An Optimistic and Conservative An Optimistic and Conservative Register Assignment Heuristic for Register Assignment Heuristic for
Chordal GraphsChordal Graphs
Philip Brisk
Ajay K. Verma
Paolo Ienne
International Conference on Compilers, Architecture, and Synthesis for Embedded Systems
October 3, 2007. Salzburg, Austria
OutlineOutline
Coalescing Problem
Biased Coloring
Chordal Graphs
Optimistic Chordal Coloring Heuristic
Experimental Results
Conclusion
Coalescing ProblemCoalescing Problem
Given: k-colorable interference graph G = (V, E)
Spilling already performed
Set of copy operations y xx and y do not interfereRuntime profiling gives execution count
Objective:Find legal k-coloring of GMinimize the number of dynamic copies
executed
ExampleExample
K = 4 RegistersInterference Edge
Copy Instruction
A
BC
DE
F
G
HI
JK
ML
A
BC
DE
F
G
HI
JK
L M
7 copies
A
BC
DE
F
G
HI
JK
ML
5 copies
Coalescing Problem: ComplexityCoalescing Problem: Complexity
SSA FormTheorem: 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]
ContributionContribution
Coloring heuristic for chordal graphs
ConservativeStronger guarantees than “Conservative
Coalescing”
OptimisticRetains the benefits of “Optimistic Coalescing”
Bottom LineNear-optimal solutions for set of benchmarks
studied+1 extra copy, across all benchmarks
Coalescing via Node MergingCoalescing via Node Merging
YXXY
[Chaitin et al., Comp. Languages 1981; SCC 1982]
Negative Side Effects of Negative Side Effects of CoalescingCoalescing
Vertex degree increases [Briggs et al., TOPLAS 1994]
[George and Appel, TOPLAS 1996]
[Hailperin, TOPLAS 2005]
YX
Deg(X) = 3Deg(Y) = 4
XY
Deg(XY) = 6
Negative Side Effects of Negative Side Effects of CoalescingCoalescing
XY
Chromatic number increases
[Briggs et al., TOPLAS 1994]
[George and Appel, TOPLAS 1996]
Triangles form
[Kaluskar, MS Thesis, GA Tech., 2003]
YX
Positive Effects of CoalescingPositive Effects of Coalescing
YXXY
Reduce the degree of neighboring vertices
[Park and Moon, PACT 1998, TOPLAS 2004]
[Vegdahl, PLDI 1999]
Z ZDeg(Z) = 2 Deg(Z) = 1
Coalescing StrategiesCoalescing 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]
Biased ColoringBiased Coloring
YXX Y
Introduced by [Briggs et al., TOPLAS 1994] For whatever reason, we couldn’t 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
Chordal GraphsChordal Graphs
No chordless cycles of length 4 or more
Non-chordal Chordal
Non-chordal Still Non-chordal Chordal ChordalNon-Chordal
Coalescing and Chordal GraphsCoalescing and Chordal Graphs
Coalescing does not preserve chordality
Non-chordal
(After coalescing)
Chordal
(Before coalescing)
Pseudo-coalescingPseudo-coalescing
Find independent sets of move-related vertices, but don’t merge themTry to assign the same color to the vertices
Chordal
(After pseudo-coalescing)
Chordal
(Before pseudo-coalescing)
Simpicial VerticesSimpicial Vertices
v is a Simplicial Vertex if the neighbors of v form a clique.
