Control-Flow Analysis and Loop Detection

Transcription

1 ! Control-Flow Anlysis nd Loop Detection!Lst time! PRE!Tody! Control-flow nlysis! Loops! Identifying loops using domintors! Reducibility! Using loop identifiction to identify induction vribles CS553 Lecture Control-Flow nd Loop Detection 1 Context!Dt-flow! Flow of dt vlues from defs to uses! Could lterntively be represented s dt dependence!control-flow! Sequencing of opertions! Could lterntively be represented s control dependence! e.g., Evlution of then-code nd else-code depends on if-test CS553 Lecture Control-Flow nd Loop Detection 2

2 Why study control flow nlysis?!finding Loops! most computtion time is spent in loops! to optimize them, we need to find them!loop Optimiztions! Loop-invrint code hoisting! Induction vrible elimintion! Arry bounds check removl! Loop unrolling! Prlleliztion!...!Identifying structured control flow! cn be used to speed up dt-flow nlysis CS553 Lecture Control-Flow nd Loop Detection 3 Representing Control-Flow!High-level representtion! Control flow is implicit in n AST!Low-level representtion:! Use Control-flow grph! Nodes represent sttements! Edges represent explicit flow of control!other options! Control dependences in progrm dependence grph (PDG) [Ferrnte87]! Dependences on explicit stte in vlue dependence grph (VDG) [Weise 94] CS553 Lecture Control-Flow nd Loop Detection 4

3 Wht Is Control-Flow Anlysis?!Control-flow nlysis discovers the flow of control within procedure (e.g., builds CFG, identifies loops) 1 := 0!Exmple b := * b!1 := 0!2 b := * b!3 L1: c := b/d!4 if c < x goto L2!5 e := b / c!6 f := e + 1!7 L2: g := f!8 h := t - g!9 if e > 0 goto L3!10 goto L1!11 L3: return 3 c := b/d c < x? 7 g := f h := t - g e > 0? CS553 Lecture Control-Flow nd Loop Detection 5 No 5 e := b / c f := e + 1 Yes 10 goto 11 return Loop Concepts!Loop: Strongly connected subgrph of CFG with single entry point (heder)!loop entry edge: Source not in loop & trget in loop!loop exit edge: Source in loop & trget not in loop!loop heder node: Trget of loop entry edge. Domintes ll nodes in loop.!bck edge: Trget is loop heder & source is in the loop!nturl loop: Associted with ech bck edge. Nodes dominted by heder nd with pth to bck edge without going through heder!loop til node: Source of bck edge!loop preheder node:!nested loop: Single node tht s source of the loop entry edge Loop whose heder is inside nother loop CS553 Lecture Control-Flow nd Loop Detection 6

4 Picturing Loop Terminology bck edge preheder entry edge hed loop til exit edge CS553 Lecture Control-Flow nd Loop Detection 7 The Vlue of Preheder Nodes!Not ll loops hve preheders! Sometimes it is useful to crete them h!without preheder node! There cn be multiple entry edges t!with single preheder node! There is only one entry edge p h pre-heder!useful when moving code outside the loop! Don t hve to replicte code for multiple entry edges t CS553 Lecture Control-Flow nd Loop Detection 8

5 ! d!! p Identifying Loops!Why?! Most execution time spent in loops, so optimizing loops will often give most benefit!mny pproches! Intervl nlysis! Exploit the nturl hierrchicl structure of progrms! Decompose the progrm into nested regions clled intervls! Structurl nlysis: generliztion of intervl nlysis! Identify domintors to discover loops!we ll focus on the domintor-bsed pproch CS553 Lecture Control-Flow nd Loop Detection 9 Domintor Terminology Domintors d dom i if ll pths from entry to node i include d Strict domintors sdom i if d dom i nd d! i entry d i d dom i entry Immedite domintors idom b if sdom b nd there does not exist node c such tht c!, c! b, dom c, nd c dom b idom b not " c, sdom c nd c sdom b Post domintors pdom i if every possible pth from i to exit includes i p (p dom i in the flow grph whose rcs re reversed nd entry nd exit re interchnged) p p pdom i b CS553 Lecture Control-Flow nd Loop Detection 10 exit

