Organization. Dominators. Control-flow graphs 8/30/2010. Dominators, control-dependence. Dominator relation of CFGs
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1 Orniztion Domintors, ontrol-pnn n SSA orm Domintor rltion o CFGs postomintor rltion Domintor tr Computin omintor rltion n tr Dtlow lorithm Lnur n Trjn lorithm Control-pnn rltion SSA orm Control-low rphs Domintors CFG is DAG Uniqu no rom whih ll nos in CFG r rhl Uniqu no rhl rom ll nos Dummy to simpliy isussion Pth in CFG: squn o nos, possily mpty, suh tht sussiv nos in squn r onnt in CFG y I is irst no in squn n y is lst no, w will writ th pth s * y I pth is non-mpty (hs t lst on ) w will writ + y In CFG G, no is si to omint no i vry pth rom to ontins. Dominn rltion: rltion on nos W will writ om i omints 1
2 Empl Computin ominn rltion D A B C E F G A B C D E F G A B C D E F G Dtlow prolm: Domin: powrst o nos in CFG N Dom_out(N) out(n) = {N} U Dom_in(N) Conlun oprtion: st intrstion Fin rtst solution Work throuh mpl on prvious sli to hk this. Qustion: wht o you t i you omput lst solution? Proprtis o ominn Dominn is rliv: om nti-symmtri: om n om = trnsitiv: om n om om tr-strutur: om n om om or om intuitivly, this mns omintors o no r thmslvs orr y ominn Empl o proo Lt us prov tht ominn is trnsitiv. Givn: om n om Consir ny pth P: + Sin om P must ontin Sin om, P must ontin. Consir pri o P = Q: + Q must ontin us om. Thror P ontins. 2
3 Domintor tr mpl Computin omintor tr Iniint wy: Solv tlow qutions to omput ull ominn rltion Buil tr top-own Root is For vry othr no Rmov rom its omintor st I no is thn omint only y itsl, no s hil o in omintor tr Kp rptin this pross in th ovious wy Chk: vriy tht rom omintor tr, you n nrt ull rltion Builin omintor tr irtly Alorithm o Lnur n Trjn Bs on pth-irst srh o rph O(E*α(E)) whr E is numr o s in CFG Essntilly linr tim Linr tim lorithm u to Buhsum t l Muh mor ompl n proly not iint to implmnt pt or vry lr rphs Immit omintors Prnt o no in tr, i it ists, is ll th immit omintor o writtn s iom() iom not in or iom not in or Intuitivly, ll omintors o othr thn itsl omint iom() In our mpl, iom() = 3
4 Usul lmm Postomintors Lmm: Givn CFG G n, iom() omints Proo: Othrwis, thr is pth P: + tht t os not ontin iom(). Contntin to pth P, w t pth rom to tht os not ontin iom() whih is ontrition. is in CFG Iom() = whih omints Givn CFG G, no is si to postomint no i vry pth rom to ontins. w writ pom to sy tht postomints Postominn is ominn in rvrs CFG otin y rvrsin irtion o ll s n intrhnin rols o n. Cvt: om os not nssrily imply pom. S mpl: om ut os not pom Ovious proprtis Postominn is tr-strutur rltion Postomintor rltion n uilt usin kwr tlow nlysis. Postomintor tr n uilt usin Lnur n Trjn lorithm on rvrs CFG Immit postomintor: ipom Lmm: i is n in CFG G, thn ipom() postomints. Control pnn Intuitiv i: no w is ontrol-pnnt on no u i no u trmins whthr w is ut Empl: S1 m S2.. i thn S1 ls S2. W woul sy S1 n S2 r ontrol-pnnt on 4
5 S1 Empls (ont.).. whil o S1;. W woul sy no S1 is ontrol-pnnt on. It is lso intuitiv to sy no is ontrol-pnnt on itsl: - ution o no trmins whthr or not is ut in. S1 Empl (ont.) S1 n S3 r ontrolpnnt on Ar thy ontrol-pnnt on? Dision t os not ully trmin i S1 (or S3 is S2 ut) sin thr is ltr S3 tst tht trmins this So w will NOT sy tht S1 n n S3 r ontrol-pnnt on Intuition: ontrol-pnn m is out lst ision point Howvr, is ontrolpnnt on, n S1 n S3 r trnsitivly (itrtivly) ontrol-pnnt on Empl (ont.) Cn no ontrolpnnt on mor thn on no? ys, s mpl n nst rpt-until loops t1 n is ontrol-pnnt on t1 n t2 (why?) t2 In nrl, ontrolpnn rltion n qurti in siz o prorm Forml inition o ontrol pnn Formlizin ths intuitions is quit triky Strtin roun 1980, lots o propos initions Commonly pt inition u to Frrn, Ottnstin, Wrrn (1987) Uss i o postominn W will us slihtly moii inition u to Bilri n Pinli whih is sir to think out n work with 5
