Dominators and CFGs

Dominators and CFGs Taken largely from University of Delaware Compiler Notes

Dominators Node (basic block) D in a CFG dominatesnode 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).

An Example Domination relation: { (1, 1), (1, 2), (1, 3), (1, 4) … (2, 3), (2, 4), … (2, 10) } S 1 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

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.) Immediate Dominators and Dominator Tree What does this mean ?

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

Question 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.

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

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);

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 the order in which we last visit the nodes in a preorder traversal” (Aho-Sethi-Ullman) • Used (sometimes) to identify loops • Later …

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

Dominators and CFGs