6 Identifying Nturl Loops with Domintors Bck edges A bck edge of nturl loop is one whose trget domintes its source Nturl loop The nturl loop of bck edge (m#n), where n domintes m, is the set of nodes x such tht n domintes x nd there is pth from x to m not contining n Exmple SCC with c nd d not loop becuse hs two entry points c b d e CS553 Lecture Control-Flow nd Loop Detection 11 t s n m The trget, c, of the edge (d#c) does not dominte its source, d, so (d#c) does not define nturl loop bck edge nturl loop b c d e Computing Domintors Input: Set of nodes N (in CFG) nd n entry node s Output: Dom[i] = set of ll nodes tht dominte node i Dom[s] = {s} for ech n $ N {s} Dom[n] = N repet chnge = flse for ech n $ N {s} D = {n} % (& p$pred(n) Dom[p]) if D! Dom[n] chnge = true Dom[n] = D until!chnge Key Ide If node domintes ll predecessors of node n, then it lso domintes node n p 1 p 2 p 3 CS553 Lecture Control-Flow nd Loop Detection 12 n pred[n] x $ Dom(p 1 ) ^ x $ Dom(p 2 ) ^ x $ Dom(p 3 ) ' x $ Dom(n)

7 Computing Domintors (exmple) Input: Set of nodes N nd n entry node s Output: Dom[i] = set of ll nodes tht dominte node i s {s} Dom[s] = {s} for ech n $ N {s} Dom[n] = N repet chnge = flse for ech n $ N {s} D = {n} % (& p$pred(n) Dom[p]) if D! Dom[n] chnge = true Dom[n] = D until!chnge {n, p, q, r, s} q r {n, p, q, r, s} n {n, p, q, r, s} Initilly Dom[s] = {s} CS553 Lecture Control-Flow nd Loop Detection 13 p Dom[q] = {n, p, q, r, s}... Finlly Dom[q] = Dom[r] = {q, s} {r, s} Dom[p] = {p, s} Dom[n] = {n, p, s} {n, p, q, r, s} Reducibility!Definitions! A CFG is reducible (well-structured) if we cn prtition its edges into two disjoint sets, the forwrd edges nd the bck edges, such tht! The forwrd edges form n cyclic grph in which every node cn be reched from the entry node! The bck edges consist only of edges whose trgets dominte their sources! A CFG is reducible if it cn be converted into single node using T1 nd T2 trnsformtions.!structured control-flow constructs give rise to reducible CFGs!Vlue of reducibility! Dominnce useful in identifying loops! Simplifies code trnsformtions (every loop hs single heder)! Permits intervl nlysis nd it is esy to clculte the CFG depth CS553 Lecture Control-Flow nd Loop Detection 14

8 T1 nd T2 trnsformtions!t1 trnsformtion! remove self-cycles!t2 trnsformtion! if node n hs unique predecessor p, then remove n nd mke ll the successors for n be successors for p b b c d c d CS553 Lecture Control-Flow nd Loop Detection 15 Hndling Irreducible CFG s!node splitting! Cn turn irreducible CFGs into reducible CFGs b b c d c d e d() e e() CS553 Lecture Control-Flow nd Loop Detection 16

9 Why Go To All This Trouble?!Modern lnguges provide structured control flow! Shouldn t the compiler remember this informtion rther thn throw it wy nd then re-compute it?!answers?! We my wnt to work on the binry code in which cse such informtion is unvilble! Most modern lnguges still provide goto sttement! Lnguges typiclly provide multiple types of loops. This nlysis lets us tret them ll uniformly! We my wnt compiler with multiple front ends for multiple lnguges; rther thn trnslte ech lnguge to CFG, trnslte ech lnguge to cnonicl LIR, nd trnslte tht representtion once to CFG CS553 Lecture Control-Flow nd Loop Detection 17 Induction vribles!induction vrible identifiction! Induction vribles! Vribles whose vlues form n rithmetic progression!why bother?! Useful for strength reduction, induction vrible elimintion, loop trnsformtions, nd utomtic prlleliztion!simple pproch! Serch for sttements of the form, i = i + c! Exmine reching definitions to mke sure there re no other defs of i in the loop! Does not ctch ll induction vribles. Exmples? CS553 Lecture Induction Vribles 18

10 ! s Exmple Induction Vribles!s = 0;!for (i=0; i<n; i++) += [i]; CS553 Lecture Induction Vribles 19 Induction Vrible Identifiction!Types of Induction Vribles! Bsic induction vribles (eg. loop index)! Vribles tht re defined once in loop by sttement of the form, i=i+c (or i=i-c), where c is constnt integer or loop invrint! Derived induction vribles! Vribles tht re defined once in loop s liner function of nother induction vrible! k = j + c 1 or! k = c 2 * j where c 1 nd c 2 re loop invrint CS553 Lecture Induction Vribles 20

