CS553 Lecture Introduction to Data-flow Analysis 1

Transcription

1 ! Ide Introdution to Dt-flow nlysis!lst Time! Implementing Mrk nd Sweep GC!Tody! Control flow grphs! Liveness nlysis! Register llotion CS553 Leture Introdution to Dt-flow Anlysis 1 Dt-flow Anlysis! Dt-flow nlysis derives informtion bout the dynmi behvior of progrm by only exmining the stti ode Exmple! How mny registers do we need for the progrm on the right?! Esy bound: the number of vribles used (3)! Better nswer is found by onsidering the dynmi requirements of the progrm!1 := 0!2 L1: b := + 1!3 := + b!4 := b * 2!5 if < 9 goto L1!6 return CS553 Leture Introdution to Dt-flow Anlysis 2

2 Liveness Anlysis!Definition!A vrible is live t prtiulr point in the progrm if its vlue t tht point will be used in the future (ded, otherwise).! To ompute liveness t given point, we need to look into the future!motivtion: Register Allotion!A progrm ontins n unbounded number of vribles!must exeute on mhine with bounded number of registers!two vribles n use the sme register if they re never in use t the sme time (i.e, never simultneously live).! Register llotion uses liveness informtion CS553 Leture Introdution to Dt-flow Anlysis 3 Control Flow Grphs (CFGs)!Definition!A CFG is grph whose nodes represent progrm sttements nd whose direted edges represent ontrol flow!exmple 1 = 0!1 := 0!2 L1: b := + 1!3 := + b!4 := b * 2!5 if < 9 goto L1!6 return 6 return No b = + 1 = + b 4 = b * 2 <9 Yes CS553 Leture Introdution to Dt-flow Anlysis 4

3 Terminology!Flow Grph Terms! A CFG node hs out-edges tht led to suessor nodes nd in-edges tht ome from predeessor nodes! pred[n] is the set of ll predeessors of node n 1 su[n] is the set of ll suessors of node n = 0!Exmples! Out-edges of node 5:! su[5] = {2,6}! pred[5] = {4}! pred[2] = {1,5} (5"6) nd (5"2) b = + 1 = + b 4 = b * 2 <9 6 return No Yes CS553 Leture Introdution to Dt-flow Anlysis 5 Liveness by Exmple!Wht is the live rnge of b?! Vrible b is red in sttement 4, so b is live on the (3 " 4) edge! Sine sttement 3 does not ssign into b, b is lso live on the (2"3) edge! Sttement 2 ssigns b, so ny vlue of b on the (1"2) nd (5"2) edges re not needed, so b is ded long these edges!b s live rnge is (2"3"4) 6 return = 0 b = = b * 2 5 No = + b <9 Yes CS553 Leture Introdution to Dt-flow Anlysis 6

4 ! Vribles Liveness by Exmple (ont)!live rnge of! is live from (1"2) nd gin from (4"5"2)! is ded from (2"3"4) 1 2 = 0 b = + 1!Live rnge of b! b is live from (2"3"4)!Live rnge of! is live from (entry"1"2"3"4"5"2, 5"6) 6 return 3 4 = b * 2 5 No = + b <9 Yes nd b re never simultneously live, so they n shre register CS553 Leture Introdution to Dt-flow Anlysis 7 Uses nd Defs!Def (or definition)! An ssignment of vlue to vrible! def_node[v] = set of CFG nodes tht define vrible v! def[n] = set of vribles tht re defined t node n = 0!Use! A red of vrible s vlue! use_node[v] = set of CFG nodes tht use vrible v! use[n] = set of vribles tht re used t node n < 9? v live # def_node[v]!more preise definition of liveness! A vrible v is live on CFG edge if $ use_node[v]!(1) % direted pth from tht edge to use of v (node in use_node[v]), nd (2) tht pth does not go through ny def of v (no nodes in def_node[v]) CS553 Leture Introdution to Dt-flow Anlysis 8

