# 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.

## Source of Name

This entry was named for Robert Clay Prim.

## Historical Note

Prim's Algorithm was initially developed in $1930$ by Czech mathematician Vojtěch Jarník.

Robert Clay Prim independently discovered it in $1957$, and in $1959$ it was rediscovered once more by Edsger Wybe Dijkstra.

For these reasons it is also known as the DJP Algorithm, the Jarník Algorithm, or the Prim-Jarník Algorithm.