Register Allocation via Coloring of Chordal Graphs

Fernando M Q Pereira

Jens Palsberg

UCLA

• But, what is a chordal graph?

95% of the interference graphs produced from Java Methods by the JoeQ compiler are Chordal.

• Java 1.5 library: 95.5% (23,681 methods).

• Public ML graphs: 94.1% (27,921 IG’s).– Not considering pre-colored registers.

Why is this good news?

• Many problems that are NP-complete for general graphs are linear time for chordal graphs. Ex. graph coloring.

• Simpler register allocation algorithms, but still competitive!

Register Allocation is Complicated…

build

simplify

coalesce

freezepotential

spill

select

actualspill

• Iterated Register Coalescing [George and Appel 96]

The Proposed Algorithm.

pre-spillingphase

coalescingphase

coloringphase

SEO

• Simple

• Modular

• Efficient

• Works with non-chordal interference graphs.

– Induced subgraphs: H = G[VH]

– Induced cycles: H is a cycle.

– Clique: H is a complete graph.

A B

DC

E

A B

DC

A B

DC

E

A B

D E

A

DC

A B

DC

E

Some terminology:

Chordal Graphs.

• A graph G is chordal iff the size of the largest induced cycle is 3 (it is a triangle).

• non-chordal: Chordal:

A B

D E

A B

D E

Why are Interference graphs Chordal?

• Chordal graphs are the intersection graphs of subtrees of a tree:

A

B C

DE F

But CFG’s are not trees…

int m(int a, int d) { int b, c; if(a > 0) { b = 1; c = a; } else { c = 2; b = d; } return b + c; }}

a b

d c

Interference graphs of programs in SSA form are chordal.

• Independently proved by Brisk[2005], Hack[2005], Bouchez[2005].

• Intuition:– The chordal graphs are the intersection graphs

of subtrees of a tree.– Live ranges in SSA are subtrees of the

dominance tree.

Why only 95% of chordal graphs?

• Executable code is not in SSA form.• SSA elimination.

– Phi-functions are abstract constructions.– In executable code, phi functions are replaced by

copy instructions.

• We call programs after SSA elimination Post-SSA programs.

• Some Post-SSA programs are non-chordal :(

The Proposed Algorithm.

pre-spillingphase

coalescingphase

coloringphase

SEO

The pre-spilling version:

post-spillingphase

coalescingphase

coloringphase

SEO

The post-spilling version:

The Example.

1 int B = R1;2 int A = R2;3 int F = 1;4 int E = A + F;5 int D = 0;6 int C = D;7 R2 = C + E;8 R1 = B;

A B C

F E D

Two registers available for allocation: R1 and R2

R2R1

The Simplicial Elimination Ordering.

pre-spillingphase

coalescingphase

coloringphase

SEO

Sim

plicial Elim

ination Ordering (S

EO

)

Simplicial Elimination Ordering

Neighbors of N that precede N constitute a clique:

S1 = (A, F, B, E, D, C, R2, R1)S2 = (R2, B, E, F, A, D, C, R1)But S3 = (R2, R1, D, F, E, C, A, B) is not a SEO. Why?

} are SEO’s

A B C

F E D

R2R1

Sim

plicial Elim

ination Ordering (S

EO

)

A third definition of chordal graph.

• A graph G = (V, E) is chordal if, and only if, it has a simplicial elimination ordering [Dirac 61].

• There exist O(|V| + |E|) algorithms to find a simplicial elimination ordering:– Maximum Cardinality Search,– Lexicographical Breadth First Search.

Sim

plicial Elim

ination Ordering (S

EO

)

The Pre-Spilling Phase.

The pre-spilling phase

pre-spillingphase

coalescingphase

coloringphase

SEO

The Pre-Spilling Phase

• Chromatic number = size of largest clique.1 - List all the maximal cliques in the graph.2 - Remove nodes until all maximal cliques have K

or less nodes.2.1 - Which registers to remove?– For each register r:

• n = number of big cliques that contain r.• f = frequency of use.• s = size of r’s live range.

– Spill factor = n * s / f

The pre-spilling phase

S1 = ( A, F, B, E, D, C, R2, R1 )

A A B B B B R2

F F E

Node B is present in most of the cliques, and must be removed.

The pre-spilling phase

Only look into cliques greater than K = 2.

A B C

F E D

R2R1

S1 = ( A, F, E, D, C, R2, R1 )

A B B R2F E

Node B is present in most of the cliques, and must be removed.

The pre-spilling phase

Resulting graph:

A C

F E D

R2R1

The Coloring Phase.

pre-spillingphase

coalescingphase

coloringphase

SEO

The coloring phase

Coloring Chordal Graphs.• Feed the greedy coloring with a simplicial

elimination ordering.

The coloring phase

S1 = ( A, F, E, D, C, R2, R1 )

C

F E

D

R2R1 A

Coloring Chordal Graphs.

The coloring phase

F

S1 = ( A, F, E, D, C, R2, R1 )

C

E

D

R2R1 A

Coloring Chordal Graphs.

The coloring phase

EF

S1 = ( A, F, E, D, C, R2, R1 )

C

D

R2R1 A

Coloring Chordal Graphs.

