Prim’s algorithm is a minimum spanning tree algorithm that takes a graph as input and finds the subset of the edges of that graph which

• form a tree that includes every vertex
• has the minimum sum of weights among all the trees that can be formed from the graph

How Prim’s algorithm works

It falls under a class of algorithms called greedy algorithms that find the local optimum in the hopes of finding a global optimum.

We start from one vertex and keep adding edges with the lowest weight until we reach our goal.

The steps for implementing Prim’s algorithm are as follows:

1. Initialize the minimum spanning tree with a vertex chosen at random.
2. Find all the edges that connect the tree to new vertices, find the minimum and add it to the tree
3. Keep repeating step 2 until we get a minimum spanning tree

Example of Prim’s algorithm

Prim’s Algorithm pseudocode

The pseudocode for prim’s algorithm shows how we create two sets of vertices U and V-U. U contains the list of vertices that have been visited and V-U the list of vertices that haven’t. One by one, we move vertices from set V-U to set U by connecting the least weight edge.

T = ∅;
U = { 1 };
while (U ≠ V)
    let (u, v) be the lowest cost edge such that u ∈ U and v ∈ V - U;
    T = T ∪ {(u, v)}
    U = U ∪ {v}

Python Examples

Although adjacency matrix representation of graphs is used, this algorithm can also be implemented using Adjacency List to improve its efficiency.

/* Prim's Algorithm in Python */

INF = 9999999
/* number of vertices in graph */
V = 5
# create a 2d array of size 5x5
# for adjacency matrix to represent graph
G = [[0, 9, 75, 0, 0],
     [9, 0, 95, 19, 42],
     [75, 95, 0, 51, 66],
     [0, 19, 51, 0, 31],
     [0, 42, 66, 31, 0]]
/* create a array to track selected vertex */
/* selected will become true otherwise false */
selected = [0, 0, 0, 0, 0]
/* set number of edge to 0 */
no_edge = 0
/* the number of egde in minimum spanning tree will be
   always less than(V - 1), where V is number of vertices in
   graph */
/* choose 0th vertex and make it true */
selected[0] = True

/* print for edge and weight */
print("Edge : Weightn")
while (no_edge < V - 1):
    /* For every vertex in the set S, find the all adjacent vertices
     , calculate the distance from the vertex selected at step 1.
       if the vertex is already in the set S, discard it otherwise
       choose another vertex nearest to selected vertex  at step 1. */

    minimum = INF
    x = 0
    y = 0
    for i in range(V):
        if selected[i]:
            for j in range(V):
                if ((not selected[j]) and G[i][j]):  
                    # not in selected and there is an edge
                    if minimum > G[i][j]:
                        minimum = G[i][j]
                        x = i
                        y = j
    print(str(x) + "-" + str(y) + ":" + str(G[x][y]))
    selected[y] = True
    no_edge += 1

Prim’s vs Kruskal’s Algorithm

Kruskal’s algorithm is another popular minimum spanning tree algorithm that uses a different logic to find the MST of a graph. Instead of starting from a vertex, Kruskal’s algorithm sorts all the edges from low weight to high and keeps adding the lowest edges, ignoring those edges that create a cycle.

Prim’s Algorithm Complexity

The time complexity of Prim’s algorithm is O(E log V).

Prim’s Algorithm Application

• Laying cables of electrical wiring
• In network designed
• To make protocols in network cycles