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) = 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 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  -nodes 1 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 =  (T i, …, T i+n-1 ) in B endfor

Algorithm to Insert  -nodes 2 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 =  (T i, …, T i+n-1 ) in B endfor endif endfor

Algorithm for Minimum  -nodes 3 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 =  (T i, …, T i+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” V i –V => V i 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