1. Register Allocation via Coloring of Chordal Graphs
Fernando M Q Pereira
Jens Palsberg
UCLA

2. 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.
But, what is a chordal graph?

3. 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!
Chordal graphs have been focus of research during the 70’s.
All these problems are O(|V| + |E|)

4. Register Allocation is Complicated…
Iterated Register Coalescing [George and Appel 96]
actual
spill
build
simplify
select
coalesce
potential
spill
freeze

5. The Proposed Algorithm.
pre-spilling
phase
coloring
phase
coalescing
phase
SEO
Simple
Modular
Efficient
Works with non-chordal interference graphs.

6. Some terminology: Induced subgraphs: H = G[VH]
Induced cycles: H is a cycle.
Clique: H is a complete graph. A B A B C C D E D A B A B C
Induced subgraph: delete some nodes, keep all the edges that existed among the original nodes. D E D E A B A C C D E D

7. 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 A B
There are three basic ways to define a chordal graph:
graphs without induced cycles with more than three nodes.
intersection graphs of intervals on a tree.
graphs that have a simplicial elimination ordering.
Every time there is a cycle with more than three nodes, there is an edge connecting non-adjacent vertices. D E D E

8. Why are Interference graphs Chordal?
Chordal graphs are the intersection graphs of subtrees of a tree: E D F
The second definition of chordal graphs: intersections of subtrees of a tree.
Intuitively, one can imagine the control flow graph as a tree, and the live ranges as subtrees. In this case, the interference graph would be chordal. B C A

9. 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
Control flow graphs are better described as series parallel graphs.

10. 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.

11. 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 :(

12. The Proposed Algorithm.
The pre-spilling version:
pre-spilling
phase
coloring
phase
coalescing
phase
SEO
The post-spilling version:
Three requirements:
Modularity: composed from separate building blocks;
Efficiency: polynomial complexity;
Produce good code.
Must work for chordal and non-chordal graphs.
coloring
phase
post-spilling
phase
coalescing
phase
SEO

13. The Example. Two registers available for allocation: R1 and R2 R1 R2 A
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
Solid lines denote interference between registers.
dashed lines denote move related nodes. F E D

14. The Simplicial Elimination Ordering.
pre-spilling
phase
coloring
phase
coalescing
phase
SEO
Simplicial Elimination Ordering (SEO)

15. Simplicial Elimination OrderingB C
Simplicial Elimination Ordering (SEO) F E D
The most important feature we use in our algorithm is the so called simplicial elimination ordering.
It is an ordering of the vertices of a graph.
Every chordal graph has many simplicial elimination orderings
But there are ordering that are not simplicial.
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

16. 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.
Simplicial Elimination Ordering (SEO)

17. The Pre-Spilling Phase.
coloring
phase
coalescing
phase
SEO
The pre-spilling phase
Most algorithms do spilling after at least one attempt of coloring the interference graph has been performed. But, with a chordal graph, it is possible to determine the minimum number of colors beforehand.

18. 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

19. The pre-spilling phase
Only look into cliques greater than K = 2. R1 R2 A B C
The pre-spilling phase F E D
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.

20. The pre-spilling phase
Resulting graph: R1 R2 A C
The pre-spilling phase F E D
S1 = ( A, F, E, D, C, R2, R1 ) A F E B B R2
Node B is present in most of the cliques, and must be removed.

21. The Coloring Phase. pre-spilling phase coloring phase coalescing phase
SEO
The coloring phase

22. Coloring Chordal Graphs.
Feed the greedy coloring with a simplicial elimination ordering. C
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

23. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

24. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

25. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

26. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

27. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

28. Coloring Chordal Graphs.
The coloring phase R1 R2 A F E D
S1 = ( A, F, E, D, C, R2, R1 )

29. Register Coalescing. pre-spilling phase coloring phase coalescing
SEO
The coalescing phase

30. Register Coalescing Greedy coalescing after register allocation.
Why not before graph coloring?
The coalescing phase
Algorithm: Register Coalescing
Input: (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

31. Register Coalescing R2 R1
The coalescing phase D A F E C

32. Register Coalescing R2 R1
The coalescing phase D A F E C

33. The Post-Spilling Phase.
coloring
phase
post-spilling
phase
coalescing
phase
SEO
The Post-Spilling Phase
Remove nodes assigned same color. E.g:
Remove least used color.
Remove greatest color.
Faster implementation, but generates worse code.

34. 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].

35. 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.

36. Non-transformed Programs
Chordal coloring
16 registers
18 registers
IRC
# 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

37. Post-SSA Programs
Chordal coloring
16 registers
18 registers
IRC
# 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

39. 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.

40. Time and Complexity: G = (V, E)
SEO: O(|V| + |E|);
Pre-spilling: O(|V| + |E|);
Coloring: O(|E|);
Coalescing: O(|V|3);
coloring spilling coalescing
pre-spilling
Largest color
Least used color

41. 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.

42. 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:

43. 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.

44. 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.

45. Non-1-Perfect Example d a b e c 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; d a b e c

46. The Post SSA Interference Graph.d a x e2 b
b = 0;
c1 = d;
e1 = b;
e2 = 0;
c2 = d; c2 c1
e = e2;
c = c2;
c = c1;
e = e1; e e1
return c + e; c

47. 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.