Fast optimal instruction scheduling for single-issue processors with arbitrary latencies

Peter van Beek, University of WaterlooKent Wilken, University of California, DavisCP 2001 · Paphos, CyprusNovember 2001

2

Local instruction scheduling

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

Single-issue pipelined processors·single instruction can begin execution each clock cycle·delay or latency before result is available

Classic problem·lots of attention in literature

Remains important·single-issue RISC processors used in embedded systems

3

Example: evaluate (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

4

Example: evaluate (a + b) + c

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

5

Example: evaluate (a + b) + c

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

6

Local instruction scheduling problem

Given a labeled dependency DAG G = (N, E) for a basic block, find a schedule S that specifies a start time S( i ) for each instruction such that

S( i ) S( j ), i, j N, i j,

and

S( j ) S( i ) + latency( i, j ), ( i, j ) E,

and

max{ S( i ) | i N } is minimized.

7

Previous work

NP-Complete if arbitrary latencies (Hennessy & Gross, 1983; Palem & Simons,

1993)

Polynomial special cases(Bernstein & Gertner, 1989; Palem & Simons,

1993; Wu et al., 2000)

Optimal algorithms·dynamic programming (e.g., Kessler, 1998)·integer linear programming (e.g., Wilken et al., 2000)·constraint programming (e.g., Ertl & Krall, 1991)

8

Minimal constraint model

variables A, B, C, D, E

domains {1, …, m}

constraints D A + 3D B + 3E C + 3E D + 1all-diff(A, B, C, D, E)

A B

D C

E

3 3

31

dependency DAG

9

Bounds consistency

[1, 3]

[4, 6]

variable ABCDE

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

D A + 3constraints

D B + 3 E C + 3 E D + 1 all-diff(A, B, C, D, E)

[4, 5] [3, 3]

[6, 6]

[1, 2] [1, 2]

For each constraint C and for each variable x in C, min has a support in C and max has a support in C

10

Three improvements to minimal model

1. Initial distance constraints •defined over nodes which define regions

2. Improved distance constraints for small regions

3. Predecessor and successor constraints•defined over nodes with multiple predecessors

or multiple successors

11

Three improvements to minimal model

1. Initial distance constraints •defined over nodes which define regions

2. Improved distance constraints for small regions

3. Predecessor and successor constraints•defined over nodes with multiple predecessors

or multiple successors

12

Distance constraints: RegionsA 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.

13

Distance constraints: Initial estimate

A

B

ED

H

F G

C

1

1

1

33

1

3

1

3

14

Distance constraints: Initial estimate

A

B

ED

H

F G

C

1

1

1

33

1

3

1

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

5

A F

15

Distance constraints: Initial estimate

A

B

ED

H

F G

C

1

1

1

33

1

3

1

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

E H

5

16

Distance constraints: Initial estimate

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

17

Three improvements to minimal model

1. Initial distance constraints •defined over nodes which define regions

2. Improved distance constraints for small regions

3. Predecessor and successor constraints•defined over nodes with multiple predecessors

or multiple successors

18

Improved distance constraints for small regions

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

• Extract region from DAG

• Post constraints

• Test consistency of A 1

H 10

• Given H A + 9

19

Improved distance constraints for small regions

• Repeat with H A + 10

• Extract region from DAG

• Post constraints

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

• Test consistency of A 1

H 10

• Given H A + 9

propagate all-diff

inconsistent

20

Three improvements to minimal model

1. Initial distance constraints •defined over nodes which define regions

2. Improved distance constraints for small regions

3. Predecessor and successor constraints•defined over nodes with multiple predecessors

or multiple successors

21

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

22

Predecessor constraints

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

23

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

[9,12]

[12,14]

9 10 11 12

24

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

25

Solving instances of the model

Use constraints to establish:·lower bound on length m of optimal schedule·lower and upper bounds of variables

Backtracking search·maintains bounds consistency

Puget’s (1998) all-diff propagator and optimizationsLeconte’s (1996) optimizations

·branches on lower(x), lower(x)+1, …

If no solution found, increment m and repeat search

26

Experimental results

Embedded in Gnu Compiler Collection (GCC)

Compared with:·GCC’s critical path list scheduling·ILP scheduler (Wilken et al., 2000)

SPEC95 floating point benchmarks·compiled using highest level of optimization (-O3)

Target processor:·single-issue·latency of 3 for loads, 2 for floating point, 1 for integer ops

27

Experimental results: SPEC95 floating point benchmarks

Total basic blocks (BB)

BB passed to CSP scheduler

BB solved optimally by CSP scheduler

BB with improved schedule

Static cycles improved

Total benchmark cycles

CSP scheduling time (sec.)

Baseline compile time (sec.)

7,402

517

517

29

66

107,245

4.5

708

28

Scheduling time for CSP and ILP schedulers

29

Quantifying contributions ofthree model improvements

0 5 10 15

full constraint model

mm + improved distance

mm + initial distance

mm + pred + succ

mm = latency + alldiff

10

100

1000

Problems solved (/15)

30

Conclusions

CP approach to local instruction scheduling·single-issue processors·arbitrary latencies

Optimal and fast on very large, real problems

·experimental evaluation on SPEC95 benchmarks·20-fold improvement over previous best approach

Key was an improved constraint model

31

Good ideas not included

Cycle cutsets (e.g., Dechter, 1990)·most larger problems had small cutsets (2 to 20 nodes) that split problem into equal-sized independent subproblems

Singleton consistency (e.g., Prosser et al., 2000)·often reduced domains dramatically prior to search

Symmetry breaking constraints·many symmetric (non) schedules