Register Allocation. Goal: replace temporary variable accesses by register accesses Why? Constraints:

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1 Rgistr Allotion Gol: rpl tmporry vril sss y rgistr sss Why? Constrints: Fix st of rgistrs Two simultnously liv vrils nnot llot to th sm rgistr A vril is liv if it will us gin for ing rfin.

2 1. Intify Liv Vril Rngs Bsi rul: Tmporris t1 n t2 n shr th sm rgistr if t ny point in th progrm t most on of t1 or t2 is liv!

3 Rgistr Intrfrn Grph

4 Wht is th Rgistr Infrn Grph for this xmpl?

5 Rgistr Intrfrn Grph 1. Wht nnot ssign sm rgistr? 2. Wht n ssign th sm rgistr?

6 Your Turn Writ own th liv vrils ftr h sttmnt." Hint: Strt t th ottom. Instrutions Liv vrs = + 2 = * = + 1 rturn *

7 Liv Vrils Instrutions Liv vrs = + 2 = * = + 1 rturn *,

8 Liv Vrils Instrutions Liv vrs = + 2 = * = + 1 rturn *,,

9 Liv Vrils Instrutions Liv vrs = + 2 = * = + 1 rturn *,,,

10 Instrutions Liv vrs = + 2, = *, = + 1, rturn * Liv Vrils

11 Intrfrn grph n Rgistr Allotion Nos of th grph = vrils Egs onnt vrils tht intrfr with on nothr Nos will ssign olor orrsponing to th rgistr ssign to th vril Two olors n t nxt to on nothr in th grph

12 Rgistr Allotion = Grph Coloring

13 Coloring th RIG olor rgistr Instrutions Liv vrs = + 2, = *, = + 1, rturn * R1 R2

14 Coloring th RIG olor rgistr Instrutions Liv vrs = + 2, = *, = + 1, rturn * R1 R2

15 How to o th Grph oloring Qustions: Cn w ffiintly fin oloring of th grph whnvr possil? Cn w ffiintly fin th optimum oloring of th grph? How o w hoos rgistrs to voi mov instrutions? Wht o w o whn thr rn t nough olors (rgistrs) to olor th grph?

16 Coloring grph Kmp s lgorithm [1879] for fining K- oloring of grph Assum K=3 Stp 1 (simplify): fin no with t most K-1 gs n rmov from th grph (with its gs). (Rmmr this no on stk for ltr stgs.)

17 Coloring grph On oloring is foun for th simplr grph, w n lwys olor th no w sv on th stk Stp 2 (olor): whn th simplifi sugrph hs n olor, k th no on th top of th stk n ssign it olor not tkn y on of th jnt nos

18 Coloring with K=2 olor rgistr R1 R2 stk:

19 Coloring olor rgistr R1 R2 stk:

20 Coloring olor rgistr R1 R2 stk:

21 Coloring olor rgistr R1 R2 stk:

22 Coloring olor rgistr R1 R2 stk:

23 Coloring olor rgistr R1 R2 stk:

24 Coloring olor rgistr R1 R2 stk:

25 Coloring olor rgistr R1 R2 stk:

26 Coloring olor rgistr R1 R2 stk:

27 Coloring olor rgistr R1 R2 stk:

28 Coloring olor rgistr R1 R2 stk:

29 Filur If th grph nnot olor, it will vntully simplifi to grph in whih vry no hs t lst K nighors Somtims, th grph is still K-olorl! Fining K-oloring in ll situtions is n NP-omplt prolm W will hv to pproximt to mk rgistr llotors fst nough

30 Coloring with K=2 olor rgistr R1 R2 stk:

31 Coloring olor rgistr R1 R2 stk: ll nos hv 2 nighours!

32 Coloring olor rgistr R1 R2 stk:

33 Coloring olor rgistr R1 R2 stk:

34 Coloring olor rgistr R1 R2 stk:

35 Coloring olor rgistr R1 R2 stk:

36 Coloring olor rgistr R1 R2 stk:

37 Coloring olor rgistr R1 R2 stk:

38 Coloring olor rgistr R1 R2 stk: W got luky!

39 Try to Color this with 4 olors? " 3 olors?

40 On Possil 4 oloring

41 Th o woul look lik this

42 Coloring with K=2 olor rgistr R1 R2 Som grphs n t olor in K olors: stk:

43 Coloring olor rgistr R1 R2 Som grphs n t olor in K olors: stk:

44 Coloring olor rgistr R1 R2 Som grphs n t olor in K olors: stk:

45 Coloring olor rgistr R1 R2 Som grphs n t olor in K olors: stk: no olors lft for!

46 Spilling Stp 3 (spilling): on ll nos hv K or mor nighors, pik no for spilling Stor on th stk Thr r mny huristis tht n us to pik no E.g., not in n innr loop

47 Spilling: Insrting Co

48 Exmpl

49 Romputing Vril Livnss

50 Romput th RIG ftr spilling This is 3-olorl!

51 Ovrll Algorithm Simplify Livnss Mrk possil spills Color & tt tul spills Rwrit o to implmnt tul spills

52 Summry Rgistr llotion hs thr mjor prts Livnss nlysis Grph oloring Progrm trnsformtion (spilling) For mor informtion, hptr in Appl

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