# Kruskal's Algorithm

## Algorithm

The purpose of this algorithm is to produce a minimum spanning tree for any given weighted graph $G$.

Step 1: Start with the edgeless graph $T$ whose vertices correspond with those of $G$.
Step 2: Choose an edge $e$ of $G$ such that:
a): Adding $e$ to $T$ would not make a cycle in $T$;
b): $e$ has the minimum weight of all the edges remaining in $G$ that fulfil the condition in a).
Step 3: Add $e$ to $T$.
Step 4: If $T$ spans $G$, stop. Otherwise, go to Step 2.

The above constitutes an algorithm, for the following reasons:

### Finiteness

For each iteration through the algorithm, step 3 is executed, which increases the number of edges in $T$ by 1.

As a tree with $n$ nodes has $n-1$ edges, the algorithm will terminate after $n-1$ iterations.

### Definiteness

Step 1: Trivially definite.
Step 2: As the edges of a graph can be arranged in order of weight, there is a definite edge (or set of edges) with minimal weight. It is straightforward to select an edge $e$ which does not make a cycle in $T$, by ensuring that at least one end of $e$ is incident to a vertex which has not so far been connected into $T$.
Step 3: Trivially definite.
Step 4: It is straightforward to determine whether all the vertices are connected.

### Inputs

The input to this algorithm is the weighted graph $G$.

### Outputs

The output to this algorithm is the minimum spanning tree $T$.

### Effective

Each step of the algorithm is basic enough to be done exactly and in a finite length of time.

## Note

It is clear that this is a greedy algorithm: at each stage the minimum possible weight is chosen, without any analysis as to whether there may be a combination of larger weights which may produce a smaller-weight spanning tree.

For this reason, it is sometimes called Kruskal's greedy algorithm.

In this case, the greedy algorithm does produce the minimum spanning tree.

## Source of Name

This entry was named for Joseph Bernard Kruskal.