A new approach to incremental cycle detection and related problems

We consider the problem of detecting a cycle in a directed graph that grows by arc insertions and the related problems of maintaining a topological order and the strong components of such a graph. For these problems, we give two algorithms, one suited to sparse graphs, the other to dense graphs. The former takes O(min{m^1/2, n^2/3}m) time to insert m arcs into an n-vertex graph; the latter takes O(n² log n) time. Our sparse algorithm is substantially simpler than a previous O(m^3/2)-time algorithm; it is also faster on graphs of sufficient density. The time bound of our dense algorithm beats the previously best time bound of O(n^5/2) for dense graphs. Our algorithms rely for their efficiency on vertex numberings weakly consistent with topological order: we allow ties. Bounds on the size of the numbers give bounds on running time.