Register Allocation. Register Allocation

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1 Rgistr Allocation Jingk Li Portlan Stat Univrsity Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 1 / 28 Rgistr Allocation Assign an unboun numbr of tmporaris to a fix numbr of rgistrs. Exampl: loa a, %r1 t1 := a-b; loa b, %r2 t2 := c+b; sub %r1, %r2, %r1 t3 := t1*; loa c, %r3 t4 := t2+t3; a %r2, %r3, %r2 t5 := t1*t3; loa, %r3 t6 := t4-t5; mul %r1, %r3, %r a %r2, %r3, %r2 mul %r1, %r2, %r1 sub %r2, %r1, %r1 Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 2 / 28

2 Rgistr Allocation Approachs A Naiv First-Com-First-Srv Approach Trat all rgistrs qual; allocat rgistrs along co-gnration; first-com-first-srv; if run out of rgistrs, ump som to mmory. Just kp track of th usag of ach rgistr an hanl rqusts; vry simpl an fast. Catgory-Bas Approach Classify program valus into catgoris (i.. basic arsss, arithmtic computations, tc.); an assign a group of rgistrs to ach catgory. From thr, follow th naiv approach. But ths approachs ar in gnral not goo nough. In allocating rgistrs, w hav an optimization goal: to minimiz mmory oprations (loas & stors) an rgistr movs, which motivats th following approach: Usag-Bas Approach Analyz program to collct valus-usag information; thn try to put (an kp) frquntly-us valus in rgistrs. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 3 / 28 Usag-Bas Rgistr Allocation Th usag analysis can b prform at iffrnt scop lvls: Local Rgistr Allocation Analyz variabl usags in a basic block, an optimiz bas on th information. It is simpl an fast. Rgistr Allocation for Loops Focus usag analysis on a loop. N to intify loops; othrwis still simpl an fast. Global Rgistr Allocation Analyz variabl usags across a whol procur (via ataflow analysis), an optimiz bas on th information. Can provi rally fficint usag of rgistrs. Vry usful for machins with only a small numbr of rgistrs availabl. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 4 / 28

3 Local Rgistr Allocation Consir a singl basic block. Assign rgistrs bas on usag count of tmps. Rsrv a fw rgistrs (2-3) to hanl mmory accsss (in cass whr th numbr of rgistrs is smallr than th numbr of tmps). Algorithm: 1. Comput a priority for ach tmp In a linar pass ovr th instructions in th block, tally th numbr of occurrncs of ach tmp. Th occurrnc count for a tmp bcoms its priority. 2. Sort th tmps into priority orr. 3. Assign rgistrs in priority orr If thr ar k rgistrs availabl, th first k tmps gt assign to thm. 4. Rwrit co Walk ovr th block a scon tim. Rplac ach rg-boun tmp with th rgistr s nam, an rplac othr tmps with loa/stor instructions. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 5 / 28 An Exampl Assum two rgs ar rsrv for mmory accsss, an two rgs ar availabl for assignmnt t1 := a-b; loa a, %r1 loa b, %r2 t2 := c+b; sub %r1, %r2, %r3 t3 := t1*; loa c, %r1 loa b, %r2 t4 := t2+t3; a %r1, %r2, %r1 stor %r1, mm[0] // spill t2 t5 := t1*t3; loa, %r1 t6 := t4-t5; mul %r3, %r1, %r4 loa mm[0], %r1 // rloa t a %r1, %r4, %r1 Usag counts: stor %r1, mm[1] // spill t4 mul %r3, %r4, %r1 a b c t1 t2 t3 t4 t5 t6 stor %r1, mm[2] // spill t loa mm[1], %r1 // rloa t4 loa mm[2], %r2 // rloa t5 Assignmnts: sub %r1, %r2, %r1 stor %r1, mm[3] // spill t6 t1: %r3, t3: %r4 Obsrvation: Furthr optimization possibl, but ns tmps livnss information. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 6 / 28

