Organization. Control Dependence, Program Analyses and The Roman Chariots Problem. Part 1: What is an Optimal Representation of Control Dependence?

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1 Orgniztion. Optiml Rprsnttion o Control pnn - Dinition - Is th ontrol pnn grph (O( E * V ) sp/tim) optiml? 2. Our pproh: - Ru prolm to ROMAN CHARIOTS PROBLEM - Buil APT t strutur in O( E + V ) sp/tim => APT is n optiml rprsnttion o ontrol pnn 3. Othr pplitions o APT: - SSA omputtion in linr tim pr vril - SDEG omputtion in linr tim pr prolm - DFG omputtion in linr tim pr vril 4. Conlusions: - APT is tor orm o th CDG whih rquirs iltr srh to nswr quris Control pnn: (Frrnt,Ottnstin,Wrrn 987) No w is ontrol pnnt on g (u -> v) i - w postomints v - i w == u, w os not postomint u. V E -> -> Control Flow Grph Postomintor Tr Control Dpnn Rltion 4 2 Control Dpnn, Progrm Anlyss n Th Romn Chriots Prolm Prt : Kshv Pingli Cornll Univrsity Ginrno Bilri Univrsit i Pov, Itly Wht is n Optiml Rprsnttion o Control Dpnn? 3

2 Optiml Control Dpnn Computtion Control Dpnn Grph (CDG) IF A > THEN A = A + CDR iprtit grph twn gs n nos onnt no v to g i no v is ontrol pnnt on g onnt nos in sm CDEQUIV lss into rings (not shown) Prprossing CDR ons quiv -> -> Qury V E -> -> Control Dpnn Rltion -> -> Control Dpnn Grph Qury tim or CD, CONDS,CDEQUIV sts is proportionl to st siz Sp n tim or prprossing shoul miniml. Qury tim: Proportionl to siz o output Prprossing : O( E * V ) sp n tim 8 6 Quris on Control Dpnn Rltion: Worst-s siz o ontrol pnn rltion: - (): st o nos ontrol pnnt on g - ons(v): st o ontrol pnns o no v - quiv(v): st o nos with sm ontrol pnns s no v (in sm quivln lss s v) V E -> -> V E -> -> -> -> Control Dpnn Rltion Control Flow Grph n nst rpt-until loops => siz o CDR is n(n+3) Applitions: progrm nlysis, shuling or piplins, prllliztion Th siz o th CDR n grow qurtilly with progrm siz. 5 7

3 Ky I (I): Exploit strutur o rltion Anlogy: Postomintor rltion Prt II: APT n th Romn Chriots Prolm - quris: immit pom o no, ll poms o no 2 - siz o rltion is O( V ) - rltion is trnsitiv, so uil trnsitiv rution (pom tr) in O( E ) tim Hrl,Trjn - qury tim using pom tr is optiml => Thr is no point in onstruting th ntir rltion Wht strutur is thr in th ontrol pnn rltion? Control pnn rltion: - nos tht r ontrol pnnt on n g orm simpl pth in th postomintor tr - in tr, simpl pth is uniquly spii y its npoints Postomintor tr + npoints o h ontrol pnn pth n uilt in O( E ) sp n tim 2 Our Solution: Thr hv n mny unsussul orts to ru th siz o th CDG. W thror onjtur tht to numrt ons sts in tim proportionl to th siz o th st rquirs t strutur o qurti siz." - ru ontrol pnn omputtion to grph prolm ll Romn Chriots Prolm - sign t strutur ll APT (ugmnt postomintor tr) () whih n uilt in O( E ) sp n tim, n () whih n us to nswr CD,CONDS n CDEQUIV quris in tim proportionl to output siz. Cytron,Frrnt,Srkr, PLDI 99 APT is t strutur or optiml ontrol pnn omputtion. 9

