Prim's Algorithm

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


 * Step 1: Choose any vertex of $G$, and add it to $T$.


 * Step 2: Add an edge of minimum weight $e$ to join a vertex in $T$ to one not in $T$.


 * Step 3: If $T$ spans $G$, stop. Otherwise, go to Step 2.

Proof
The above constitutes an algorithm, for the following reasons:

Finiteness
For each iteration through the algorithm, step 2 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 connecting $T$ to the remaining vertices can be arranged in order of weight, there is a definite edge (or set of edges) with minimal weight.


 * Step 3: 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 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 Prim's greedy algorithm.

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