4 Rgistr Allocation for Loops Usag Counts: us(x,b) th numbr of tims variabl x is us in basic block B prior to any finition of x. liv(x,b) is 1 if x is aliv on xit from B an is assign a valu in B; is 0 othrwis. b,c,,f a := b+c := -b := a+f B1 us liv a,c,, f := a- B2 b,c,,,f b := +c a,c,,f b := +f := a-c B4 B3 b,,,f B1 B2 B3 B4 B1 B2 B3 B4 a b c f b,c,,,f Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 7 / 28 Calculating th Bnfits: RgAlloc for Loops (cont.) Th bnfit from allocating a rgistr to x within loop L: cost l us(x, B) + cost st liv(x, B) B L Th first trm is th cost of arssing valu in mmory an th scon trm is th cost of storing valu into mmory. Assum cost l = cost st = 1, thn bnfit(a, L) = 3 bnfit(b, L) = 4 bnfit(c, L) = 3 bnfit(, L) = 5 bnfit(, L) = 1 bnfit(f, L) = 3 Assum that w hav two rgistrs availabl for this loop. Thn b an woul b th winnrs. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 8 / 28

5 RgAlloc for Loops (cont.) Rgistr Assignmnt: b an ar kpt in rgistrs. mov b,r1 mov,r2 mov R1,R0 a c,r0 sub R1,R2 mov R0,R3 a f,r3 mov R1, B1 mov R0,R3 sub R2,R3 mov R3,f B2 mov R2,R1 a f,r1 mov R0,R3 sub c,r3 mov R3, B3 mov R2,R1 a c,r1 B4 mov R1,b a R2, mov R1,b a R2, Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 9 / 28 Rgistr Allocation by Coloring Prform ataflow livnss analysis ovr a CFG: collct variabl livnss info at vry program point Buil an intrfrnc graph: ach no rprsnts a tmporary valu ach g rprsnts an intrfrnc btwn two tmporaris thy can t b assign to th sam rgistr Color th intrfrnc graph: colors = rgistrs; want to us as fw colors as possibl for a machin with K rgistrs, ci whthr th intrfrnc graph is K-colorabl if ys, th coloring givs a rgistr assignmnt if no, n to spill som tmporaris to mmory Can b us ithr at th local scop or at th global scop. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 10 / 28

6 An Exampl Livnss Analysis Rsult an Intrfrnc Graph: Statmnt Liv Variabls t1 := a-b; a, b t2 := c+b; t1, c, b t3 := t1*; t1, t2, t4 := t2+t3; t1, t2, t3 t5 := t1*t3; t1, t3, t4 t6 := t4-t5; t4, t5 t6 Coloring How many colors o w n? Two Problms Rmaining: K-colorability problm is NP-complt! How to hanl spilling, if n? 3 2 t2 b 2 3 t3 1 t1 t4 2 c t5 3 a 1 1 t6 1 Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 11 / 28 Coloring by Simplification To ovrcom th NP-complt problm, a huristic algorithm is us. Fact Lt m b a no in G with fwr than K nighbors; lt G b th graph G {m}. If G is K-colorabl, thn so is G. A Huristic Coloring Algorithm:[.5x] Rpatly simplify th graph by rmoving (an push on a stack) nos of gr lss than K; until ithr (1) no mor no is lft, which mans th graph is K-colorabl; or (2) all rmaining nos ar of gr mor than K, which mans th graph is not K-colorabl. Exampl: Is th first graph 3-colorabl? Not that this algorithm may giv wrong answrs: Is th scon graph 2-colorabl? Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 12 / 28

