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Kruskal's algorithm

Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component). Kruskal's algorithm is an example of a greedy algorithm.

It works as follows:
T = {};
while (T contains less than n-1 edges && E is not empty)
  choose a least cost edges (v,w) from E;
  delete (v,w) from E;
  if ((v,w) does not create a cycle in T)
    add (v,w) in T
    discard (v,w);
if (T contains fewer than n-1 edges)
  printf("no spanning tree \n");
Image Description
This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted.
AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted.
CE is now the shortest arc that does not form a cycle, with length 5, so it is highlighted as the second arc.
The next arc, DF with length 6, is highlighted using much the same method.
The next-shortest arcs are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The arc BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen.
The process continues to highlight the next-smallest arc, BE with length 7. Many more arcs are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD.
Finally, the process finishes with the arc EG of length 9, and the minimum spanning tree is found.

Where E is the number of edges in the graph and V is the number of vertices, Kruskal's algorithm can be shown to run in O(Elog E) time, or equivalently, O(E log V) time, all with simple data structures. These running times are equivalent because:

  • E is at most V2 and logV2 = 2logV is O(log V).
  • If we ignore isolated vertices, which will each be their own component of the minimum spanning forest, VE+1, so log V is O(log E).

We can achieve this bound as follows: first sort the edges by weight using a comparison sort in O(E log E) time; this allows the step "remove an edge with minimum weight from S" to operate in constant time. Next, we use a disjoint-set data structure to keep track of which vertices are in which components. We need to perform O(E) operations, two 'find' operations and possibly one union for each edge. Even a simple disjoint-set data structure such as disjoint-set forests with union by rank can perform O(E) operations in O(E log V) time. Thus the total time is O(E log E) = O(E log V).