Register Allocationafter Classical SSA

Elimination is NP-complete

Fernando M Q Pereira

Jens Palsberg

- UCLA -

The University of California, Los Angeles

The Core Register Allocation Problem

• Instance: a program P and a number K of available registers.

• Problem: can each of the temporaries of P be mapped to one of the K registers such that temporary variables with interfering live ranges are assigned to different registers?

Chaitin’s Proof

• CHAITIN, G., AUSLANDER, M., CHANDRA, A., COCKE, J., HOPKINS, M., AND MARKSTEIN, P. Register allocation via coloring. Computer Languages 6, 1, 47--57, 1981.

• Graph Coloring Register Allocation

Register Allocation in SSA-form

• Instance: a program P, in SSA form, and a number K of available registers.

• Problem: can each of the temporaries of P be mapped to one of the K registers such that temporary variables with interfering live ranges are assigned to different registers?

Polynomial Time RA

• Sebastian Hack, Daniel Grund, and Gerhard Goos, Register Allocation for Programs in SSA-form, University of Karlsruhe, Germany, CC, 2006

• CC, 16:30 - 18:00, Friday, Session 4

Polynomial Time RA?

The coreRegister

AllocationProblem.

The coreRegister

AllocationProblem.

- SSA form -

Polynomial mapping

???

PolynomialTime

NP-complete PolynomialIff P = NP

1 2

SSA-Elimination

1) int m(int p1,int p2){2) int v1 = p1;3) int i = p2;4) while(i < 10){5) i = i+1;6) if(v1 > 11)7) break;8) v1 = i+2;9) }10) return v1; 11) }

int m(int p1, int p2) { int v11 = p1; int i1 = p2; int v1 = v11; int i = i1; while (i < 10) { int i2 = i + 1; if (v1 > 11) break; int v12 = i2 + 2; v1 = v12;

i = i2; } return v1; }

Target program

SSA-form ProgramPost-SSA Program

Targetprogram

SSA-form

Code Code

We’ve shown thatthe core register

allocation problemis NP-complete.

The Main Conclusion

NP-complete register allocation

Polynomialregister

allocation

Hack et alCC’06

ClassicalSSA

elimination

What about Chaitin’s Proof?

G G’

C(G) = C(G’) - 1

Chaitin’s Proof does not Work for SSA-form Programs

Three Registers are Enough.

Circular-Arc Graphs

• Theorem: to find a minimal coloring for a circular-arc graph is NP-complete.– Garey, Johnson and Stockmeyer. Some simplified

NP-complete problems, ACM Symposium on Theory of Computing, 47-63, 1974.

Arcs and Loops

1) int d = …;2) int c = …;3) while ( … ) {4) int a = c;5) int b = d;6) c = a+1;7) d = b+1;8) }

d1 = …;c1 = …;

a = c;b = d;c2 = a+1;d2 = b+1;

dc

d1,d2

c1,c2=

c

b

c

d

b

c2

d2

a

ac2

d1

d1

d

d2

c1

c1

Non-SSA-form SSA-form

Value Related Live Ranges

1) int d1 = …;2) int c1 = …;3) d = d1;4) c = c1;5) while ( … ) {6) int a = c;7) int b = d;8) c2 = a+1;9) d2 = b+1;10) d = d2;11) c = c2;12) }

b

c2

b

ca

a

c=c2

d2

d

d=d2

d1 = …;c1 = …;

a = c;b = d;c2 = a+1;d2 = b+1;

dc

d1,d2

c1,c2=

c

b

c

d

b

c2

d2

a

ac2

d

d2

Post-SSAprogram

Post-SSA graph

Color Mapping

b

c2

b

ca

a

c=c2

d2

d

d=d2

We want to show that coloringcircular-arc graphs, andpost-SSA graphs is equally hard.

Can G be colored with K colors?

b

c

b

ca

a

c=c2

d2

d

d=d2

n K-n

n = variables crossing the loop; K = number of colors

Example of Post-SSA-Graphn = 2; K = 3

• Circular-arc graph has K coloring iff post-SSA-graph has K coloring.

• It is possible to map post-SSA-graphs to post-SSA programs.

Mapping Graphs to Programs

int a = C1 + i;

If(a > C2) break;

• Post-SSA-graph function with loop

• Arc Live range of variable.

• How to avoid compiler optimizations?

A Simple Example

int m(int a,int e,int i){ while(i < 100) { i = i + 1; if(e > 10) break; int b = i + 11; if (a > 11) break; int c = i + 12; if(b > 12) break; int d = i + 13; if(c > 13) break; e = i + 14; if(d > 14) break; a = i + 15; } return a;}

K = 3, n = 2 one auxiliary tint m(int a,int e,int t,int i){ while(i < 100) { i = i + 1; if(t > 9) break; if(e > 10) break; int b = i + 11; if (a > 11) break; int c = i + 12; if(b > 12) break; int d = i + 13; if(c > 13) break; e = i + 14; if(d > 14) break; a = i + 15; t = i + 16; } return a;}

The Example in SSA-forma1 = …;e1 = …;t1 = …;i1 = …;

i2 = i + 1;if(t > 9) break;if(e > 10) break;b = i2 + 11;if(a > 11) break;c = i2 + 12;if(b > 12) break;d = i2 + 13;if(c > 13) break;e2 = i2 + 14;if(d > 14) break;a2 = i2 + 15;t2 = i2 + 16;

acti

a1,a2

e1,e2

t1,t2

i1,i2

=

After SSA-elimination …int m(int a1, int e1, int t1, int i1) { int a = a1; int e = e1; int t = t1; int i = i1; while(i > 10) { int i2 = i + 1; << main loop >> i = i2; a = a2; t = t2; e = e2; } return a;}

if(t > 9) break;if(e > 10) break;int b = i2 + 11;if (a > 11) break;int c = i2 + 12;if(b > 12) break;int d = i2 + 13;if(c > 13) break;int e2 = i2 + 14;if(d > 14) break;int a2 = i2 + 15;int t2 = i2 + 16;

K + 1 register assignment

K coloring

Summary of the Reduction

Circular-arcgraph

Post-SSAgraph

SimplePost-SSAProgram

RegisterAssignment

RegisterTo colors

Arcs to arcs

int m(int a1,int e1, int t1,int i1){ int a = a1; int e = e1; int t = t1; int i = i1; while(i < 100) { int i2 = i + 1; << main loop >> i = i2; a = a2; t = t2; e = e2; } return a;}

Other NP-completeness Proofs

• Ravi Sethi, Complete register allocation problems, 73.– NP-complete for strait line code with rescheduling.

• Farach and Liberatore, On Local Register Allocation, 98.– which register to spill is NP-complete.

• Bodlaender and Gustedt, Linear Register Allocation for a Fixed Number of Registers, 98.

• SSA-form programs have chordal interference graphs: three different proofs: Hack et al, Brisk et al, Bouchez, 05.

• Polynomial register allocation: Hack 06.

Conclusions

• Register allocation is NP-complete if classical algorithms are used to eliminate -functions.

• Proof is independent on the order in which copy instructions are inserted.

• Post-SSA programs can be built with a simple if-then-else statement.