7 A Coloring Exampl Statmnt g := mm[j+12] h := k - 1 f := g * h := mm[j+8] m := mm[j+16] b := mm[f] c := + 8 := c k := m + 4 j := b Liv Variabls k j 2 k g j f g h j 3 j f j f f m j k b m m b 1 3 c m b c m b 1 2 h g b k j k Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 13 / 28 Lt K = 4. A Coloring Exampl (cont.) f j k b m c h g Rmov nos c, f, g, an h: Rmov nos, j, k, an m: b j k b m All rmaining nos hav a gr lss than 4. Th graph is 4-colorabl. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 14 / 28

8 Spilling If thr ar k rgistrs but th intrfrnc graph is not k-colorabl, w n to spill tmporaris to stack. Th Spilling Action: Aftr th tmp s finition, it is stor to stack. Bfor ach us, th tmp ns to b loa back to a rg. Effcts of Spilling: To hanl ata on stack, 2 or 3 rgistrs ar n. Th no corrsponing to th spill tmp can b rmov from th intrfrnc graph. How to Slct a Variabl: Choos last frquntly accss variabls. Choos nos with th most conflicts in th intrfrnc graph. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 15 / 28 Rgistr Allocation with Coloring (Chaitin s Algorithm) 1. Buil Construct th intrfrnc graph. 2. Simplify Rpatly simplify th graph. 3. Spill Whn no mor nos can b simplifi, choos a no to spill; a it to th spill st, an rmov it from th graph Rpat Stps 2&3 until no mor nos lft in th graph. 4. Start Ovr If th spill st is not mpty, insrt spill co for all spill nos (i.. loa bfor ach us, stor aftr ach finition); rconstruct th intrfrnc graph. Rpat th algorithm on this rwrittn program. 5. Slct Assign colors to nos. Starting with an mpty graph, rpatly aing a no from th top of th stack. Thr shoul always b a nw color availabl. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 16 / 28

9 An Exampl 1 b 2 m 1 k 3 j 4 3 h 1 g 2 f 2 c 3 stack color 4 2 f j k b m 1 3 c 1 2 h g Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 17 / 28 Dlaying Spilling [Brigg] Rcall that whn th huristic algorithm says that a graph is not k-colorabl, it may not b tru. Whn th algorithm gts stuck in th simplification stp, w o not hav to spill right away. 1. Buil Construct th intrfrnc graph. 2. Simplify Rpatly simplify th graph. 3. Spill Whn no mor nos can b simplifi, choos a no for potntial spill; rmov it from th graph an push it on th stack. Rpat Stps 2&3 until all nos ar on th stack. 4. Slct Assign colors to nos. Starting with an mpty graph, rpatly aing a no from th top of th stack. For a simplifi no, thr is always a nw color availabl; for a potntial spill no, thr may still b a nw color availabl. 5. Start Ovr If th Slct stp is unabl to fin a color for som no(s), thn th program must b rwrittn to ftch thm from mmory just bfor ach us, an stor thm back aftr ach finition. Th algorithm is rpat on this rwrittn program. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 18 / 28

10 No Coalscing Fact If thr is no g in th intrfrnc graph btwn th sourc an stination of a MOVE instruction, thn th sourc an stination nos can b coalsc into a nw no whos gs ar th union of th two. (Effctivly lt th MOVE instruction.) In principl, any pair of nos not connct by an intrfrnc g coul b coalsc. Howvr, coalscing can mak a K-colorabl graph not K-colorabl. Consrvativ Coalscing Stratgis Guarants that coalscing will prsrv th graph s k-colorability proprty. Briggs Nos a an b can b coalsc if th rsulting no ab will hav fwr than k nighbors of significant gr (i.. having K gs). Gorg & Appl Nos a an b can b coalsc if, for vry nighbor t of a, ithr t alray intrfrs with b or t is of insignificant gr. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 19 / 28 Rgistr Allocation with Coloring (Improv Vrsion) 1. Buil Construct th intrfrnc graph, an catgoriz ach no as ithr mov-rlat or non-mov-rlat. 2. Simplify Rpatly simplify th graph by rmoving non-mov-rlat nos of < K gr. 3. (Consrvativ) Coalsc Coalsc two nos accoring to Brigg s or Gorg&Appl s conitions. Rpat Simplify an Coalsc stps until only K-gr or mov-rlat nos rmain. 4. Frz Look for a mov-rlat no of low gr, an frz th movs, i.. giving up futur coalsc attmpts on thm. Rsum simplify an coalsc. 5. Spill If thr is no low-gr nos, slct a high-gr no for potntial spill; rmov it from th graph an push it on th stack. 6. Slct Pop th ntir stack, assigning colors. 7. Start Ovr If an actual spill occurs, rwrit th program an rrun th algorithm. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 20 / 28

