Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

We present two online algorithms for maintaining a topological order of a directed n-vertex acyclic graph as
arcs are added, and detecting a cycle when one is created. Our first algorithm handles m arc additions in
O(m3/2) time. For sparse graphs (m/n = O(1)), this bound improves the best previous bound by a logarithmic
factor, and is tight to within a constant factor among algorithms satisfying a natural locality property.
Our second algorithm handles an arbitrary sequence of arc additions in O(n5/2) time. For sufficiently dense
graphs, this bound improves the best previous bound by a polynomial factor. Our bound may be far from
tight: we show that the algorithm can take (n22
√
2 lgn) time by relating its performance to a generalization
of the k-levels problem of combinatorial geometry. A completely different algorithm running in (n2 log n)
time was given recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to the maintenance
of strong components, without affecting the asymptotic time bounds.