Theorems:1. Every chordal graph has at
least one simplicial vertex [Lekkerkerker and Boland, Fund. Math 1964]
2. Every induced subgraph of a chordal graph is chordal
Perfect Elimination OrderPerfect Elimination Order
Undirected graph: G = (V, E)
Elimination Order One-to-one and onto function: σ: V {0, …, |V| - 1} Rename vertices: vi satisfies σ(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
Coloring Chordal GraphsColoring 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
Correctness ArgumentCorrectness Argument
Invariant:When we assign a color to vi, Gi is legally
colored
Optimistic extensions for copy eliminationExamples given in the next 3 slides Invariant above is always maintained
Color assignment is conservativeThe chromatic number is known, and never
changes
Simple Color Assignment RulesSimple Color Assignment Rules
PEO order: 1 2 3
That’s a really bad ideaNo harm doneBut that isn’t how it really happens…Let’s assume nodes 1 and 3 are pseudo-coalesced
Propagate color RED to node 3Try to respect the fact that node 3 has been optimistically colored
Only choose RED if no other color is available
If RED is the only option… then we’ll re-color node 3 later
Undo Pseudo-coalescingUndo Pseudo-coalescing
PEO order:
1
2 3
45
I can’t pseudo-coalesce all the nodes…
Assign colorPropagate colorLegal coloring invariant still holds
Pre-assigned color is legalPre-assigned color is illegalCannot stay pseudo-coalesced with node 2Undo pseudo-coalescingPseudo-coalesce with nodes 3 and 4 instead
Negative Biasing (Same Example)Negative Biasing (Same Example)
PEO order:
1
2 3
45
Select a Color for Node 2
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 REDAssign Color BLUE to Node 1 instead
Maximum Cardinality SearchMaximum Cardinality Search
Compute PEO order in O(|V| + |E|) time[Tarjan and Yannakakis, Siam J. Comput, 1984]
Color move-related nodes earlyA legal color is always found for each node…So coloring non-move-related node …
Constrains the move-related-nodes
Biased MCSPush move-related vertices toward the front of the
PEO
SimplificationSimplification If Deg(v) < k, remove v from G
[Kempe, American J. Math. 1876] A color can always be found for v later Don’t do this for move-related nodes
Removing them doesn’t help to eliminate moves
Repeat until all remaining nodes v… Have Deg(v) > k, or Are move-related
Justification Fewer coloring constraints on move-related vertices [Hack and Grund, CC 2007]
In the context of chordal coloring
RefinementRefinement
Sometimes heuristics just don’t do the right thing…Can’t hurt, right?
Optimistic Chordal Coloring Optimistic Chordal Coloring HeuristicHeuristic
1. Simplification
2. (Biased) Maximum Cardinality Search
3. Aggressive Pseudo-Coalescing
4. Optimistic Chordal Color Assignment
5. Refinement
6. Unsimplify
Register Allocation for ASIPsRegister Allocation for ASIPs
Proc1 Proc2 Procm…
Profile, Convert to Pruned SSA, Build Interference Graph
G1 G2 Gm
Chordal Color [Gavril, Siam J. Comp. 1972]
χ(G1) χ(G2) χ(Gm)
MAX
χmax – Number of registers to allocate (only spill at procedure calls)
Input Data
ExperimentsExperiments
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
ResultsResults
Benchmark
Dynamically Executed Copies
OPT IRC OCC OCC++
adpcm_rawcaudioadpcm_rawdaudioblowfishcrc32dijkstrafftg721_decodeg721_encodegsmjpeg_cjpegjpeg_djpegmpeg2decmpeg2encpatriciapegwitshasusan
69950166995016
053322406
08209
00
1909187541326272636
211595197
182048
4422
--------
+ 2990-
+ 6283---
+ 24--
--------
+ 2990-
+ 6283---
+ 1--
----------------
+ 1
4 graphs were solved sub-optimally by OCC -> motivated OCC++
ConclusionConclusion OCC Heuristic for chordal graphs
OptimisticPseudo-coalescing instead of node mergingPreserves the chordal graph property
ConservativeStronger 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 propagationShares principal similarities to biased coloring
sym_decryptsym_decrypt
1
2
3
4
56
89
10
11
12
7 Propagate
Color is illegal
13
Weakness in aggressive pseudo-coalescing phase
Rules for Color AssignmentRules for Color Assignment
If vi is move-related…and vi is uncolored
Be judicious when selecting a colorPropagate 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 nodesColor assignment affects move-related nodes
