Bidirectional pruned tree-based efficient minimum cut acceleration in dense graph

The minimum cut (min-cut) problem has been extensively investigated, and the corresponding algorithms have been used in many related problems. The recently proposed acceleration strategy based on tree-cut mapping has been shown to be an effective alternative, with a slight loss in acceleration accuracy. However, the existing method requires a large number of ineffective traversal passes in the high-overhead preprocessing step, which has the potential to be significantly improved in mapping effective graphs, i.e. dense graphs. To solve the problem, we propose to use bidirectional pruned tree for tree-cut mapping, which combines pruned depth-first traversal tree and bidirectional traversal paths to enumerate as many different tree instances as possible. Thus, it can help find the min-cut of any node pair efficiently. Theoretical analysis also shows its potential. For its serial implementation in a graph with M edges, after a preprocessing step with a time complexity as low as O(M), 99.9% of sampled node pairs can obtain their exact min-cut value in various types of dense graphs with degrees greater than 8 in experiments. In addition, the same precision can be obtained in sparse graphs in case of unit capacity. Per-pair running time in a million-node graph can be several orders of magnitude faster than existing methods and as low as several microseconds, so can be used as an effective complement to existing methods.