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.

Kruskal's algorithm

Animation of Kruskal's algorithm on a complete graph with weights based on Euclidean distance
Class	Minimum spanning tree algorithm
Data structure	Graph
Worst-case performance	${\displaystyle O(|E|\log |V|)}$

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 Borůvka's algorithm, Jarník's algorithm, and the reverse-delete algorithm.