Prim’s Algorithms

Definition

Prim’s algorithm is a greedy graph algorithm that constructs a minimum spanning tree for a connected, weighted, undirected graph by starting from any vertex and repeatedly selecting the minimum-weight edge that connects a vertex in the current tree to a vertex outside the tree, without forming a cycle.

In simple words, it grows a tree by always choosing the cheapest valid next edge.

Main Content

1. Minimum Spanning Tree (MST)

A spanning tree is a subgraph that:
includes all vertices of the graph,
is connected,
contains no cycles.
A minimum spanning tree is a spanning tree whose total edge weight is as small as possible.

For example, consider a graph with vertices A, B, C, D and weighted edges:

A-B = 1
A-C = 4
B-C = 2
B-D = 5
C-D = 3

One possible MST would choose:

A-B = 1
B-C = 2
C-D = 3

Total weight = 1 + 2 + 3 = 6

This is smaller than any other spanning tree for the same graph.

Prim’s algorithm is specifically designed to find such a tree efficiently.

2. Greedy Strategy in Prim’s Algorithm

Prim’s algorithm uses the greedy approach, meaning it makes the best local choice at each step.
At every stage, it selects the minimum-cost edge that connects:
one vertex already in the growing tree, and
one vertex not yet in the tree.

This works because:

the algorithm never allows a cycle,
it always expands the tree using the cheapest available connection,
it eventually covers all vertices.

Example idea

Suppose the current tree contains vertex A. The available edges from A are:

A-B = 2
A-C = 6

Prim’s algorithm chooses A-B because 2 < 6.

Now suppose the tree contains A, B, and the outgoing edges are:

A-C = 6
B-C = 3
B-D = 5

It chooses B-C = 3 next.

This repeated “choose the cheapest edge that expands the tree” rule is the core of Prim’s algorithm.

3. Graph Representation and Data Structures

Prim’s algorithm can be implemented using different graph representations:
Adjacency matrix
Adjacency list
The choice of representation affects efficiency.

With adjacency matrix

Easy to understand and implement.
Best for dense graphs.
Time complexity is usually O(V^2).

With adjacency list + priority queue

More efficient for sparse graphs.
Uses a min-priority queue to quickly select the smallest edge.
Time complexity can be O(E log V).

Common supporting data structures

key[]: stores the minimum edge weight needed to connect each vertex to the MST.
parent[]: stores the predecessor of each vertex in the MST.
mstSet[]: marks whether a vertex is already included in the MST.
Priority queue: helps repeatedly extract the vertex with the smallest key.

Small visual example

For the graph:

A --1-- B --2-- C
|      /         |
4     3          5
|   /            |
D ------------- E
       6

Prim’s algorithm may start at A, then choose the smallest edge from the tree outward at each step until all vertices are connected.

Working / Process

1. Start with any one vertex

Mark this vertex as part of the MST.
Set its key value to 0 so it is chosen first.
Set all other vertices’ keys to infinity initially.

2. Choose the minimum edge repeatedly

From all edges connecting the current tree to unvisited vertices, pick the one with the smallest weight.
Add the corresponding vertex and edge to the MST.
Update the keys of neighboring vertices if a cheaper connection is found.

3. Continue until all vertices are included

Repeat the selection and update process until every vertex is part of the MST.
The result is a spanning tree with minimum total cost.

Step-by-step example

Graph edges:

A-B = 2
A-C = 3
B-C = 1
B-D = 4
C-D = 5

Start from A:

Include A
Compare edges from A: choose A-B = 2
Tree now has A, B
From A, B, outgoing edges are:
A-C = 3
B-C = 1
B-D = 4
Choose B-C = 1
Tree now has A, B, C
Outgoing edges to D:
B-D = 4
C-D = 5
Choose B-D = 4

Final MST edges:

A-B = 2
B-C = 1
B-D = 4

Total weight = 7

ASCII view of the chosen MST

Advantages / Applications

Finds the minimum spanning tree, which minimizes total connection cost.
Useful in network design, such as connecting computers, roads, pipelines, or electrical grids with minimum expense.
Efficient and practical for both small and large graphs when implemented with the right data structures.

Additional applications

Telecommunication networks

: to connect routers/cities at minimum cost.

Transportation planning

: to design roads or railway links efficiently.

Circuit design

: to reduce wiring length.

Clustering and image processing

: MST-based methods are used in data grouping and segmentation.

Approximation algorithms

: MSTs help build solutions for harder optimization problems.

Summary

Prim’s algorithm is a greedy method for building a minimum spanning tree.
It grows the tree by repeatedly adding the smallest edge that connects a new vertex.
It is useful for minimizing total cost in connected weighted graphs.

Important terms to remember

spanning tree, minimum spanning tree, greedy algorithm, edge weight, priority queue, adjacency list, adjacency matrix