An Optimistic and Conservative Register Assignment Heuristic for Chordal Graphs

1
An Optimistic and Conservative Register
Assignment Heuristic for Chordal Graphs
Philip Brisk
Ajay K. Verma
Paolo Ienne
International Conference on Compilers,
Architecture, and Synthesis for Embedded
Systems October 3, 2007. Salzburg, Austria
2
Outline

• Coalescing Problem
• Biased Coloring
• Chordal Graphs
• Optimistic Chordal Coloring Heuristic
• Experimental Results
• Conclusion

3
Coalescing Problem

• Given
• k-colorable interference graph G (V, E)
• Spilling already performed
• Set of copy operations y ? x
• x and y do not interfere
• Runtime profiling gives execution count
• Objective
• Find legal k-coloring of G
• Minimize the number of dynamic copies executed

4
Example

• K 4 Registers

Interference Edge
Copy Instruction
5
Coalescing Problem Complexity

• SSA Form
• Theorem SSA interference graphs are chordal
• Bouchez et al., MS Thesis, ENS-Lyon, 2005
• Brisk et al., IWLS 2005, TCAD 2006
• Hack et al., Tech. Report, U. Karlsruhe, 2005
  IPL, 2006
• Coalescing is NP-Complete, even for chordal
  graphs
• Bouchez et al., CGO 2007, Best Paper Award

6
Contribution

• Coloring heuristic for chordal graphs
• Conservative
• Stronger guarantees than Conservative
  Coalescing
• Optimistic
• Retains the benefits of Optimistic Coalescing
• Bottom Line
• Near-optimal solutions for set of benchmarks
  studied
• 1 extra copy, across all benchmarks

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Coalescing via Node Merging

• Chaitin et al., Comp. Languages 1981 SCC 1982

8
Negative Side Effects of Coalescing

• Vertex degree increases
• Briggs et al., TOPLAS 1994
• George and Appel, TOPLAS 1996
• Hailperin, TOPLAS 2005

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Negative Side Effects of Coalescing

• Chromatic number increases
• Briggs et al., TOPLAS 1994
• George and Appel, TOPLAS 1996

Triangles form Kaluskar, MS Thesis, GA Tech.,
2003
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Positive Effects of Coalescing
Deg(Z) 2
Deg(Z) 1
XY

• Reduce the degree of neighboring vertices
• Park and Moon, PACT 1998, TOPLAS 2004
• Vegdahl, PLDI 1999

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Coalescing Strategies

• Conservative
• Coalesce if you can prove that the resulting
  graph is still k-colorable
• Optimistic
• Aggressively coalesce as many moves as possible
• De-coalesce, when necessary, to avoid spilling
• Coalesce non-move-related nodes to exploit
  positive benefit
• Complexity
• Conservative, aggressive, optimistic coalescing
  are all NP-Complete
• Bouchez et al., CGO 2007

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Biased Coloring

• Introduced by Briggs et al., TOPLAS 1994
• For whatever reason, we couldnt coalesce X and Y
• Assume X gets a color before Y
• If Color(X) is available when we assign a color
  to Y
• Then assign Color(X) to Y

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Chordal Graphs

• No chordless cycles of length 4 or more

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Coalescing and Chordal Graphs

• Coalescing does not preserve chordality

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Pseudo-coalescing

• Find independent sets of move-related vertices,
  but dont merge them
• Try to assign the same color to the vertices

Chordal (After pseudo-coalescing)
Chordal (Before pseudo-coalescing)
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Simpicial Vertices

• v is a Simplicial Vertex if the neighbors of v
  form a clique.
• Theorems
• Every chordal graph has at least one simplicial
  vertex Lekkerkerker and Boland, Fund.
  Math 1964
• Every induced subgraph of a chordal graph is
  chordal

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Perfect Elimination Order

• Undirected graph G (V, E)
• Elimination Order
• One-to-one and onto function s V ? 0, , V -
  1
• Rename vertices vi satisfies s(vi) i
• Vi v1, , vi
• Gi (Vi, Ei) is the subgraph of G induced by Vi
• Perfect Elimination Order (PEO)
• vi is a simplicial vertex in Gi, for i 1 to n
• G is chordal iff G has a PEO

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Coloring Chordal Graphs

• Optimal, O(V E)-time Algorithm
• Gavril, Siam J. Comp, 1972
• Process vertices in PEO order
• Give vi the smallest color not assigned to its
  neighbors
• Works because vi is simplicial in Gi
• Extensions for copy elimination (this paper)
• Pseudo-coalescing
• Bias color assignment

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Correctness Argument

• Invariant
• When we assign a color to vi, Gi is legally
  colored
• Optimistic extensions for copy elimination
• Examples given in the next 3 slides
• Invariant above is always maintained
• Color assignment is conservative
• The chromatic number is known, and never changes

