Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights. In real world applications, network designers often seek to quickly find a replacement edge for each edge in the MST. This code finds the lowest cost replacement edge for each edge of the MST, based on this paper:

David A. Bader and Paul Burkhardt, "A Linear Time Algorithm for Finding Minimum Spanning Tree Replacement Edges", arXiv:1908.03473v3, 2020.

• MST-ReplacementEdges-Tarjan: This subdirectory contains a superlinear implementation using the Tarjan approach for disjoint set unions.

• MST-ReplacementEdges-Gabow-Tarjan: This subdirectory implements the linear-time Bader-Burkhardt algorithm using the Gabow-Tarjan approach for disjoint set unions when the union tree is known in advance.

This code can also find the most vital edge -- the tree edge whose removal causes the highest cost -- in linear time.