Analysis of a simple algorithm for global data flow problems

TLDR: There is an ordering of the nodes of a flow graph G which topologically sorts the dominance relation and can be found in 0(edges) bit vector steps and it follows that there is a very simple bit propagation algorithm which also uses the above ordering, and is at least as good as the interval algorithm for solving all known global data flow problems.

Abstract: There is an ordering of the nodes of a flow graph G which topologically sorts the dominance relation and can be found in 0(edges) steps. This ordering is the reverse of the order in which a node is last visited while growing any depth-first spanning tree of G. Moreover, if G is reducible, then this ordering topologically sorts the "dag" of G. Thus, for a reducible flow graph (rfg) there is a simple algorithm to compute the dominators of each node in 0(edges) bit vector steps.The main result of this paper relates two parameters of an rfg. If G is reducible, d is the largest number of back edges found in any cycle-free path in G, and k is the length of the interval derived sequence of G, then k≥d. From this result it follows that there is a very simple bit propagation algorithm (indeed, the obvious one) which also uses the above ordering, and is at least as good as the interval algorithm for solving all known global data flow problems such as "available expressions" and "live variables."