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Simple Color Assignment Rules
3

• PEO order

• 1

2
Thats a really bad idea
No harm done
But that isnt how it really happens
Lets assume nodes 1 and 3 are pseudo-coalesced
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Undo Pseudo-coalescing

• PEO order

2
3
4
5

• 1

Assign color
Propagate color
Pre-assigned color is legal
Pre-assigned color is illegal
Undo pseudo-coalescing
Legal coloring invariant still holds
Cannot stay pseudo-coalesced with node 2
Pseudo-coalesce with nodes 3 and 4 instead
I cant pseudo-coalesce all the nodes
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Negative Biasing (Same Example)

• PEO order

2
3
4
5
Select a Color for Node 2

• 1

Node 2 is pseuo-coalesced with Node 5
Node 5 interferes with Node 1, which has Color RED
So bias Node 1, AWAY FROM Color RED
Assign Color BLUE to Node 1 instead
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Maximum Cardinality Search

• Compute PEO order in O(V E) time
• Tarjan and Yannakakis, Siam J. Comput, 1984
• Color move-related nodes early
• A legal color is always found for each node
• So coloring non-move-related node
• Constrains the move-related-nodes
• Biased MCS
• Push move-related vertices toward the front of
  the PEO

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Simplification

• If Deg(v) lt k, remove v from G
• Kempe, American J. Math. 1876
• A color can always be found for v later
• Dont do this for move-related nodes
• Removing them doesnt help to eliminate moves
• Repeat until all remaining nodes v
• Have Deg(v) gt k, or
• Are move-related
• Justification
• Fewer coloring constraints on move-related
  vertices
• Hack and Grund, CC 2007
• In the context of chordal coloring

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Refinement
Sometimes heuristics just dont do the right
thing
Cant hurt, right?
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Optimistic Chordal Coloring Heuristic

1. Simplification
2. (Biased) Maximum Cardinality Search
3. Aggressive Pseudo-Coalescing
4. Optimistic Chordal Color Assignment
5. Refinement
6. Unsimplify

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Register Allocation for ASIPs
Proc1
Proc2
Procm

Input Data

• Profile, Convert to Pruned SSA, Build
  Interference Graph

G1
G2
Gm
Chordal Color Gavril, Siam J. Comp. 1972
?(G1)
?(G2)
?(Gm)
MAX
?max Number of registers to allocate (only
spill at procedure calls)
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Experiments

• Benchmarks
• MediaBench, MiBench in SSA Form
• Color Assignment
• Optimal Coalescing via ILP (OPT)
• Hack and Grund CC, 2007
• Iterated Register Coalescing (IRC)
• George and Appel, TOPLAS 1996
• Optimistic Chordal Coloring (OCC)
• OCC (Not in the paper)
• Illegal pseudo-coalescing
• More aggressive color propagation
• Enhanced refinement stage

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Results
 Benchmark Dynamically Executed Copies Dynamically Executed Copies Dynamically Executed Copies Dynamically Executed Copies
 Benchmark OPT IRC OCC OCC
adpcm_rawcaudio adpcm_rawdaudio blowfish crc32 dijkstra fft g721_decode g721_encode gsm jpeg_cjpeg jpeg_djpeg mpeg2dec mpeg2enc patricia pegwit sha susan 6995016 6995016 0 53322406 0 8209 0 0 1909187 541326 272636 2115 95197 1820 48 442 2 - - - - - - - - 2990 - 6283 - - - 24 - - - - - - - - - - 2990 - 6283 - - - 1 - - - - - - - - - - - - - - - - - - 1
4 graphs were solved sub-optimally by OCC -gt
motivated OCC
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Conclusion

• OCC Heuristic for chordal graphs
• Optimistic
• Pseudo-coalescing instead of node merging
• Preserves the chordal graph property
• Conservative
• Stronger guarantees than conservative coalescing
  
• Chromatic number is easy to compute for chordal
  graphs
• Same correctness invariant as chordal graph
  coloring
• Choice of color assignment/color propagation
• Shares principal similarities to biased coloring

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sym_decrypt

• Weakness in aggressive pseudo-coalescing phase

3
5
6
Color is illegal
2
4
13
10
12
1
7
Propagate
8
9
11
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Rules for Color Assignment

• If vi is move-related
• and vi is uncolored
• Be judicious when selecting a color
• Propagate the color to vertices pseudo-coalesced
  with vi
• or, vi has optimistically been assigned a color
  already
• Is this color still legal with respect to Gi?
• If not, re-color vi
• Undo/redo pseudo-coalescing
• Non-move-related nodes
• Color assignment affects move-related nodes