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An Optimistic and Conservative Register Assignment Heuristic for Chordal Graphs

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### 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

Outline

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

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

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]

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

Deg(X) = 3

Y

X

Deg(XY) = 6

Deg(Y) = 4

Negative Side Effects of CoalescingVertex degree increases

- [Briggs et al., TOPLAS 1994]
- [George and Appel, TOPLAS 1996]
- [Hailperin, TOPLAS 2005]

Y

XY

Negative Side Effects of CoalescingChromatic number increases

[Briggs et al., TOPLAS 1994]

[George and Appel, TOPLAS 1996]

Triangles form

[Kaluskar, MS Thesis, GA Tech., 2003]

X

Z

Z

Positive Effects of CoalescingReduce the degree of neighboring vertices

- [Park and Moon, PACT 1998, TOPLAS 2004]
- [Vegdahl, PLDI 1999]

Deg(Z) = 2

Deg(Z) = 1

XY

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]

X

X

Y

Biased Coloring- 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

Non-chordal

Still Non-chordal

Chordal

Chordal

Non-Chordal

Chordal Graphs- No chordless cycles of length 4 or more

(Before coalescing)

Non-chordal

(After coalescing)

Coalescing and Chordal Graphs- Coalescing does not preserve chordality

Pseudo-coalescing

- Find independent sets of move-related vertices, but don’t merge them
- Try to assign the same color to the vertices

Chordal

(Before pseudo-coalescing)

Chordal

(After pseudo-coalescing)

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

Perfect 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 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 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

Try 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

Propagate color RED to node 3

Simple Color Assignment Rules1

3

PEO order:

2

That’s a really bad idea

No harm done

But that isn’t how it really happens…

Let’s assume nodes 1 and 3 are pseudo-coalesced

Undo Pseudo-coalescing

1

PEO order:

2

3

5

4

Pre-assigned color is legal

Pre-assigned color is illegal

Assign color

Propagate color

Undo pseudo-coalescing

Cannot stay pseudo-coalesced with node 2

Pseudo-coalesce with nodes 3 and 4 instead

Legal coloring invariant still holds

I can’t pseudo-coalesce all the nodes…

Negative Biasing (Same Example)

1

PEO order:

2

3

5

4

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 RED

Assign Color BLUE to Node 1 instead

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

Simplification

- 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

Optimistic Chordal Coloring Heuristic

- Simplification
- (Biased) Maximum Cardinality Search
- Aggressive Pseudo-Coalescing
- Optimistic Chordal Color Assignment
- Refinement
- Unsimplify

Register Allocation for ASIPs

Profile, Convert to Pruned SSA, Build Interference Graph

Input

Data

Proc1

Proc2

…

Procm

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)

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

Results

4 graphs were solved sub-optimally by OCC -> motivated OCC++

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

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

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