Write a program that takes an undirected graph and finds the minimum cut, i.e., the set of edges that, if removed, would disconnect the graph into two or more connected components. The program should have a time complexity of \$O(n^2m)\$, where n is the number of vertices and m is the number of edges in the graph.
One algorithm to solve this problem is the Karger's algorithm, which is a randomized algorithm that finds the minimum cut with high probability. Here is a high-level overview of the algorithm:
Initialize a "contracted graph" that is a copy of the original graph.
While the contracted graph has more than 2 vertices:
- Choose an edge at random from the contracted graph.
- Contract the two vertices connected by the chosen edge into a single vertex,
- removing the chosen edge.
- Repeat steps 1 and 2 until only 2 vertices remain.
The minimum cut is the set of edges that connect the two remaining vertices in the contracted graph.
This algorithm works by repeatedly contracting edges in the graph until only 2 vertices remain. The intuition behind the algorithm is that, as edges are contracted, the size of the cut decreases until there are only 2 vertices left, at which point the cut is the set of edges that connect those 2 vertices.
Karger's algorithm has a time complexity of \$O(n^2m)\$, which makes it relatively efficient for small to medium-sized graphs. However, it may not be practical for very large graphs.