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Register Allocation after Classical SSA Elimination is NP-complete Fernando M Q Pereira Jens Palsberg - UCLA - The University of California, Los Angeles

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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?

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Chaitin’s Proof CHAITIN, G., AUSLANDER, M., CHANDRA, A., COCKE, J., HOPKINS, M., AND MARKSTEIN, P. Register allocation via coloring. Computer Languages 6, 1, , Graph Coloring Register Allocation

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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?

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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: :00, Friday, Session 4

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Polynomial Time RA? The core Register Allocation Problem. The core Register Allocation Problem. - SSA form - Polynomial mapping ??? Polynomial Time NP-complete Polynomial Iff P = NP 12

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SSA-Elimination 1) int m(int p 1,int p 2 ){ 2) int v 1 = p 1 ; 3) int i = p 2 ; 4) while(i < 10){ 5) i = i+1; 6) if(v 1 > 11) 7) break; 8) v 1 = i+2; 9) } 10) return v 1 ; 11) } int m(int p 1, int p 2 ) { int v 11 = p 1 ; int i 1 = p 2 ; int v 1 = v 11 ; int i = i 1 ; while (i < 10) { int i 2 = i + 1; if (v 1 > 11) break; int v 12 = i 2 + 2; v 1 = v 12; i = i 2 ; } return v 1 ; } Target program SSA-form Program Post-SSA Program

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Target program SSA-form Code We’ve shown that the core register allocation problem is NP-complete. The Main Conclusion NP-complete register allocation Polynomial register allocation Hack et al CC’06 Classical SSA elimination

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What about Chaitin’s Proof? G G’ C(G) = C(G’) - 1

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

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Three Registers are Enough.

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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.

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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) } d 1 = … ; c 1 = … ; a = c; b = d; c 2 = a+1; d 2 = b+1; dcdc d 1,d 2 c 1,c 2 = c b c d b c2c2 d2d2 a a c2c2 d1d1 d1d1 d d2d2 c1c1 c1c1 Non-SSA-formSSA-form

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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 c2c2 b c a a c=c 2 d2d2 d d=d 2 d 1 = … ; c 1 = … ; a = c; b = d; c 2 = a+1; d 2 = b+1; dcdc d 1,d 2 c 1,c 2 = c b c d b c2c2 d2d2 a a c2c2 d d2d2 Post-SSA program Post-SSA graph

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Color Mapping b c2c2 b c a a c=c 2 d2d2 d d=d 2 We want to show that coloring circular-arc graphs, and post-SSA graphs is equally hard.

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Can G be colored with K colors? b c b c a a c=c 2 d2d2 d d=d 2 nK-n n = variables crossing the loop; K = number of colors

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Example of Post-SSA-Graph n = 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.

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Mapping Graphs to Programs int a = C 1 + i; If(a > C 2 ) break; Post-SSA-graph function with loop Arc Live range of variable. How to avoid compiler optimizations?

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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; }

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K = 3, n = 2 one auxiliary t int 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; }

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The Example in SSA-form a 1 = … ; e 1 = … ; t 1 = … ; i 1 = … ; i 2 = i + 1; if(t > 9) break; if(e > 10) break; b = i ; if(a > 11) break; c = i ; if(b > 12) break; d = i ; if(c > 13) break; e 2 = i ; if(d > 14) break; a 2 = i ; t 2 = i ; actiacti a 1,a 2 e 1,e 2 t 1,t 2 i 1,i 2 =

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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; > 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

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Summary of the Reduction Circular-arc graph Post-SSA graph Simple Post-SSA Program Register Assignment Register To 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; > i = i2; a = a2; t = t2; e = e2; } return a; }

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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.

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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.

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