possibleto color G such that x and y have the same color?

This problem is polynomial!

Moreover, if the answer is yes, it is possible to modify G so that x and y are

merged and G stays chordal.

The same question with greedy-k-colorable graphs is still open.

BU MEU GA

28 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Example on a chordal graph

Let us consider a 3-colorable chordal graph.

Demo: graph in ubigraph

BU MEU BU

29 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs as subtrees of a tree

BU MEU ZO

30 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs as subtrees of a tree

BU MEU ZO

30 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Chordal graphs as subtrees of a tree

BU MEU ZO

30 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

xy

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Finding a path on the interval graphOnce the branches are pruned, an interval graph remains.

x y

BU MEU MEU

31 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

A simple algorithmic strategy for chordal incrementalcoalescing

Building the representation of a chordal graph as subtrees of a tree ispainful.

We have devised an algorithmic strategy that works directly on the graph,using the same ideas as in Chaitin et al.’s greedy coloring algorithm.

ZO GA GA

32 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Demonstration of coalescingDemonstration of conservative coalescing on graph #311 of the“Coalescing Challenge.” (Appel&George)

Demo: conservative coalescing ZO GA BU

33 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Questions raised by register allocation in two phases

SSA = chordal(2002)

spillingsimplifiedby SSA?

Coalescing(out-of-SSA)

Chaitin’sproof

revisitChaitin’s

proof

criticalflow-edgesplitting

swapproblem

# simult.defs

criticalflow-edgesplitting

π-motion

spilling NOTsimplifiedby SSA

Coalescing(out-of-SSA)

edgesplitting

π-motion

heuristicscomplexity

WDDD’06

LCPC’06 (ext)LCTES’07 CGO’07

(best paper) CASES’08

not published

ZO GA ZO

34 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Conclusion: practical work going on

This work was part of a collaboration with STMicroelectronics in Grenoble.Practical experiments were made during this thesis (coalescing). But a lotmore are going on, including:

including Hack’s spilling in LAO

experimenting new coalescing strategies in LAO

experimenting the permutation motion to avoid splitting edges

ZO GA MEU

35 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Conclusion

Why are those results important for register allocation?The idea of separating register allocation in two phases has been aroundfor a while:

spilling is the most important issue

splitting helps (Briggs, Cooper) but copies are hard to remove

greedy coloring is not a good indicator for spilling

Assuming that some copies are always better than a spill:

Maxlive is an “oracle” for spilling

spilling can be optimized first so that Maxlive ≤ R

good coalescing strategies: do not worry about splitting anymore

ZO BU GA

36 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

Remaining problems to work on

continuing the work on permutation motion

improving the spilling as a first phase

having a better “oracle” that handles registers constraints(Pereira&Palsberg)

register allocation in two phases for JIT

Maxlive indicator can be used for better instruction scheduling

Bonus: what will be written left to the pumping shadoks in next slide?

ZO BU BU

37 / 39

Introduction Difficulty of register allocation The coalescing problem Conclusion

That’s all for today

Answer: ZO BU ZO

ZO BU ZO

38 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

i

j

c ′

d ′

g ′

1

2

34

2

332

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

i

j

j

c ′

d ′

g ′ 2

24

2

332

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

g

i

j

j

c ′

d ′

g ′

13

2

332

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

g

i

i

j

j

c ′

d ′

g ′

2

2

332

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e

f

f

g

g

i

i

j

j

c ′

d ′

g ′1

232

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

132

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d

e

e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

22

4 4

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

2

3 3

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′2 2

3

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′1

2

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

1

1

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

jc ′

c ′

d ′ d ′

g ′g ′

Hi-degree nodeLow-degree node

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

jc ′

c ′

d ′ d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

b

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

b

c

c

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′ d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

dd

e

e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d

e

e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e

f

f

g

g

i

i

j

j

c ′

d ′

g ′g ′

Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e

f

f

g

g

i

i

j

j

c ′

d ′

g ′Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

g

i

i

j

j

c ′

d ′

g ′Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

g

i

j

j

c ′

d ′

g ′Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

i

j

j

c ′

d ′

g ′Return

ZO BU MEU

39 / 39

Greedy coloring scheme of Chaitin et al.

Greedy-k-colorable-graphs: simplify nodes with < k neighbors.

Sta

ck

a

bc

d e f

g

i

j

c ′

d ′

g ′Return

ZO BU MEU

39 / 39