Identifying loops in almost linear time

Loop identification is an essential step in performing various loop optimizations and transformations. The classical algorithm for identifying loops is Tarjan's interval-finding algorithm, which is restricted to reducible graphs. More recently, several people have proposed extensions to Tarjan's algorithm to deal with irreducible graphs. Havlak presents one such extension, which constructs a loop-nesting forest for an arbitrary flow graph. We show that the running time of this algorithm is quadratic in the worst-case, and not almost linear as claimed. We then show how to modify the algorithm to make it run in almost linear time. We next consider the quadratic algorithm presented by Sreedhar et al. which constructs a loop-nesting forest different from the one constructed by the Havlak algorithm. We show that this algorithm too can be adapted to run in almost linear time. We finally consider an algorithm due to Steensgaard, which constructs yet another loop-nesting forest. We show how this algorithm can be made more efficient by borrowing ideas from the other algorithms discussed earlier.