The coloring phaseD

EF

S1 = ( A, F, E, D, C, R2, R1 )

C

R2R1 A

Coloring Chordal Graphs.

The coloring phase

C

D

EF

S1 = ( A, F, E, D, C, R2, R1 )

R2R1 A

Coloring Chordal Graphs.

The coloring phase

R2

C

D

EF

S1 = ( A, F, E, D, C, R2, R1 )

R1 A

Coloring Chordal Graphs.

The coloring phase

R2

C

D

EF

S1 = ( A, F, E, D, C, R2, R1 )

AR1

Register Coalescing.

pre-spillingphase

coalescingphase

coloringphase

SEO

The coalescing phase

Register Coalescing

• Greedy coalescing after register allocation.– Why not before graph coloring?

Algorithm: Register CoalescingInput: (G, color(G))Output: (G, color’(G))begin for every non-interfering move instruction (x := y) do let color(x) = color(y) = unused color(N(x) U N(y));end

The coalescing phase

Register Coalescing

The coalescing phase

D

EFA

R2

C

R1

Register Coalescing

The coalescing phase

D

EFA

R2

C

R1

The Post-Spilling Phase.

post-spillingphase

coalescingphase

coloringphase

SEO

The P

ost-Spilling P

hase

• Remove nodes assigned same color. E.g:– Remove least used color.– Remove greatest color.

• Faster implementation, but generates worse code.

What about a Non-Chordal Graph?

• Coloring is no longer optimal.

• The number of colors will be between the optimal, and twice the optimal for almost every possible graph [Bollobas 1988].

Benchmark

• The Java 1.5 standard libraries.– 23,681 methods.

• Algorithms implemented in the JoeQ framework.

• Two test cases:– Code without any transformation: 90% chordal.– Programs in Post-SSA form: 95% chordal.

Non-transformed Programs

Chordal coloring

16 registers

Chordal coloring

18 registers

IRC

18 registers

# registers / method 4.13 4.20 4.25

# spills / method 0.0055 0.0044 0.0050

Total # spill 131 105 115

Maximum # spill 17 15 16

Coalescing / moves 0.29 0.34 0.31

Post-SSA Programs

Chordal coloring

16 registers

Chordal coloring

18 registers

IRC

18 registers

# registers / method 4.12 4.13 4.17

# spills / method 0.0053 0.0040 0.0049

Total # spill 125 94 118

Maximum # spill 16 17 27

Coalescing / moves 0.68 0.72 0.70

Registers per Method - Post-SSA Program

0

1000

2000

3000

4000

5000

6000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Number of registers

Number of methods

MCS Iterated Coalescing

Methods in Java 1.5

• 23,681 methods; 22,544 chordal methods.

• 85% methods could be colored with 6 regs.

• 99.8% could be colored with 16 regs.

• 28 methods demanded more than 16 regs.

Time and Complexity: G = (V, E)

• SEO: O(|V| + |E|);

• Pre-spilling: O(|V| + |E|);

• Coloring: O(|E|);

• Coalescing: O(|V|3);

colo

ring

spil

ling

coa

lesc

ing

pre-s

pillin

g

Larges

t colo

r

Least

used

color

Related Work.

• All the 27,921 public ML interference graphs are 1-perfect [Andersson 2003].– Structured programs have 1-perfect IG?

• Polynomial register allocation [Brisk 2005], [Hack 2005]. – SSA-Interference graphs are chordal.

Conclusions

• Many interference graphs of structured programs are chordal;

• New algorithm:– Modular;– Efficient;– Competitive;

• We have an extended version of the algorithm implemented on top of GCC:– http://compilers.cs.ucla.edu/fernando/projects/

Are Java Interference Graphs 1-Perfect?

• 1-Perfect graph: minimum coloring equals largest clique.– It is different of perfect graphs.

• All the 27,921 IG of the ML compiler compiling itself are 1-perfect [Andersson, 2003].– Not considering pre-colored registers: 94.5% of

chordal graphs.

SSA and Post-SSA Graphs.

• SSA interference graphs:– Chordal– Perfect– 1-Perfect

• Post SSA graphs:– If phi functions are replaced by copy

instructions, than register allocation is NP-complete.

Non-1-Perfect Example

int m(it a, int d) { int e, c; if(in() > 0) { e = 0; c = d; } else { b = 0; c = a; e = b; } return c + e;}

ad

e

b

c

The Post SSA Interference Graph.

ad

e2 b

c1

e1

c

c2

e

e2 = 0;c2 = d;

b = 0;c1 = d;e1 = b;

e = e2;c = c2;

c = c1;e = e1;

return c + e;

x

References

• [Andersson 2003] Christian Andersson, Register Allocation by Optimal Graph Coloring, 12th Conference on Compiler Construction

• [Brisk 2005] Philip Brisk and Foad Dabiri and Jamie Macbeth and Majid Sarrafzadeh, Polynomial-Time Graph Coloring Register Allocation, 14th International Workshop on Logic and Synthesis

• [Hack 2005] Sebastian Hack and Daniel Grund and Gerhard Goos, Towards Register Allocation for Programs in SSA-form.