Constraint Programming for Compiler Optimization March 2006

Constraint Programming for Compiler Optimization

March 2006

2

Acknowledgements

Joint work with:

Alexander Golynski

Alejandro López-Ortiz

Abid Malik

Jim McInnes

Claude-Guy Quimper

John Tromp

Kent Wilken

Funding:

NSERC

IBM Canada

3

Optimization problems in compilers

Instruction selection

Instruction scheduling·basic-block instruction scheduling·super-block scheduling ·software pipelining & loop unrolling

Register allocation

Memory hierarchy optimizations

4

Basic-block instruction scheduling

Schedule basic-block·straight-line sequence of code with single entry, single exit

Multiple-issue pipelined processors·multiple instructions can begin execution each clock cycle

·delay or latency before results are available

Find minimum length schedule

Classic problem·lots of attention in literature

5

Example: (a + b) + c

instructions

A r1 a

B r2 b

C r3 c

D r1 r1 + r2

E r1 r1 + r3

3 3

31

A B

D C

E

dependency DAG

6

Single-issue pipelined processor

non-optimal schedule

A r1 aB r2 b

nopnop

D r1 r1 + r2C r3 c

nopnop

E r1 r1 + r3

A B

D C

E

3 3

31

dependency DAG

7

Single-issue pipelined processor

optimal schedule

A r1 aB r2 bC r3 c

nopD r1 r1 + r2E r1 r1 + r3

A B

D C

E

3 3

31

dependency DAG

8

Multiple-issue pipelined processor

A B

D C

E

3 3

31

dependency DAG

A

issue width is 2

1

5

4

3

2

B

C

D

E

9

Multiple-issue pipelined processor

A B

D C

E

3 3

31

dependency DAG

A

issue width is 1+1

1

5

4

3

2

C

B

D

E6

10

Production compilers

“At the outset, note that basic-block scheduling is an NP-hard problem, even with a very simple formulation of the problem, so we must seek an effective heuristic, rather than exact, approach.”

Steven Muchnick,

Advanced Compiler Design

& Implementation, 1997

11

Optimal approachesstate-of-the-art

Single-issue

Previous·10-40 instructions

ILP (Arya, 1985)CP (Ertl & Krall, 1991)

·up to 1000 instructionsILP (Wilken et al, 2000)

Our work·up to 2600 instructions·20 × faster

Multiple-issue

Previous·10-40 instructions

ILP (Chang et al., 1997)DP (Kessler, 1998)

·up to 1000 instructionsB&B (Heffernan et al., 2005)

Our work·up to 2600 instructions·50-fold improvement

12

Constraint programming methodology

Model problem·specify in terms of constraints on acceptable solutions

·define/choose constraint model: variables, domains, constraints

Solve model·define/choose search algorithm·define/choose heuristics

13





14

Minimal constraint model

variables A, B, C, D, E

domains {1, …, m}

constraints D A + 3D B + 3E C + 3E D + 1gcc(A, B, C, D, E, width)

A B

D C

E

3 3

31

dependency DAG

15

Bounds consistency constraint propagation

[1, 3]

[4, 6]

variable ABCDE

domain [1, 6][1, 6][1, 6][1, 6][1, 6]

D A + 3constraints

[4, 5] [1, 3]

[4, 6]

[1, 3] [1, 2]

D B + 3 E C + 3 E D + 1 gcc(A, B, C, D, E, 1)

[5, 6]

[1, 2] [3, 3]

[6, 6]

16

Improvements to constraint model

1. Distance constraints •constraints over nodes which define regions

2. Predecessor and successor constraints•constraints over nodes with multiple

predecessors or multiple successors

3. Safe pruning constraint•global constraint

4. Dominance constraints•constraints based on graph isomorphism

17







18

Distance constraints: Regions

A pair of nodes i, j define a region in a DAG G if:

(i) there is more than one path from i to j, and

(ii) not all paths from i to j go through some node k distinct from i and j.

i

j

19

Distance constraints: Estimate

A

B

ED

H

F G

C

1

1

1

33

1

3

1

3

20


A

B

ED

H

F G

C

1

1

1

33

1

3

1

3j j+1j+2j+3j+4j+5

5

A F

21


A

B

ED

H

F G

C

1

1

1

33

1

3

1

3j j+1j+2j+3j+4j+5

E H

5

22


A

B

ED

H

F G

C

1

1

1

33

1

3

1

3

9

A

j j+1j+2j+3j+4j+5

j+6j+7j+8j+9

H

23

Distance constraints: Optimal

A

B

ED

H

F G

C

1

1

1

33

1

3

1

3

[1,1]

[10,10]

[2,3]

[5,6] [5,6]

[6,7] [6,7]

[2,3]

propagate latency

propagate all-diff

• Not optimal: A 1

H 10

• Estimate: H A + 9

24

Distance constraints: Optimal

• Optimal: H A + 10

A

B

ED

H

F G

C

1

1

1

33

1

3

1

3

[1,1]