4 CD(n): Whih itis r srv y hriot n? Qury prour: (similr to FOW 87) How n w us th ompt rprsnttion o th CDR to nswr quris or CD,CONDS n CDEQUIV sts in tim proportionl to output siz? - Look up ntry or hriot n in Rout Arry (sy it is x,y) - Trvrs nos in tr T, strting t x n ning t y - Output ll nos nountr in trvrsl (. CDG: mny routs n shr tr nos/gs) CD qury tim is proportionl to output siz. 4 6 Exmpl: Control Flow Grph V E -> -> Control Dpnn Rltion Pth E ->, ->, MILANO NAPOLI POMPEII ROMA Romn Chriots Prolm BOLOGNA VERONA CORLEONE Tr: - nos r itis - gs r ros VENEZIA Pth Rout # I MILANO,ROMA II POMPEII,BOLOGNA III VENEZIA,ROMA Citis on rout orr y nstor rltion In rout x,y, x is snnt o y Postomintor Tr Pth Arry A O( E ) Rprsnttion o th Control Dpnn Rltion Givn tr T, n n rry A o hriot routs spii y npoints, sign t strutur to nswr th ollowing quris in optiml tim. () CD(n): Whih itis r srv y hriot n? () CONDS(w): Whih hriots srv ity w? () CDEQUIV(w): Whih itis r srv y th sm hriots tht srv w? 3 5

5 Ky I (II): Ch rout inormtion in tr Stp 3: Ch rout t multipl nos. At h no n in th tr, kp list o hriot # s whos ottom no is n. {I} Rout Chriot # I, II, IV III II I {IV,III,II,I} IV III II I {IV,III,II,I} {IV} {IV,III} {IV,III,II} {IV,III,II,I} {II} Qury prour: CONDS(w) or h snnt o w o or h rout = x,y in list t o i w is snnt o y thn output ; i o o Qury tim is proportionl to # o snnts + siz o ll lists t snnts Two xtrms: () Chriot # stor only t ottom no o rout Sp : O( V + A ) Qury Tim: O( V + Output ) (2) Chriot # stor t ll nos on rout Sp: O( V * A ) Qury Tim: O( Output ) Cn w hv isiplin hing poliy to hv linr sp n optiml qury tim? 8 2 CONDS(w): Whih hriots srv ity w? Rinmnt: Sort h list y rsing lngth. Rout Chriot # I, II, IV III II I {IV,III,II,I} Rout Chriot # I, II, III, IV, Qury prour: or h hriot in Rout Arry o lt rout o x,y; i w is n nstor o x n w is snnt o y o thn output ; i Cn w voi xmining ll routs in Rout Arry? Qury prour: CONDS(w) or h snnt o w o or h rout = x,y in list t o i w is snnt o y thn output ; ls BREAK; i o o At most on non-ovrlpping pth is xmin t snnt => Qury tim is proportionl to siz o output + # o snnts 7 9

6 How o w onstrut zons? I Invrint: For ny no v, Z v < α Α v + whr α is sign prmtr. Qury tim or CONDS(v) = O( A v + Z v ) = O( ( α + ) A v + ) = O( A v ) II Buil zons ottom-up, mking thm s lrg s possil w/o violting invrint v is l no => mk v ounry no v is n intrior no => i ( + Σ Z ) > ε α A v + u hilrn(v) u thn mk v ounry no ls mk v n intrior no Ky i (III): Ch rout t multipl nos Ching Rul: Divi tr into ZONES Qury prour: Visit only nos low qury no n in th sm zon s qury no Zon onstrution: For ll nos v, Z v < A v + => Qury tim A + Z (α + ) A v v v - Nos r prtition into - ounry nos: lowst nos in zon - intrior nos: ll othr nos - Ching rul: - ounry no: stor ll hriots srving no - intrior no: stor ll hriots whos ottom no is tht no - Our lgorithm: ottom-up, gry zon onstrution => sp rquirmnts < A + V / α α α = (som hing) h 3 g α Α + v {} {} {->, h->, g->,->} {} {->} {->} {g -> } { ->, h -> } 2 23

