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This paper 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 paper 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.
Though linear time algorithms are already known for the two applications we consider, we believe that our approach itself is of interest because of its potential applicability to other problems, such as elimination-based dataflow analysis.
By: G. Ramalingam
Published in: ACM Transactions on Programming Languages and Systems , volume 24, (no 5), pages 455-90 in 2002
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