Generating SSA Form (mostly from Morgan)

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 bpostdominates 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) = Ø foreach X  SUCC(B) do if idom(X)  B then add X to DF(B) endif endfor foreach C  children(B) do foreach X  DF(C) do if idom(X)  B then add X to DF(B) endif endfor endfor 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, S Ouput: The set DF+(S) Worklist = Ø DF+ = Ø foreach B  S do DF+(S) = DF+(S)  {B} Worklist = Worklist  {B} endfor while 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 endfor endwhile

Algorithm to Insert -nodes1Morgan, 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 endfor endfor

Algorithm to Insert -nodes2Morgan, 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 endif endfor

Algorithm for Minimum -nodes3Morgan, 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 endif endfor

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

Generating SSA Form (mostly from Morgan)