1. Optimizing Compilers CISC 673 Spring 2009 More Control Flow
John Cavazos
University of Delaware

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

3. 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

4. Class Problem: Identify Edges
procedure 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 = 0
InTree(root)
DFST(root)
Work out answer on the board..
Tree Edges:
Advancing Edges:
Back Edges:
Cross Edges:
Etc…

5. Class Problem: Answer procedure 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 = 0
InTree(root)
DFST(root)
notes

6. Reducibility Natural loops: Reducible: hierarchical, “well-structured”
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

7. Reducibility Example
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
reducible graph
irreducible graph

8. 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

9. 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

10. Dominance Tree 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
control-flow graph
dominator tree

11. Dominator Tree?

12. Dominator Tree

13. Graph is reducible iff …
Reducible Graph Test
Graph is reducible iff …
all back edges are ones whose head dominates its tail

14. Why is this not a Reducible Graph?

15. Why is this not a Reducible Graph?
Remember: head must dominate tail of back edge.

16. B (head) does not dominate C (tail)
Nonreducible Graph
Spanning Tree
B (head) does not dominate C (tail)

17. Reducible Graph? Build Spanning Tree
Back edges: head must dominate tail

18. 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

19. Find Natural Loops?

20. 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}

21. 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.

22. Strongly-connected component
SCC Example
Entry
Maximal strongly-connected component B1 B2
Strongly-connected component B3

23. 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

24. 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

25. Next Time
Dataflow analysis
Read Marlowe and Ryder paper