SSA. Stanford University CS243 Winter 2006 Wei Li 1

Transcription

1 SSA Wei Li 1

2 Overview SSA Representation SSA Construction Converting out of SSA 2

3 Static Single Assignment Each variable has only one reaching definition. When two definitions merge, a Ф function is introduced to with a new definition of the variable. First consider SSA for alias free variables. 3

4 Example: CFG a= a= = a+5 a= = a+5 Multiple reaching definitions = a+5 4

5 Example: SSA Form a 1 = a 3 = = a 1 +5 a 2 = = a 2 +5 a 4 = Ф(a 1,a 3 ) = a 4 +5 Single reaching definition 5

6 Ф Functions A Ф operand represents the reaching definition from the corresponding predecessor. The ordering of Ф operands are important for knowing from which path the definition is coming from. 6

7 SSA Conditions 1. If two nonnull paths X + Z and Y + Z converge at node Z, and nodes X and Y contains (V =..), then V = Ф(V,.., V) has been inserted at Z. 2. Each mention of V has been replaced by a mention of V i 3. V and the corresponding V i have the same value. 7

8 Overview SSA Representation SSA Construction Step 1: Place Ф statements Step 2: Rename all variables Converting out of SSA 8

9 Ф Placement a = a = Ф(a,a) Place minimal number of Ф functions a = Ф(a,a) 9

10 Renaming a 1 = a 2 = a 1 +5 a 4 = a 4 +5 a 3 = Ф(a 1,a 2 ) a

11 SSA Construction (I) Step 1: Place Ф statements by computing iterated dominance frontier 11

12 CFG A control flow graph G = (V, E) Set V contains distinguished nodes START and END every node is reachable from START END is reachable from every node in G. START has no predecessors END has no successors. Predecessor, successor, path 12

13 Dominator Relation If X appears on every path from START to Y, then X dominates Y. Domination is both reflexive and transitive. idom(y): immediate dominator of Y Dominator Tree START is the root Any node Y other than START has idom(y) ) as its parent Parent, child, ancestor, descendant 13

14 Dominator Tree Example START START a b c d CFG END DT 14

15 Dominator Tree Example START a a START b c d CFG END DT 15

16 Dominator Tree Example START a a START b c b c d CFG END DT 16

17 Dominator Tree Example START a a START b c b c d d CFG END DT 17

18 Dominator Tree Example START a a START END b c b c d d CFG END DT 18

19 Dominance Frontier Dominance Frontier DF(X) for node X Set of nodes Y X dominates a predecessor of Y X does not strictly dominate Y 19

20 DF Example START a a START END b c b c d DT d DF(c) =? CFG END DF(a) =? 20

21 DF Example START a a START END b c b c d DT d DF(c) = {d} CFG END DF(a) =? 21

22 DF Example START a a START END b c b c d DT d DF(c) = {d} CFG END DF(a) = {END} 22

23 Computing DF DF(X) is the union of the following sets DF local (X), a set of successor nodes that X doesn t t strictly dominate E.g. DF DF local (c (c)) = {d} DF up (Z) for all Z є Children(X) DF up DF up (Z) ) = {Y є DF(Z) idom(z) ) doesn t t strictly dominate Y} E.g. X = a, Z = d, Y = END 23

24 Iterated Dominance Frontier DF(SET) is the union of DF(X), where X є SET. Iterated dominance frontier DF + (SET) is the limit of DF 1 = DF(SET) and DF i+1 = DF(SET U DF i ) 24

25 Computing Joins J(SET) of join nodes Set of all nodes Z There are two nonnull CFG paths that start at two distinct nodes in SET and converge at Z. Iterated join J + (SET) is the limit of J 1 = J(SET) and J i+1 = J(SET U J i ) J + (SET) = DF + (SET) 25

26 Placing Ф Functions For each variable V Add all nodes with assignments to V to worklist W While X in W do For each Y in DF(X) do If no Ф added in Y then Place (V = Ф (V,,V)),V)) at Y If Y has not been added before, add Y to W. 26

27 Computational Complexity S a b c d E Constructing SSA takes O(A tot * avrgdf), where (A tot A tot : total number of assignments avrgdf: : weighted average DF size The computational complexity is O(n 2 ). e.g. nested repeat-until loops 27

28 Ф Placement Example a = a = Ф(a,a) Place Ф at Iterative Dominance Frontiers a = Ф(a,a) 28

29 SSA Construction (II) Step 2: Rename all variables in original program and Ф functions, using dominator tree and rename stack to keep track of the current names. 29

30 Variable Renaming Rename from the START node recursively For node X For each assignment (V = )) in X Rename any use of V with the TOS of rename stack Push the new name V i on rename stack i = i + 1 Rename all the Ф operands through successor edges Recursively rename for all child nodes in the dominator tree For each assignment (V = )) in X Pop V i in X from the rename stack 30

31 Renaming Example a 1 = a 1 +5 TOS Rename expr a= a = a+5 a= Ф(a 1,a) a+5 31

32 Overview SSA Representation SSA Construction Converting out of SSA 32

33 Converting Out of SSA Mapping all V i to V? 33

34 Overlapping Live Ranges Simply mapping all V i to V may not work a 1 = b 1 a 1 +5 b 2 = a 1 = b 1 b 1 +5 b 2 = a 1 = b 1 b 1 +5 b 2 = a 1 +5 a 1 +5 b

35 Converting Out of SSA Option 1: coloring Compute live ranges, and assign a unique variable name for each live range Similar techniques used in register allocation to be covered next week. Option 2: simply remove all Ф functions Every optimization in SSA needs to guarantee not to generate overlapping live ranges 35

36 Reference Efficient Computing Static Single Assignment Form and the Control Dependence Graph,, R. Cytron,, J. Ferrante,, B. Rosen, M. Wegman,, and F. K. Zadeck,, Transactions on Programming Languages and Systems (TOPLAS), Oct ently.html 36

37 Backup 37

38 Handling Arrays Difficult to treat A[i] ] as a variable = A[i] A[j] = V = A[k] = R(A,i) A = W(A,j,V) = R(A,k) = R(A 8,i 7 ) A 9 = W(A 8,j 6,V 5 ) = R(A 9,k 4 ) The entire array can be treated like a scalar. 38

39 Unnecessary Liveness W operator may introduce unnecessary liveness for A. Introduce HW (HiddenW( HiddenW). repeat A[i] = i i = i +1 until i>10 repeat i 2 = Ф(i 1,i 3 ) A 1 = Ф(A 0,A 2 ) A 2 = W(A 1,i 2,i 2 ) i 3 = i 2 +1 until i 3 >10 repeat i 2 = Ф(i 1,i 3 ) A 2 = HW(i 2,i 2 ) i 3 = i 2 +1 until i 3 >10 39