A Coalescing Algorithm for Aliased Registers

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A Coalescing Algorithm for Aliased Registers

• Mariza A S Bigonha
• Fabrice Rastello
• Fernando M Q Pereira
• Roberto S Bigonha

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Our Objectives
We want to design and implement a new coalescing
algorithm that takes register aliasing into
consideration.

• The most popular computer architectures have
  aliasing.
• Current coalescing schema are oblivious to
  aliasing.

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Introduction

• International cooperation Brazil-France
• ENS Lyon (France) and UFMG (Brazil)
• Code generation and optimization
• Our end goal is to contribute in the construction
  of compilers that help programmers to produce
  efficient machine code, while still using high
  level programming languages.

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Register Allocation

• Where to store the variables used in the program?
• Memory lots of space, but slow access.
• Registers lightning fast, but very few

a
a
a 2
a 2
b
b a
a
b
b 3
a
b
b
c a b
print(b)
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Register Coalescing

• Register coalescing is an optimization on top of
  register allocation. The objective is to map both
  variables used in copies to the same register.

R1 0 R1 R1 print(R1)
R1 0 R2 R1 print(R2)
a ? R1 b ? R2
int a 0 int b a print(b)
a ? R1 b ? R1
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The Importance of Register Coalescing

• Size reduction

Execution speedup
Energy saving
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SSA Based Register Allocation

• 2005 SSA form programs have chordal
  interference graphs Bouchez05, Hack06.
• Chordal graphs can be optimally colored in
  polynomial time.
• Minimum register assignment has polynomial time
  solution.
• The SSA representation is a form of live range
  splitting.
• It inserts many copies in the source program

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Live Range Splitting

• Live range splitting is the inverse of register
  coalescing.
• A good coalescer is paramount to maximize the
  benefits of live range splitting.

a
a1
a 2
a1 2
a1
a2 a1
a2
a
print(a)
print(a2)
a2
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Register Aliasing
Two registers alias when an assignment to one may
change the value of the other.
AX
BX
CX
DX
Ex. x86
AH
AL
BH
BL
CH
CL
DH
DL
D0
D15
Ex. PowerPC

F10
F1
F30
F31
And also ARM, UltraSparc, ST240, etc
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Aliasing and Coalescing
a ? AH b ? AL c ? AH d ? AL
AH 0 AL 1 out(AH, AL)
int a 0 int b 1 c a d b out(c, d)
(ideal assignment)
AH 0 AL 1 BX AX out(BH, BL)
a ? AH b ? AL c ? BH d ? BL
(acceptable assignment)
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The Coalescing Zoo

• Global the whole program.
• Local one basic block.
• Punctual two consecutive instructions.

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Complexity Results

• Global coalescing NP complete
• Local coalescing polynomial without aliasing,
  but NP complete with aliasing, via reduction to
  aligned coloring.
• Punctual coalescing polynomial.

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Punctual lt Local lt Global
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Coalescing in Practice

• Long history Chaitin et al proposed aggressive
  coalescing Chaitin 81
• Briggs et al brought in conservative coalescing
  Briggs 94
• George and Appel invented incremental
  conservative coalescing George 96
• Park et al came with optimistic coalescing Park
  04
• These heuristics do not talk about aliasing.

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The Cooperating Parties

• Ecole Normale Superieure de Lyon
• Chair Fabrice Rastello
• Especially coalescing

• Universidade Federal de Minas Gerais
• Chair Mariza Bigonha
• Especially aliasing

Eg Florent Bouchez, Alain Darge,
Fabrice Rastello On the Complexity of
Register Coalescing. CGO 2007 102-114
Eg Fernando Magno Quintão Pereira, Jens
Palsberg Register allocation by puzzle solving.
PLDI 2008 216-226
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What we have so far

• We have an integer linear programming (ILP)
  solution to global coalescing with aliasing.
• We have been able to find optimal solutions to
  more than 90 of the functions in SPEC CPU 2000.
• Our model
• Live range splitting at each program point.
• Two level aliasing, like in floating point
  registers of PowerPC.
• We handle only register assignment no spilling.

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What we have so far

• Optimal solution to punctual coalescing for
  elementary programs.

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The Plan

• Adapt existing coalescing approaches to handle
  aliasing. Two candidates
• Bouchezs incremental coalescing Cases 08.
• Hacks coalescing by graph recoloring PLDI 08.
• Compare with the punctual coalescing used in
  register allocation by puzzle solving PLDI 2008.

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Candidate 1 incremental coalescing

• Greedy-K-colorable graph (GKC) a graph that can
  be trivially colored by a greedy coloring
  algorithm.
• Let G be a GKC graph. Let G be the result of
  coalescing two nodes of G. Is G still GKC?
• Polynomial test to prove GKC property.
• Conservative coalescing.
• About 15 faster than optimistic approaches.

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Candidate 2 recoloring

• Color the graph without worrying about
  coalescing.
• Recolor as many nodes as possible, to satisfy as
  many affinities as possible.

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The Big Picture

• ILP for global coalescing
• Optimal, but very slow
• Baseline for comparisons

• Heuristics for global coalescing
• Sub-Optimal, but Polynomial
• Default for compilation

• Punctual coalescing
• Optimal, fast, but restricted
• Just in time compilation

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The Cooperation

• French side
• Designed and implemented base coalescer.
• Design the new coalescing algorithm.
• Help in the implementation effort.
• Brazilian side
• Designed and implemented ILP and punctual
  coalescing with aliasing.
• Help in designing the algorithm.
• Implement the algorithm.

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