Generating SSA Form(mostly from Morgan)

Why is SSA form useful?

• For many dataflow problems, SSA form enables sparse dataflow analysis that– yields the same precision as bit-vector CFG-based

dataflow analysis – but is asymptotically faster since it permits the

exploitation of sparsity

• SSA has two distinct features– factored def-use chains– renaming– you do not have to perform renaming to get

advantage of SSA for many dataflow problems

Summary of dependences

• Dependence– Data-dependence: relation between nodes

• Flow- or read-after-write (RAW)• Anti- or write-after-read (WAR)• Output- or write-after-write (WAW)

– Control-dependence: relation between nodes and edges (of CFG)

Postdominators

• Given a CFG G, node b postdominates node a each path (a … END) contains b.

b pdom a => b postdominates a

• Postdominance is dominance in “reverse CFG” (reverse direction of all CFG edges).

• Immediate Postdominator (ipdom) of B is the parent of B in the postdominator tree.

Dominance Frontier(from Morgan p. 75)

• “Just beyond” nodes dominated by block

• Dominance Frontier DF(B), set of blocks C– B dominates predecessor of C, but– B == C, OR, B does not dominate C

• Used to find “merge” points for -nodes

Dominance frontier

• Dominance frontier of node w– Node u is in dominance frontier of node w if w

• dominates a CFG predecessor v of u, but• does not strictly dominate u

• Dominance frontier = control dependence in reverse graph!

FindDominanceFrontier (B:Block)(from Morgan p. 76)

foreach C children(B) do FindDominanceFrontier(C)endfor

DF(B) = Øforeach X SUCC(B) do if idom(X) B then add X to DF(B) endifendfor

foreach C children(B) do foreach X DF(C) do if idom(X) B then add X to DF(B) endif endforendfor

Succ(B) is a CFG successor of B

children(B) are dominator tree children of B.

Iterated Dominance Frontier(from Morgan p. 171)

Input: Set of blocks, SOuput: The set DF+(S)

Worklist = ØDF+ = Øforeach B S do DF+(S) = DF+(S) {B} Worklist = Worklist {B}endforwhile Worklist Ø remove B from Worklist foreach C DF(B) do if C DF+ (S) then DF+(S) = DF+(S) {C} Worklist = Worklist {C} endif endforendwhile

Algorithm to Insert -nodes1

Morgan, p. 172

foreach T Variables do S = (B | B contains definition of T) {Entry} Compute DF+(S) foreach B DF+(S) n = |pred(B)| Insert an n-operand -node, T = (Ti, …, Ti+n-1) in B endforendfor

Algorithm to Insert -nodes2

Morgan, p. 173

foreach T Variables do if T Globals then S = (B | B contains definition of T) {Entry} Compute DF+(S) foreach B DF+(S) n = |pred(B)| Insert an n-operand -node, T = (Ti, …, Ti+n-1) in B endfor endifendfor

Algorithm for Minimum -nodes3

Morgan, p. 173

foreach T Variables do if T Globals then S = (B | B contains definition of T) {Entry} Compute DF+(S) foreach B DF+(S) if T LiveIn(B) n = |pred(B)| Insert an n-operand -node, T = (Ti, …, Ti+n-1) in B endif endfor endifendfor

Renaming Variables

• After inserting -nodes

• Match each use with the “new” name for definition

• Requires walk of the Dominator Tree– Each def of V, in block B, gets new “name” Vi

– V => Vi for all blocks dominated by B

• See handout for algorithm, page 175 of Morgan for details

Computing SSA form

• Cytron et al algorithm– compute DF relation (see slides on computing control-

dependence relation)– find irreflexive transitive closure of DF relation for set

of assignments for each variable

• Computing full DF relation– Cytron et al algorithm takes O(|V| +|DF|) time– |DF| can be quadratic in size of CFG

• Faster algorithms– O(|V|+|E|) time per variable: see Bilardi and Pingali

What’s Next?

• So, we’ve converted CFG to SSA form

• Coming Attractions– Optimizations on SSA– Converting SSA back to CFG