7 α = << (ull hing) h g { ->, h -> } {->, h->,g -> } {->, h->, g->, ->} {->, h->, g->, ->, ->} {->, h->, g->, ->, ->} {->, h->, g->, ->} {->, h->,g -> } { ->, h -> } α Α + v α = >> (no hing) h {} g {} Summry o CONDS Approh: Qury Tim: ( α +) A v Sp : A + V / α α {} {} Α + v {->} {->} {g -> } { ->, h -> } - Prmtr α is us to prtition tr into zons α << : lowr qury tim, inrs sp rquirmnts α >> : highr qury tim, lowr sp rquirmnts - Nos r prtition into - ounry nos: lowst nos in zon - intrior nos: ll othr nos - Ching rul: - ounry no: stor ll hriots srving no - intrior no: stor ll hriots whos ottom no is tht no Qury prour: - Visit only nos low qury no n in th sm zon s qury no 25 27

8 APT. Postomintor tr with iirtionl gs 2. s-numrv: intgr - us or nstorship trmintion in CONDS qury 3. ounry?v: ooln - tru i v is ounry no, ls othrwis - us in CONDS qury 4. Lv: list o hriots # s/ontrol pnns - ounry no: ll hriots srving v (ll ontrol pnns o v) - intrior no: ll hriots whos ottom no is v (ll immit ontrol pnns o v) - us in CONDS qury 5. Rv: pointr to CDEQUIV quivln lss - us in CDEQUIV qury Qury tim: ( α+) * output-siz Sp: E + V / α Storg CDG ALPHA = /32 ALPHA = /6 5 ALPHA = Nsting Dpth 3 32 Whih itis r srv y sm hriots tht srv v? 3 - Frrnt, Ottnstin, Wrrn 87: O( E ) using hshing or st qulity CDEQUIV(v): - Cytron, Frrnt, Srkr 9: O( E 2 ) - Bll 92: O( E ) or strutur progrms - Pogurski 93: O( E ) or orwr ontrol pnn in gnrl grphs - Johnson, Prson, Pingli 94: O( E ) or gnrl grphs (optiml) CDEQUIV or Romn Chriots Prolm - ln-up vrsion o JPP94 lgorithm - omput two ingr prints or CONDS sts. siz o CONDS st. Lo:lowst no ontin in ll routs o CONDS st Exprimntl Rsults r r2 Lo(CONDS()) = Lo(CONDS()) = Lo(CONDS()) = Two CONDS sts r qul i thy hv th sm ingr-prints. Cn omput ingr-prints in O( V + A ) sp n tim 29 3

9 Storg Storg 2 pth = tul prit pth = 64 2 pth = log(alpha) ALPHA = 5 ALPHA = 4 ALPHA = 32 5 ALPHA = >> Nsting Dpth log(alpha) pth = 4 5 Worst Cs Qury Tim pth = 64 pth = 32 4 pth = 45 5 Log(ALPHA) pth = 4 PDOM: pth = 32 2 pth = 32 Prprossing Tim (ss) 6 pth = PDOM: pth = x 3

10 Nsting Dpth ALPHA = Prprossing Tim (ss) ALPHA = /32 ALPHA = /6 PDOM Tim..2 2 qn qntott Storg 5 s 4 li p 3 6 No Ching 7 Som Ching: ALPHA = 8 Full Ching 9 sprsso Ching in APT or SPEC Intgr Bnhmrks Storg Storg 8 7 +: Full Ching *: Som Ching: ALPHA = o: No Ching Progrm Siz: Nos 2 spi Full Ching Som Ching: ALPHA = 8 No Ching 6 wv ou 4 2 mljp tomtv or mljsp swm su2or hyro2 ns7 pppp r lvinn SPEC Floting Point Bnhmrks

