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Register Allocation via Coloring of Chordal Graphs

Register Allocation via Coloring of Chordal Graphs. Fernando M Q Pereira Jens Palsberg. UCLA. 95% of the interference graphs produced from Java Methods by the JoeQ compiler are Chordal. But, what is a chordal graph?. Java 1.5 library: 95.5% (23,681 methods).

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Register Allocation via Coloring of Chordal Graphs

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  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. • But, what is a chordal graph? • Java 1.5 library: 95.5% (23,681 methods). • Public ML graphs: 94.1% (27,921 IG’s). • Not considering pre-colored registers.

  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!

  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. coloring phase coalescing phase pre-spilling phase SEO • Simple • Modular • Efficient • Works with non-chordal interference graphs.

  6. Some terminology: A B A B • Induced subgraphs: H = G[VH] • Induced cycles: H is a cycle. • Clique: H is a complete graph. C C D E D A B A B C 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 D E D E

  8. Why are Interference graphs Chordal? • Chordal graphs are the intersection graphs of subtrees of a tree: E D F 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

  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: coloring phase coalescing phase pre-spilling phase SEO The post-spilling version: post-spilling phase coalescing phase coloring phase SEO

  13. The Example. Two registers available for allocation: R1 and R2 R1 R2 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

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

  15. Simplicial Elimination Ordering R1 R2 A B C Simplicial Elimination Ordering (SEO) F E D 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 pre-spilling phase SEO The pre-spilling phase

  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. 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. 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. coloring phase coalescing phase pre-spilling 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. C The coloring phase R1 R2 A F E D S1 = ( A, F, E, D, C, R2, R1 )

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

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

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

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

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

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

  37. Post-SSA Programs

  38. 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.

  39. 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

  40. 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.

  41. 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/

  42. 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.

  43. 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.

  44. 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; } d a b e c

  45. 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

  46. 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.

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