A Dynamic Batch Algorithm for Maintaining a Topological Order

The dynamic topological order problem is that of efficiently updating a topological order after an edge insertion. This can be solved using a standard topological sort algorithm in O(v+e+b) time, for a batch of b edge insertions. However, this always traverses the entire graph when processing a batch of insertions, even if only a few edges are added. Dynamic topological order algorithms traverse only those regions of the graph where the ordering is actually affected by the new edges — usually much less than the whole graph. While these outperform the standard topological sort on relatively small insertion batches, they have a sub-optimal worst-case bound of O(b(v+ e)). We present, for the first time, a dynamic algorithm with an optimal O(v+e+ b) bound on its runtime, which only recomputes those region(s) of the order affected by the new edges. Thus, unlike all other algorithms for this problem, it is efficient on both small and large batches. To support this claim, we empirically evaluate it against the standard topological sort algorithm and a related dynamic topological order algorithm over a large number of random DAGs. The results confirm that our new algorithm is always the best choice.