[10,10]

[2,3]

[5,6] [5,6]

[6,7] [6,7]

[2,3]

propagate latency

• Not optimal: A 1

H 10

• Estimate: H A + 9

propagate all-diff

inconsistent

25







26

Predecessor constraints

[4, ]

3

1A

B

DC E

H

F G

3 3

22

11

1

[ ,14]

[5,9]

[8,12][9,12]

[5,9][6,9]

[5,8]

7

11

27


D E

G

A

B

C

H

F

[4, ]

11

3

1

2

1

2[ ,14]

3 3

[5,9]

[8,12][9,12]

[5,9][6,9]

[5,8]

7

11

[9,12]

5 6 7 8 9

28


[4, ]

3

1A

B

DC E

H

F G

3 3

22

11

1

[ ,14]

[5,9]

[8,12][9,12]

[5,9][6,9]

[5,8]

7

11

[9,12]

[12,14]

9 10 11 12

29

Successor constraints

[4, ]

3

1A

B

DC E

H

F G

3 3

22

11

1

[ ,14]

[5,9]

[8,12][9,12]

[5,9][6,9]

[5,8]

7

11

[9,12]

[12,14]

[4,6]

6 7 8 9

30





31

Solving instances of the model

Use constraints to establish:·lower bound on length m of optimal schedule·min and max of domains of variables

Backtracking search·branches on min(x), min(x)+1, … ·interleave with bounds consistency constraint propagation

·fallback: singleton consistency on bounds

If no solution found, increment m and repeat search

32


A

B

C

D

1 2 4 5

E

A B

D C

E

3 3

31

[1,5] [1,5]

[1,5] [1,5]

[1,5]

33


A

B

C

D

1 2 4 5

E

A B

D C

E

3 3

31

[ ] [ ]

[ ] [ ]

[ ]

34


A

B

C

D

1 2 5 6

E

A B

D C

E

3 3

31

[1,6] [1,6]

[1,6] [1,6]

[1,6]

35


A

B

C

D

1 2 5 6

E

A B

D C

E

3 3

31

[1,2] [1,2]

[5,5] [3,3]

[6,6]

36

Improvements to constraint solver

Design special purpose constraint propagators

·commonly occurring constraints·significantly improve efficiency

Improved algorithms for bounds consistency·all-diff constraint ·gcc constraint

37

Comparing all-diff propagators (prototype)

n DC MT BC 69 0.0 0.0 0.0 70 0.1 0.0 0.0

111 0.1 0.1 0.1 211 >600.0 >600.0 >600.0 214 1.2 0.9 0.5 216 1.1 0.7 0.4 220 0.9 0.7 0.3 690 26.9 4.0 1.6 856 17.1 10.9 5.5

1006 87.2 15.9 6.0

Time (sec.) to solve instruction scheduling problems; model includes latency, distance, and all-diff constraints.

DC: Régin, 1994; MT: Mehlhorn & Thiel, 2000; BC: IJCAI-

2003

38

Comparing gcc propagators (prototype)

n DC vH BC 69 0.0 0.0 0.0 70 0.1 0.0 0.0

111 0.8 0.0 0.0 211 9.2 0.5 0.1 214 9.3 0.6 0.1 216 124.1 2.7 0.3 220 285.9 5.1 0.5 690 493.2 1.3 1.7 856 471.2 >600.0 3.8

1006 >600.0 >600.0 8.7

Time (sec.) to solve instruction scheduling problems; model includes latency and gcc constraints; width is 2.

DC: Régin, 1996; vH: van Hentenryck et al., 1992; BC: CP-

2003

39

Putting it all together: Experimental results

Architectures

1-issue 2-issue 4-issue 6-issue

Improved 4383 5498 6031 2759

Timed out 2 25 18 14

SPEC 2000 & MediaBench Benchmarks

Total of 352,111 basic blocks of size 3 or greater

Improved = improved schedule over heuristic scheduler

Timed out = not solved within 10 minutes

40

Putting it all together: Experimental results

Architectures Percentage Improvement 1-issue 2-issue 4-issue 6-issue

Average 5.3 5.5 5.9 5.0

Maximum 17.6 25.0 28.0 25.0

SPEC 2000 & MediaBench Benchmarks

For basic blocks with improved schedules

41

Conclusions

CP approach to instruction scheduling·Single-issue processors

20-times faster than previous best optimal approach

·Multiple-issue processorslarger and more difficult problems50-fold reduction in number of problems that

cannot be solved

Constraint propagators·faster all-diff and gcc constraint propagators·useful in many problems

42

Current and future work: Expand scope of problem

Instruction selection

Instruction scheduling·basic-block instruction scheduling·super-block scheduling ·software pipelining & loop unrolling

Register allocation

Memory hierarchy optimizations

Documents

Constraint Programming for Compiler Optimization March 2006