Optimizing Compilers CISC 673 Spring 2009 More Control Flow

UUNIVERSITYNIVERSITY OFOF D DELAWARE ELAWARE • • C COMPUTER & OMPUTER & IINFORMATION NFORMATION SSCIENCES CIENCES DDEPARTMENTEPARTMENT

Optimizing CompilersCISC 673

Spring 2009More Control Flow

John CavazosUniversity of Delaware

UUNIVERSITYNIVERSITY OFOF D DELAWARE ELAWARE • • C COMPUTER & OMPUTER & IINFORMATION NFORMATION SSCIENCES CIENCES DDEPARTMENTEPARTMENT 2

Clarification

Leaders are The entry point of the function Any instruction that is a target

of a branch Any instruction following a

(conditional or unconditional) branch


Additional Clarification

Tree edge:in CFG & ST

Advancing edge:(v,w) not tree edge but w is descendant of v in ST

Back edge:(v,w): v=w or w is proper ancestor of v in ST

Cross edge:(v,w): w neither ancestor nor descendant of v in ST


Class Problem: Identify Edgesprocedure DFST (v) pre(v) = vnum++ InStack(v) = true for w in Succ(v) if not InTree(w) add v→w to TreeEdges InTree(w) = true DFST(w) else if pre(v) < pre(w) add v→w to AdvancingEdges else if InStack(w) add v→w to BackEdges else add v→w to CrossEdges InStack(v) = false

for v in V do inTree(v) = false vnum = 0InTree(root)DFST(root)


Class Problem: Answerprocedure DFST (v) pre(v) = vnum++ InStack(v) = true for w in Succ(v) if not InTree(w) add v→w to TreeEdges InTree(w) = true DFST(w) else if pre(v) < pre(w) add v→w to AdvancingEdges else if InStack(w) add v→w to BackEdges else add v→w to CrossEdges InStack(v) = false

for v in V do inTree(v) = false vnum = 0InTree(root)DFST(root)


Reducibility

Natural loops: single entry node: no jumps into middle of

loop requires back edge into loop header (single

entry point)

Reducible: hierarchical, “well-structured” flowgraph reducible iff all loops in it

natural


Reducibility Example

A

B

C

D

E

F G

Entry

A

B

C

D

E

F G

Entry

reducible graph irreducible graph

Some languages only permit procedures with reducible flowgraphs (e.g., Java)

“GOTO Considered Harmful”:introduces irreducibility FORTRAN C C++

DFST does not find unique header in irreducible graphs


Dominance Node d dominates node i (“d dom i” )

if every path from Entry to i includes d Reflexive: a dom a Transitive:if a dom b and b dom c then

a dom c Antisymmetric: if a dom b and b dom a

then b=a


Immediate Dominance

a idom b iff a dom b there is no c such that a dom c, c dom b (c a, c b)

Idom’s: each node has unique idom relation forms a dominator tree


Dominance Tree

A

B

C

D

E

F G

Entry

Immediate and other dominators:(excluding Entry) a idom b; a dom a, b, c,

d, e, f, g b idom c; b dom b, c,

d, e, f, g c idom d; c dom c, d, e,

f, g d idom e; d dom d, e, f,

g e idom f, e idom g; e

dom e, f, g

A

B

C

D

E

F G

Entry

control-flow graph dominator tree


Dominator Tree?


Dominator Tree


Reducible Graph Test

Graph is reducible iff …all back edges are ones whose

head dominates its tail


Why is this not a Reducible Graph?


Why is this not a Reducible Graph?

Remember: head must dominate tail of back edge.


Nonreducible Graph

Spanning TreeB (head) does not dominate C

(tail)


Reducible Graph?

Build Spanning TreeBack edges: head must dominate tail


Natural Loops

Now we can find natural loops Given back edge m → n, natural

loop is n (loop header) and nodes that can reach m without passing through n


Find Natural Loops?


Natural Loops

Back Edge Natural Loop J → G {G,H,J} G → D {D,E,F,G,H,J} D → C {C,D,E,F,G,H,J} H → C {C,D,E,F,G,H,J} I → A {A,B,C,D,E,F,G,H,I,J}


Strongly-Connected Components

What about irreducible flowgraphs? Most general loop form = strongly-

connected component (SCC): subgraph S such that every node in S

reachable from every other node by path including only edges in S

Maximal SCC: S is maximal SCC if it is the largest SCC

that contains S.


SCC Example

Entry

B1

B2

B3

Strongly-connected component

Maximal strongly-connected component


Computing Maximal SCCs

Tarjan’s algorithm: Computes all maximal SCCs Linear-time (in number of nodes and

edges) CLR algorithm:

Also linear-time Simpler:

Two depth-first searches and one “transpose”:reverse all graph edge


Conclusion

Introduced control-flow analysis Basic blocks Control-flow graphs

Discussed application of graph algorithms: loops

Spanning trees, depth-first spanning trees Reducibility Dominators Dominator tree Strongly-connected components


Next Time

Dataflow analysis Read Marlowe and Ryder paper

Documents

Optimizing Compilers CISC 673 Spring 2009 More Control Flow