Kruskal's algorithm

Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. The key steps of the algorithm are sorting and the use of a disjoint-set data structure to detect cycles. Its running time is dominated by the time to sort all of the graph edges by their weight.

A minimum spanning tree of a connected weighted graph is a connected subgraph, without cycles, for which the sum of the weights of all the edges in the subgraph is minimal. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.

This algorithm was first published by Joseph Kruskal in 1956, and was rediscovered soon afterward by Loberman & Weinberger (1957). Other algorithms for this problem include Prim's algorithm, Borůvka's algorithm, and the reverse-delete algorithm.

The algorithm performs the following steps:

At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.

The following code is implemented with a disjoint-set data structure. It represents the forest F as a set of undirected edges, and uses the disjoint-set data structure to efficiently determine whether two vertices are part of the same tree.

For a graph with E edges and V vertices, Kruskal's algorithm can be shown to run in time O(E log E) time, with simple data structures. This time bound is often written instead as O(E log V), which is equivalent for graphs with no isolated vertices, because for these graphs V / 2 ≤ E < V² and the logarithms of V and V² are again within a constant factor of each other.

To achieve this bound, first sort the edges by weight using a comparison sort in O(E log E) time. Once sorted, it is possible to loop through the edges in sorted order in constant time per edge. Next, use a disjoint-set data structure, with a set of vertices for each component, to keep track of which vertices are in which components. Creating this structure, with a separate set for each vertex, takes V operations and O(V) time. The final iteration through all edges performs two find operations and possibly one union operation per edge. These operations take amortized time O(α(V)) time per operation, giving worst-case total time O(E α(V)) for this loop, where α is the extremely slowly growing inverse Ackermann function. This part of the time bound is much smaller than the time for the sorting step, so the total time for the algorithm can be simplified to the time for the sorting step.