Why does Prim and Kruskal's minimum spanning tree algorithm fail in directed graphs?

Prim's method and Kruskal's algorithm are two common methods for locating MST (minimum spanning tree) in undirected graphs. However, these techniques cannot generate correct MST for directed graphs. This is because directed graphs do not fit the basic assumptions and methods used by Prim and Kruskal's algorithms.

Prim Algorithm

First, there is Prim's algorithm, which involves adding edges to an expanding minimum spanning tree in a greedy manner until all vertices are covered. Vertices inside the MST are connected to vertices outside the MST through the edge with the lowest weight. Since all edges in an undirected graph can move in any direction, the shortest path from the MST to external vertices is easy to find. However, in a directed graph, the edges always point in one direction and there may not be a straight line connecting the MST and the external vertices. This contradicts the basic principle of Prim's algorithm.

An example of this is a directed edge (u,v) that connects vertex u in the MST to vertex v in the MST external graph. Since the MST in Prim's method must be connected to external vertices through direct edges, edges (u, v) are ignored, resulting in MST that may be inaccurate or insufficient.

Kruskal’s method

Kruskal's method is a weighted edge sorting technique that repeatedly adds minimum weight edges that do not generate cycles to the graph. This method works best for undirected graphs because the edges point in two directions, so cycles can be easily detected. Since the direction of edges matters in directed graphs, the concept of cycles becomes more subtle. Kruskal's approach ignores this complexity.

Assume there is a directed loop in the MST you are building. When applied to directed graphs, Kruskal's technique can generate trees containing directed cycles. This method produces inaccurate MST because its undirected edge-based cycle detection mechanism fails to properly capture cycles in directed graphs.

in conclusion

It can be concluded that although Prim and Kruskal's techniques are useful for locating MSTs in undirected graphs, they are not applicable to directed graphs. These methods produce inaccurate or inadequate MSTs because the underlying assumptions and mechanisms on which they rely do not hold in the setting of directed graphs. Directed graphs have their own unique properties and complexities, so it is important to employ digraph-specific techniques (such as the Chu−Liu/Edmonds method) to obtain minimum spanning trees.

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