11 Induction Vrible Triples!Ech induction vrible k is ssocited with triple (i, c 1, c 2 )! i is bsic induction vrible! c 1 nd c 2 re constnts such tht k = c 1 + c 2 * i when k is defined! k belongs to the fmily of i!bsic induction vribles! their triple is (i, 0, 1)! i = 0 + 1*i when i is defined CS553 Lecture Induction Vribles 21 Algorithm for Identifying Loop Invrint Code Input:!A loop L consisting of bsic blocks. Ech bsic block contins sequence of 3-ddress instructions. We ssume reching definitions hve been computed.! Output:!The set of instructions tht compute the sme vlue ech time through the!!!loop! Informl Algorithm:! 1.! Mrk invrint those sttements whose opernds re either!! Constnt!! Hve ll reching definitions outside of L! 2.! Repet until fixed point is reched: mrk invrint those unmrked sttements whose opernds re either!! Constnt!! Hve ll reching definitions outside of L!! Hve exctly one reching definition nd tht definition is in the set mrked invrint! Is this lst condition too strict?! CS553 Lecture Loop Invrint Code Motion 22

12 Algorithm for Identifying Loop Invrint Code (cont)!is the Lst Condition Too Strict?! No! If there is more thn one reching definition for n opernd, then neither one domintes the opernd! If neither one domintes the opernd, then the vlue cn vry depending on the control pth tken, so the vlue is not loop invrint Invrint sttements x = c 1 x = c 2 = x CS553 Lecture Loop Invrint Code Motion 23 Algorithm for Identifying Induction Vribles Input:!A loop L consisting of 3-ddress instructions, reching defs, nd loop-invrint informtion.! Output:!A set of induction vribles, ech with n ssocited triple.! Algorithm:! 1.!For ech stmt in L tht mtches the pttern i = i+c or i=i-c crete the triple (i, 0, 1).! 2.!Derived induction vribles: For ech stmt of L,!! If the stmt is of the form k=j+ c 1 or k=j* c 2!! nd j is n induction vrible with the triple (x, p, q)!! nd c 1 nd c 2 re loop invrint!! nd k is only defined once in the loop!! nd if j is derived induction vrible belonging to the fmily of i then!!the only def of j tht reches k must be in L!!nd no def of i must occur on ny pth between the definition of j nd k!! then crete the triple (x, p+ c 1, q) for k=j+ c 1 or (x, p* c 2, q* c 2 ) for k=j* c 2! CS553 Lecture Induction Vribles 24

13 Exmple: Induction Vrible Detection Picture from Prof Dvid Wlker s CS320 slides CS553 Lecture Induction Vribles 25 Algorithm for Strength Reduction Input:!A loop L consisting of 3-ddress instructions nd induction vrible triples.! Output:!A modified loop with new preheder.! Algorithm:! 1.!For ech derived induction vrible j with triple (i, p, q)!! crete new j!! fter ech definition of i in L, where i = i+c insert j = j + t!! put computtion t=q*c in preheder!! initilize j t the end of the preheder to j = p+q*i!! replce the definition of j with j=j! Note:!! j lso hs triple (i, p, q)!! multipliction hs been moved out of the loop! CS553 Lecture Induction Vribles 26

14 Exmple: Strength Reduction Picture from Prof Dvid Wlker s CS320 slides CS553 Lecture Induction Vribles 27 Algorithm for Induction Vrible Elimintion Input:!A loop L consisting of 3-ddress instructions, reching definitions, loopinvrint informtion, nd live-vrible informtion.! Output:!A revised loop.! Algorithm:! 1.! Apply copy propgtion followed by ded code elimintion to eliminte copies introduced by strength-reduction.! 2.! Remove ny induction vrible definitions where the induction vrible is only used nd defined within tht definition. (useless vrs)! 3.! For ech induction vrible i (lmost useless vrs)!! If only uses re to compute other induction vribles in its fmily nd in conditionl brnches, then mrk s eliminted!!use triple (j, c, d) in fmily ssocited with vrible k!!modify ech conditionl involving i so tht k is used insted, uses reltionships set up with triples!! Delete ll ssignments to the eliminted induction vrible! CS553 Lecture Induction Vribles 28

15 Concepts!Control-flow nlysis, Control-flow grph (CFG), Loop terminology, Identifying loops, Domintors, Reducibility!Induction vrible detection nd elimintion require loop identifiction! Induction vrible detection uses! strength reduction nd induction vrible elimintion! dt dependence nlysis, which cn then be used for prlleliztion! Strength reduction! removes multiplictions! the definition for some derived induction vribles no longer depend directly on bsic induction vrible! Induction vrible elimintion! removes unnecessry induction vribles CS553 Lecture Control-Flow nd Loop Detection 29 Next Time!Lecture! SSA CS553 Lecture Control-Flow nd Loop Detection 30