11 Comprison with toring: - Ftoring ttmpts to ru siz o CDG y mking nos shr ontrol pnns in th rprsnttion (CFS 9) SSA Computtion - phi-plmnt = itrt ominn rontir omputtion - xploit th t tht ons rltion is sm s g ominn rontir rltion in rvrs grph Solution: Us APT on rvrs grph = ADT on CFG Nos Egs Nos Egs mrg point - Our hing pproh n viw s toring in whih iltr srh is us to nswr quris (Chzll) I {I} {I} - First, look t DF(S) whr S is givn olin Algorithm: Sort S y lvl, n qury in ottom-up orr Two nos in S longing to sm zon ADT Zon - to omput DF(), visit su-zon low - tr this, to omput DF(), no n to visit suzon low! Hight o Postomintor Tr Othr Applitions o APT Control Dpnn CONDS CDEQUIV CD ADT n APT itrt itrt - n us to uil SSA orm in O( E ) pr vril - susums lgorithm o Cytron t l ( α <<) - susums lgorithm o Srhr n Go ( α >>) Dtlow Anlysis SSA,GSA DFG,PDW,VDG,... ADT : ugmnt omintor tr (APT on rvrs CFG) Progrm Siz - n us to uil DFG in O( E ) tim pr vril - SESE trmintion in O( E ) tim - s Johnson, Prson, Pingli (PLDI 94) Johnson s thsis t Cornll 4 43

12 48 46 Algorithm: - Sort nos in S y lvl. - Rmov nos rom sort list y rsing lvl orr, n qury in ADT - Atr no is quri, mrk it in ADT so urthr quris tht rh v o not look low v. Tim = O( V + A ) (O E ) in CFG trms Wht i st or qurying is givn onlin? - W n us sm strtgy provi nos r prsnt or qurying in ottom-up orr. - Hppily, i n is in DF(m), thn lvl(n) <= lvl(m)!! => us priority quu or ynmi sorting - Priority quu implmnttion: (k = # o kys = hight o ADT ) - vn Em Bos: O(log(log(k))) pr insrtion n ltion - Srhr n Go: us n rry o siz k Exmpl: x y CFG y x y x Domintor tr E V y -> x -> -> y -> phi({,x}) = {,,} phi({}) = {} x y DF(no) = stintion(edf(no)) DF({}) = {,,,}) DF({}) = {,,} DF({}) = {,} EDF Dominn Frontir 45 47

13 Rpt until Loop: Nsting = 2. APT t strutur: Conlusions Tim or phi untion plmnt Qury tim: ( α+) * output-siz Prprossing Sp n Tim: O( E + V / α ) Control Dpnn CONDS (v): optiml CDEQUIV(v): optiml CD(): optiml Dtlow Anlysis SSA: O( E ) pr vril SDEG: O( E ) pr prolm DFG: O( E ) pr vril 4 No Numr Log2(ALPHA) Ky onpts - xploit strutur o ontrol pnn rltion - intllignt hing o inormtion 5 52 Rmrks: - Tim to uil SSA orm: O( E ) pr vril 4.5 Tim or phi untion Plmnt - Susums lgorithms o Cytron tl n Srhr n Go α << : Cytron t l 9 - O( E * V ) pr vril α >> : Srhr n Go (PLDI 95) - O( E ) pr vril - Sm i n us to uil sprs tlow vlutor grphs or othr tlow prolms - Wht is st vlu o α? Intrsting tro - smll vlu: rptly isovr tht som no is in trnsitiv losur - lrg vlu: tim to omput iniviul DF sts my lrg - intrmit vlu my st! Tim (ss) log2(alpha) 49 5

14 Applitions o Thnology DCPI: Digitl Continuous Proling Inrstrutur uss ontrol quivln lgorithm to ru ovrh o pnn proling progrm IBM VLIW Compilr: Eioglu t l us Dpnn Flow (DFG) s thir IF in VLIW ompilr work Grph Aristotl Anlysis Systm: Ohio Stt Univrsity // uss wk pnn lgorithms ontrol Toy ompilr (IBM), Intl,...: us som o th ontrol pnn lgorithms 53