On the Complexity of Register Coalescing F . Bouchez, A. Darte, and - - PowerPoint PPT Presentation

Coalescing

On the Complexity of Register Coalescing

F . Bouchez, A. Darte, and F . Rastello

LIP UMR CNRS — Inria — ENS Lyon — UCBL France CGO’07, San Jose

Coalescing

Outline

1

What, Why, and How to Coalesce Basic Formulation The Different Approaches

2

A Hard Optimization Problem

3

Conclusion: What should we Implement Now?

Coalescing What, Why, and How to Coalesce

Outline

1

What, Why, and How to Coalesce Basic Formulation The Different Approaches

2

A Hard Optimization Problem

3

Conclusion: What should we Implement Now?

Coalescing What, Why, and How to Coalesce Basic Formulation

Coalescing: Coloring the Interference/Affinity Graph

Coalescing is rename 2 variables into a unique representant MOVE A, B: an affinity between A and B A and B cannot share the same ressource: an interference between A and B

b = . . . c = . . . · · · = c a = . . . d = b d = a · · · = d · · · = a

Coalescing What, Why, and How to Coalesce Basic Formulation

Coalescing: Coloring the Interference/Affinity Graph

Coalescing is rename 2 variables into a unique representant MOVE A, B: an affinity between A and B A and B cannot share the same ressource: an interference between A and B

b = . . . c = . . . · · · = c a d = . . . a d = b a d = a d · · · = a d · · · = a d

Coalescing What, Why, and How to Coalesce Basic Formulation

Coalescing: Coloring the Interference/Affinity Graph

Coalescing is rename 2 variables into a unique representant MOVE A, B: an affinity between A and B A and B cannot share the same ressource: an interference between A and B

b = . . . c = . . . · · · = c a d = . . . a d = b a d = a d · · · = a d · · · = a d

b c a d Coalesce b c ad

Coalescing What, Why, and How to Coalesce Basic Formulation

Many MOVE...

Many MOVE instructions due to register constraints (function call, 2 address instructions, etc.) SSA construction followed by basic SSA destruction

A = . . . B = . . . MOVE R0, A MOVE R1, B D = f(A, B) MOVE D, R0

C2 = . . . C1 = . . . MOVE C, C1 MOVE C, C2 C = φ(C1, C2) use C

Coalescing What, Why, and How to Coalesce Basic Formulation

Many MOVE...

Many MOVE instructions due to register constraints (function call, 2 address instructions, etc.) SSA construction followed by basic SSA destruction

A = . . . B = . . . MOVE R0, A MOVE R1, B call f MOVE D, R0

C2 = . . . C1 = . . . MOVE C, C1 MOVE C, C2 C = φ(C1, C2) use C

Coalescing What, Why, and How to Coalesce Basic Formulation

Many MOVE...

Many MOVE instructions due to register constraints (function call, 2 address instructions, etc.) SSA construction followed by basic SSA destruction

A = . . . B = . . . MOVE R0, A MOVE R1, B call f MOVE D, R0

C2 = . . . C1 = . . . MOVE C, C1 MOVE C, C2 C = φ(C1, C2) use C

Coalescing What, Why, and How to Coalesce Basic Formulation

Many MOVE... to remove

Many MOVE instructions due to register constraints (function call, 2 address instructions, etc.) SSA construction followed by basic SSA destruction Our past experience

pre-sched. (Iterared)

• reg. alloc.

sched. (aggressive) coalescing during SSA destruction

• n most benchmarks, a good speedup
• n some of them, slow down!

Coalescing What, Why, and How to Coalesce The Different Approaches

Analysing the slow down...

Aggressive coalescing may lead to spilling. Coalescing aware

• f colorability is conservative.

b c a d Aggressively coalesce

Coalescing What, Why, and How to Coalesce The Different Approaches

Analysing the slow down...

Aggressive coalescing may lead to spilling. Coalescing aware

• f colorability is conservative.

b c a d Aggressively coalesce b c ad

Coalescing What, Why, and How to Coalesce The Different Approaches

Analysing the slow down...

Aggressive coalescing may lead to spilling. Coalescing aware

• f colorability is conservative.

b c a d 2-colorable Aggressively coalesce b c ad Not 2-colorable

Coalescing What, Why, and How to Coalesce The Different Approaches

Analysing the slow down...

