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arXiv · 1112.0784

A New Approach to Incremental Cycle Detection and Related Problems

Michael A. Bender·Jeremy T. Fineman·Seth Gilbert·Robert E. Tarjan
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Abstract

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, and the other to dense graphs. The former takes the minimum of O(m^{3/2}) and O(mn^{2/3}) time to insert m arcs into an n-vertex graph; the latter takes O(n^2 log(n)) time. Our sparse algorithm is considerably 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 topologically ordered vertex numberings; bounds on the size of the numbers give bound on running times.

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Michael A. Bender, Jeremy T. Fineman, Seth Gilbert, Robert E. Tarjan. 2011-12-04. A New Approach to Incremental Cycle Detection and Related Problems. https://arxiv.org/abs/1112.0784

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