1. Taken largely from University of Delaware Compiler Notes
Dominators and CFGs
Taken largely from University of Delaware Compiler Notes

2. Dominators
Node (basic block) D in a CFG dominates node N if every path from the start node to N goes through D. We say that node D is a dominator of node N.
Define dom(N) = set of node N’s dominators, or the dominator set for node N.
Note: by definition, each node dominates itself i.e., N  dom(N).

3. An Example Domination relation: { (1, 1), (1, 2), (1, 3), (1, 4) …
(2, 3), (2, 4), …
(2, 10) } 1 S 2 3
Direct domination: 4
1 <d 2, 2 <d 3, … 5 6 7
DOM: 8
DOM(1) = {1}
DOM(2) = {1, 2}
DOM(3) = {1, 2, 3}
DOM(10) = {1, 2, 10) 9 10

4. Immediate Dominators and Dominator Tree
Node M is the immediate dominator of node N ==> Node M must be the last dominator of N on any path from the start node to N.
Therefore, every node other than the start node must have a unique immediate dominator (the start node has no immediate dominator.)
What does this mean ?

5. Dominator Tree A flowgraph (left) and its dominator tree (right) S 1 23 4 5 6 7 8 9 10 S 1 2 3 4 5 7 8 6 9 10
A flowgraph (left) and its dominator tree (right)

6. Question Answer: NO! Example: consider nodes 5 and 8.
Assume an immediate dominator n’ of a node n, is n’ necessarily an immediate predecessor in the flow graph?
Answer: NO!
Example: consider nodes 5 and 8.

7. An Example (Dominators)1 2 3 4 5 6 7 8 9 10

8. Depth-First Search (DFS)
An “ordering” of nodes of CFG
BBs dfs_nums 1..N
Each BB dfs_num represents when that BB first encountered in DFS
Write dfs(BB root);

9. DFS and Depth-First Order
CONFUSING: They are NOT the same
DFS as defined: order of first visitation during depth-first search
Depth-First Order
“The depth-first ordering of the nodes is the reverse of a postorder traversal” (Aho-Sethi-Ullman)
Used (sometimes) to identify loops
Later …

10. An Example DFS 1 2 3 4 5 6 7 8 9 10