5 The Flow of Liveness!Dt-flow! Liveness of vribles is property tht flows through the edges of the CFG!Diretion of Flow! Liveness flows bkwrds through the CFG, beuse the behvior t future nodes determines liveness t given node! Consider! Consider b! Lter, we ll see other properties tht flow forwrd 1 := b := + 1 := + b := b * 2 No 6 return < 9? Yes CS553 Leture Introdution to Dt-flow Anlysis 9 Liveness t Nodes!We hve liveness on edges! How do we tlk bout liveness t nodes? = 0 edges progrm points just before omputtion just fter omputtion!two More Definitions! A vrible is live-out t node if it is live on ny of tht node s out-edges n live-out out-edges! A vrible is live-in t node if it is live on ny of tht node s in-edges n live-in in-edges CS553 Leture Introdution to Dt-flow Anlysis 10

6 ! (1)! (2)! then! (3)! then Computing Liveness!Rules for omputing liveness Generte liveness: If vrible is in use[n], it is live-in t node n!dt-flow equtions n live-in use Push liveness ross edges: If vrible is live-in t node n it is live-out t ll nodes in pred[n] live-out Push liveness ross nodes: If vrible is live-out t node n nd not in def[n] the vrible is lso live-in t n live-out n live-in n live-out live-in live-out pred[n]! (1) in[n] = use[n] & (out[n] def[n]) (3)! out[n] = & in[s] s $ su[n] (2) CS553 Leture Introdution to Dt-flow Anlysis 11 Solving the Dt-flow Equtions!Algorithm for eh node n in CFG in[n] = '; out[n] = ' repet for eh node n in CFG in [n] = in[n] out [n] = out[n] in[n] = use[n] & (out[n] def[n]) out[n] = & in[s] s $ su[n] until in [n]=in[n] nd out [n]=out[n] for ll n initilize solutions sve urrent results solve dt-flow equtions test for onvergene!this is itertive dt-flow nlysis (for liveness nlysis) CS553 Leture Introdution to Dt-flow Anlysis 12

7 Exmple node # 1 2 b 3 b 5 1st 2nd 3rd 4th 5th 6th 7th use def in out in out in out in out in out in out in out 4 b 6 b b b b b b b b b b b b b b b b b b b b b b b b b b 1 := 0 2 b := := + b 4 := b * 2 Dt-flow Equtions for Liveness in[n] = use[n] & (out[n] def[n]) out[n] = & in[s] s $ su[n] No 6 return 5 < 9? Yes CS553 Leture Introdution to Dt-flow Anlysis 13 Liveness Anlysis in the MiniJv ompiler!currently! Prse into AST! Allote spe on stk for lols nd prmeters nd spe in hep for member vribles! Use stk for expression evlution! Generte MIPS ode from AST!To perform dt-flow nlysis! Need intermedite representtion like 3-ddress ode! Use temporries for prmeters, lols, nd expression results! Indite uses nd defs of temporries in eh 3-ddress ode instrution! Crete ontrol-flow grph with eh 3-ddress ode instrution s node CS553 Leture Introdution to Dt-flow Anlysis 14

8 ! Problem! Gol! Expression! Lol! Loop! Globl! Interproedurl Register Allotion! Assign n unbounded number of symboli registers to fixed number of rhiteturl registers! Simultneously live dt must be ssigned to different rhiteturl registers! Minimize overhed of essing dt! Memory opertions (lods & stores)! Register moves CS553 Leture Register Allotion I 15 Sope of Register Allotion CS553 Leture Register Allotion I 16