Aggressive coalescing may lead to spilling. Coalescing aware

• f colorability is conservative.

b c a d 2-colorable Conservatively coalesce b c a bd 2-colorable

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e 2

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g 1

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c m

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c m f

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c m f l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c m f l k

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c d e f g h i m j k l j k l f m c a d b e g h i h g e b d a c m f l k j

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f df Not greedy-3-colorable g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d fg Not greedy-3-colorable j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g j k l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g jk greedy-3-colorable l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g jk l

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e h i m f d g jk jkl greedy-3-colorable

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

a b c e dfg h i m jkl kl j

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Aggressive coalescing

not k-col

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Conservative coalescing

k-greedy k-col

Aggressive coalescing

not k-col

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Incremental conservative coalescing

Conservative coalescing k-greedy k-col

Aggressive coalescing

not k-col

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Decoalescing

Incremental conservative coalescing Conservative coalescing k-greedy k-col

Aggressive coalescing

not k-col

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Decoalescing

Incremental conservative coalescing Conservative coalescing k-greedy k-col

Aggressive coalescing

not k-col

Coalescing What, Why, and How to Coalesce The Different Approaches

...and the speedup

k-colorability check is hard, but checking the Greedy-k-colorablility is easy. Still, finding the optimal subset of affinities is hard. We do Incremental coalescing... Incremental is not optimal. Decoalescing is better.

• Decoalescing

Optimistic (Park et al.) Rules (Briggs, George) Incremental conservative coalescing Conservative coalescing k-greedy k-col

Aggressive coalescing

not k-col

Coalescing A Hard Optimization Problem

Outline

1

What, Why, and How to Coalesce Basic Formulation The Different Approaches

2

A Hard Optimization Problem

3

Conclusion: What should we Implement Now?

Coalescing A Hard Optimization Problem

Coalescing is Hard

G: greedy-k-colorable interference graph, Aggressive coalescing NP-complete, even with k = 3. Conservative coalescing NP-complete even if k = 3 and only affinities can be merged. Incremental conservative coalescing (Briggs, George) NP-complete if G is arbitrary. Open if G is greedy-k-colorable. Optimistic coalescing (Park & Moon) = conservative de-coalescing NP-complete even if k = 4.

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

A little hope: Chordal Graphs

Interference graph of SSA programs k-colorability easy on chordal graphs MAXLIVE = w(G) = χ(G) k-chordal ⊂ greedy-k-colorable ⊂ k-colorable

a b c d e

Coalescing A Hard Optimization Problem

. . . but Coalescing is still Hard

G: k-chordal interference graph. Aggressive coalescing NP-complete. Conservative coalescing NP-complete. Incremental conservative coalescing (Briggs, George) Polynomial! Optimistic coalescing (Park & Moon) = conservative de-coalescing NP-complete even if k = 4.

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

NP simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

NO yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

yes/no? NO yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

NO yes/no? NO yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

YES yes/no? NO yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing A Hard Optimization Problem

“Multiple-move” incremental

?? NP P simple test k-chordal k-greedy k-col k-chordal k-greedy k-col One move to coalesce

additional coalescings YES yes/no? NO yes/no? k-chordal k-greedy k-col k-chordal k-greedy k-col

Coalescing Conclusion: What should we Implement Now?

Outline

1

What, Why, and How to Coalesce Basic Formulation The Different Approaches

2

A Hard Optimization Problem

3

Conclusion: What should we Implement Now?

Coalescing Conclusion: What should we Implement Now?

Some measurements

Aggressive+decoalescing scheme:

• ptimizing the aggressive part is important

decoalescing (optimistic) can still be improved Incremental scheme: conservative rules (Briggs, George) are far from the

• ptimal

Coalescing Conclusion: What should we Implement Now?

Some measurements

Aggressive+decoalescing scheme:

• ptimizing the aggressive part is important

decoalescing (optimistic) can still be improved Incremental scheme: conservative rules (Briggs, George) are far from the

• ptimal

Coalescing Conclusion: What should we Implement Now?

Some measurements

Aggressive+decoalescing scheme:

• ptimizing the aggressive part is important

decoalescing (optimistic) can still be improved Incremental scheme: conservative rules (Briggs, George) are far from the

• ptimal

Coalescing Conclusion: What should we Implement Now?

Conclusion

Most problems are NP-complete ⇒ heuristics! Aggressive+decoalescing scheme: Aggressive coalescing is an important issue! Still gap for improving decoalescing; Incremental scheme: A large gap for incremental; Promizing approach: multiple-move incremental on k-greedy.