11 Back to th Exampl No j an b ar MOVE rlat, so ar an c. f j k b m c h g Rmov nos f, g, an h: Rmov nos an k, an thn m: j k b m j b c c Coalsc nos j & b, an & c. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 21 / 28 Final Coloring &c 1 j&b 2 m 3 k 3 1 h 1 g 4 f 4 stack color 2 4 f j k b m 1 1 c 1 4 h g Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 22 / 28

12 Pr-Color Nos In moling a ralistic rgistr allocation, som nos in th intrfrnc graph n to b pr-color, i.. assign to spcific machin rgistrs (for xampl, th fram pointr rgistr or th rturn valu rgistr). Pr-color nos cannot b simplifi, nor spill. Th original rgistr allocation algorithm can b moifi to work with pr-color nos just lav pr-color nos to th n. Pr-color nos can also mak an othrwis K-colorabl graph not K-colorabl. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 23 / 28 Optimizations with Pr-Color Nos Crating tmporary copis of pr-color rgistrs: ntr: f(r7) ntr: f(r7)... r231 <- r7... => r7 <- r231 xit: us(r7) xit: us(r7) Ths copis will shortn th liv rang of pr-color rgistrs. Th xtra mov instructions will automatically b rmov if thy ar not actually n. Distinguishing callr-sav an call-sav rgistrs: Forc a f at th ntry of th subroutin an a us at th n for vry call-sav rgistrs. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 24 / 28

13 Exampl int f(int a, int b) { int =0, =a; o { = +b; = -1; } whil (>0); rturn ; } r3 r2 b c ntr: c r3 a r1 b r2 0 a loop: + b 1 if > 0 goto loop r1 r3 c rturn (r1, r3 liv out) r1 a No opportunity for simplify, frz, or coalsc, so w must spill. Th priority for spilling can b comput: c, a, b,,. So w slct no c for spilling. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 25 / 28 Exampl (cont.) r3 r2 b r3 r2 b r1 a Now w can coalsc a an. r3 r2b r1 a Two possibl coalscing pairs: a&r 1 or &r 2. Choos th lattr. r3 r2b r1 a Coalsc a&r 1. r1a Th graph can now b fully simplifi. Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 26 / 28

14 Exampl (cont.) Sinc thr is a spilling, w n to moify th program: ntr: c1 r3 M[loc] c1 a r1 b r2 0 a loop: + b 1 if > 0 goto loop r1 c2 M[loc] r3 c2 rturn c1 c2 r3 r1 r2 This graph can b fully simplifi/coalsc. Th final coloring for th nos ar: no a b c 1 c 2 color r 1 r 2 r 3 r 3 r 3 r 1 a b Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 27 / 28 Exampl (cont.) Th rsulting program: ntr: r3 r3 M[loc] r3 r1 r1 r2 r2 r3 0 r1 r1 loop: r3 r3 + r2 r1 r1 1 if r1 > 0 goto loop r1 r3 r3 M[loc] r3 r3 rturn Aftr rmoving trivial mov instructions: ntr: M[loc] r3 r3 0 r1 r1 loop: r3 r3 + r2 r1 r1 1 if r1 > 0 goto loop r1 r3 r3 M[loc] rturn Jingk Li (Portlan Stat Univrsity) CS322 Rgistr Allocation 28 / 28