9 ! Wht Grnulrity of Allotion is lloted to registers?! Vribles! Live rnges/webs (i.e., du-hins with ommon uses)! Vlues (i.e., definitions; sme s vribles with SSA) s 1 : x := 5 b 2 s 2 : y := x b 3 s 3 : x := y+1 b 4 b 1 s 6 :... x... s 4 :... x... s 5 : x := 3 Vribles: 2 (x & y) Live Rnges/Web: 3 (s 1!s 2,s 4 ; s 2! s 3 ; s 3,s 5! s 6 ) Vlues: 4 (s 1, s 2, s 3, s 5, " (s 3,s 5 )) Wht re the trdeoffs? Eh llotion unit is given symboli register nme (e.g., t1, t2, et.) CS553 Leture Register Allotion I 17 Globl Register Allotion by Grph Coloring Ide [Coke 71], First llotor [Chitin 81] 1.! Construt interferene grph G=(N,E)!Represents notion of simultneously live!nodes re units of llotion (e.g., vribles, live rnges, vlues)!% edge (n 1,n 2 ) $ E if n 1 nd n 2 re simultneously live!symmetri (not reflexive nor trnsitive) 2.! Find k-oloring of G (for k registers)!adjent nodes n t hve sme olor 3. Allote the sme register to ll llotion units of the sme olor!adjent nodes must be lloted to distint registers t2 t1 t3 CS553 Leture Register Allotion I 18

10 ! Use! Options Interferene Grph Exmple (Vribles) :=... b :=... := d :=... := :=... e b e := e b... :=... d CS553 Leture Register Allotion I 19 Computing the Interferene Grph results of live vrible nlysis for eh symboli-register/temporry t i do for eh symboli-register/temporry t j (j < i) do for eh def $ {definitions of t i } do if (t j is live out t def) then E ( E & (t i,t j )! tret ll instrutions the sme! tret MOVE instrutions speil! whih is better? CS553 Leture Register Allotion I 20

11 ! K-oloring! Alloting! But.! If! Choosing Alloting Registers Using the Interferene Grph! Color grph nodes using up to k olors! Adjent nodes must hve different olors to k registers ) finding k-oloring of the interferene grph! Adjent nodes must be lloted to distint registers..! Optiml grph oloring is NP-omplete! Optiml register llotion is NP-omplete, too (must pproximte)! Wht if we n t k-olor grph? (must spill) CS553 Leture Register Allotion I 21 Register Allotion: Spilling we n t find k-oloring of the interferene grph! Spill vribles (nodes) until the grph is olorble vribles to spill! Choose rbitrrily or! Choose lest frequently essed vribles! Brek ties by hoosing nodes with the most onflits in the interferene grph! Yes, these re heuristis! CS553 Leture Register Allotion I 22

12 Spilling (Originl CFG nd Interferene Grph) :=... b :=... := d :=... f := f :=... e b e := f e b... :=... f d CS553 Leture Register Allotion I 23 Spilling (After spilling b ) :=... b 1 :=... M[fp+4] := b 1 := d :=... f := f :=... e b 1 b 2 e := f e... b 2 = M[fp+4]... b 2... :=... f d CS553 Leture Register Allotion I 24

13 ! Attempt Simple Greedy Algorithm for Register Allotion for eh n $ N do { selet n in deresing order of weight } if n n be olored then do it { reserve register for n } else Remove n (nd its edges) from grph { llote n to stk (spill) } :=... r1 :=... M[fp+4] := r1 := d :=... (After spilling b ) e := f e... r2 = M[fp+4]... r2... CS553 Leture Register Allotion I 25 e f d Exmple to 3-olor this grph (,, ) e f b d Weighted order: b d f e!wht if you use different order? CS553 Leture Register Allotion I 26

14 ! Attempt Exmple to 2-olor this grph (, ) Weighted order: b b CS553 Leture Register Allotion I 27 Conepts!Liveness! Used in register llotion! Generting liveness! Flow nd diretion! Dt-flow equtions nd nlysis!control flow grphs!register llotion! sope of llotion! grnulrity: wht is being lloted to register! order tht llotion units re visited in mtters in ll heuristi lgorithms!globl pproh: greedy oloring CS553 Leture Introdution to Dt-flow Anlysis 28

15 Next Time!Reding! Ch. 8.4, , intro to dt-flow nlysis! Ch 8.8 nd Briggs pper, register llotion!leture! Improvements to grph oloring register llotors! Register llotion ross proedure lls!suggested Exerises! See shedule on website CS553 Leture Introdution to Dt-flow Anlysis 29