Kruskal's Algorithm

Algorithm

The purpose Kruskal's 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.

Also known as

It is clear that Kruskal's Algorithm 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.

Also see

• Kruskal's Algorithm produces Minimum Spanning Tree
• Prim's Algorithm

Source of Name

This entry was named for Joseph Bernard Kruskal.

Historical Note

Kruskal's algorithm was designed by Joseph Bernard Kruskal in $1956$.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 4.1$: The Minimal Connector Problem: An Introduction to Trees
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Kruskal's algorithm
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): tree
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Kruskal's algorithm
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tree
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Kruskal's algorithm