On loops, dominators, and dominance frontiers

This article explores the concept of loops and loop nesting forests of control-flow graphs, using the problem of constructing the dominator tree of a graph and the problem of computing the iterated dominance frontier of a set of vertices in a graph as guiding applications. The contributions of this article include: (1) An axiomatic characterization, as well as a constructive characterization, of a family of loop nesting forests that includes various specific loop nesting forests that have been previously defined. (2) The definition of a new loop nesting forest, as well as an efficient, almost linear-time, algorithm for constructing this forest. (3) An illustration of how loop nesting forests can be used to transform arbitrary (potentially irreducible) problem instances into equivalent acylic graph problem instances in the case of the two problems of (a) constructing the dominator tree of a graph, and (b) computing the iterated dominance frontier of a set of vertices in a graph, leading to new, almost linear-time, algorithms for these problems.