6 Control pnn inition Control pnn inition First ut: ivn CFG G, no w is ontrolpnnt on n (u v) i w postomints v. w os not postomint u Intuitivly, irst onition: i ontrol lows rom u to v it is urnt tht w will ut son onition: ut rom u w n rh without nountrin w so thr is ision in m t u tht trmins whthr w is ut Smll vt: wht i w = u in prvious inition? S pitur: is u ontrolpnnt on u v? Intuition sys ys, ut inition on prvious slis sys u shoul not postomint u n our inition o postominn is rliv Fi: ivn CFG G, no w is ontrol-pnnt on n (u v) i w postomints v i w is not u, w os not postomint u v u Strit postominn Empl A no w is si to stritly postomint no u i w!= u w postomints u Tht is, strit postominn is th irrliv vrsion o th ominn rltion Control pnn: ivn CFG G, no w is ontrol-pnnt on n (u v) i w postomints v w os not stritly postomint u 6
7 Computin ontrol-pnn rltion Nos ontrol pnnt on (u v) r nos on pth up th postomintor tr rom v to ipom(u), luin ipom(u) W will writ this s [v,ipom(u)) hl-opn intrvl in tr Computin ontrol-pnn rltion Comput th postomintor tr Ovrly h u v on pom tr n trmin nos in intrvl [v,ipom(u)) Tim n sp omplity is O(EV). Fstr solution: in prti, w o not wnt th ull rltion, w only mk quris (): wht r th nos ontrol-pnnt on n? ons(w): wht r th s tht w is ontrol-pnnt on? quiv(w): wht nos hv th sm ontrol-pnns s no w? It is possil to implmnt simpl t strutur tht tks O(E) tim n sp to uil, n tht nswrs ths quris in tim proportionl to output o qury (optiml) (Pinli n Bilri 1997). SSA orm SSA mpl Stti sinl ssinmnt orm Intrmit rprsnttion o prorm in whih vry us o vril is rh y tly on inition Most prorms o not stisy this onition () s prorm on nt sli: us o Z in no F is rh y initions in nos A n C Rquirs insrtin ummy ssinmnts ll Φ- untions t mr points in th CFG to mr multipl initions Simpl lorithm: insrt Φ-untions or ll vrils t ll mr points in th CFG n rnm h rl n ummy ssinmnt o vril uniquly () s trnsorm mpl on nt sli A Z:= B p1 C Z:=. E p2 F print(z) D G p3 A Z0:= Z1 := Φ(Z4,Z0) B p1 C Z2:=. D Z3:= Φ(Z1,Z3) G p3 E Z4:= Φ(Z2,Z3) p2 F print(z4) 7
8 Miniml SSA orm Miniml-SSA orm Empl In prvious mpl, ummy ssinmnt Z3 is not rlly n sin thr is no tul ssinmnt to Z in nos D n G o th oriinl prorm. Miniml SSA orm SSA orm o prorm tht os not ontin suh unnssry ummy ssinmnts S mpl on nt sli Qustion: how o w onstrut miniml SSA orm irtly? Dominn rontir Dominn rontir o no w No u is in ominn rontir o no w i w omints CFG prssor v o u, ut os not stritly omint u Dominn rontir = ontrol pnn in rvrs rph! A B C D E F G Empl rom prvious sli A B C D E F G Itrt ominn rontir Irrliv losur o ominn rontir rltion Rlt notion: itrt ontrol pnn in rvrs rph Whr to pl Φ-untions or vril Z Lt Assinmnts = {} U {nos with ssinmnts to Z in oriinl CFG} Fin st I = itrt ominn rontir o nos in Assinmnts Pl Φ-untions in nos o st I For mpl Assinmnts = {,A,C} DF(Assinmnts) = {E} DF(DF(Assinmnts)) = {B} DF(DF(DF(Assinmnts))) = {B} So I = {E,B} This is whr w pl Φ-untions, whih is orrt 8
9 Why is SSA orm usul? For mny tlow prolms, SSA orm nls sprs tlow nlysis tht yils th sm prision s it-vtor CFG-s tlow nlysis ut is symptotilly str sin it prmits th ploittion o sprsity s ltur nots rom Spt 6 th SSA hs two istint turs tor -us hins rnmin you o not hv to prorm rnmin to t vnt o SSA or mny tlow prolms Computin SSA orm Cytron t l lorithm omput DF rltion (s slis on omputin ontrol-pnn rltion) in irrliv trnsitiv losur o DF rltion or st o ssinmnts or h vril Computin ull DF rltion Cytron t l lorithm tks O( V + DF ) tim DF n qurti in siz o CFG Fstr lorithms O( V + E ) tim pr vril: s Bilri n Pinli Dpnns W hv sn ontrol-pnns. Wht othr kin o pnns r thr in prorms? Dt pnns: pnns tht t ris rom rs n writs to mmory lotions Think o ths s onstrints on rorrin o sttmnts Dt pnns Flow-pnn (r-tr-writ): S1 S2 Eution o S2 my ollow ution o S1 in prorm orr S1 my writ to mmory lotion tht my r y S2 Empl:.. := 3... low-pnn whil o : = low-pnn This is ll loop-rri pnn 9
10 Anti-pnns Anti-pnn (writ-tr-r): S1 S2 Eution o S2 my ollow ution o S1 in prorm orr S1 my r rom mmory lotion tht my (ovr)writtn y S2 Empl: :=... := nti-pnn Output-pnn Output-pnn (writ-tr-writ): S1 S2 Eution o S2 my ollow ution o S1 in prorm orr S1 n S2 my oth writ to sm mmory lotion Summry o pnns Dpnn Dt-pnn: rltion twn nos Flow- or r-tr-writ (RAW) Anti- or writ-tr-r (WAR) Anti or writ tr r (WAR) Output- or writ-tr-writ (WAW) Control-pnn: rltion twn nos n s 10
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