Difference Between Prim’s Algorithm and Kruskal’s Algorithm

Imagine you're laying out a new fiber-optic network across cities. You want to connect all locations using the least amount of cable—without creating loops. How do you decide which connections to include and which to skip?

This is where Minimum Spanning Tree (MST) algorithms come into play. MST algorithms help us find the most efficient way to connect all nodes in a weighted, undirected graph—whether it's computer networks, road systems, or clustering in machine learning.

Among the many solutions, Prim’s Algorithm and Kruskal’s Algorithm are the two most widely used greedy approaches.

While Prim’s focuses on expanding the tree from a starting node, Kruskal’s builds it by selecting the smallest available edges.

The most important difference? Prim’s grows node-by-node; Kruskal’s grows edge-by-edge.

In this blog, we’ll explore both algorithms—how they work, their differences, and when to use which.

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Difference Between Prim’s Algorithm and Kruskal’s Algorithm

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What is a Minimum Spanning Tree (MST)?

A Minimum Spanning Tree (MST) is a subset of the edges in a connected, undirected, and weighted graph that connects all the vertices with the minimum possible total edge weight, without forming any cycles.

In simple terms, it’s the cheapest way to link every node in a network without redundancy.

Key Properties of MSTs:

• An MST always has (V - 1) edges, where V is the number of vertices.
• There can be multiple MSTs for the same graph if different combinations result in the same minimum total weight.
• MSTs do not contain any cycles.
• All vertices in the graph must be connected.

Real-World Use Cases of MST:

• Network Design: Building efficient road, electrical, or communication networks.
• Clustering: In machine learning, MSTs help identify natural groupings in data.
• Approximation Algorithms: MSTs are used in heuristics for NP-hard problems like the Traveling Salesman Problem (TSP).

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Overview of Prim’s Algorithm

Prim’s Algorithm is a classic greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, undirected, weighted graph.

It starts with a single node and grows the MST one vertex at a time by always choosing the edge with the lowest weight that connects a node inside the MST to a node outside it.

Key Characteristics of Prim’s Algorithm:

• Greedy in nature: always picks the local minimum edge.
• Focuses on expanding a single connected component.
• Works best on dense graphs when implemented with priority queues.

Time Complexity and Implementation of Prim’s Algorithm

The efficiency of Prim’s algorithm depends on how the graph and priority queue are implemented:

• Fibonacci heap offers the best asymptotic time but is rarely used due to implementation overhead.
• Min-Heap (priority queue) is most common in practice (like in Dijkstra’s algorithm).

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Prim’s is powerful when you need a growing tree that remains connected at all times

How Prim’s Algorithm Works (With Example)

Let’s consider a simple graph with 5 vertices: A, B, C, D, E

Graph (Edge Weights):

Step-by-Step Execution (starting from A):

Final MST Weight = 13

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Overview of Kruskal’s Algorithm

Kruskal’s Algorithm is a greedy algorithm used to construct a Minimum Spanning Tree (MST) by sorting all the edges of a graph in ascending order of weight and then adding them one by one—only if they don’t form a cycle.

Unlike Prim’s, Kruskal’s builds the MST by connecting disjoint sets (or trees), gradually merging them into one connected component.

Key Characteristics of Kruskal’s Algorithm:

• Greedy: picks the lowest-weight edge at every step.
• Does not need a starting vertex.
• Relies on disjoint-set (union-find) data structure to avoid cycles.
• Works well on sparse graphs.

How Kruskal’s Algorithm Works (With Example)

Let’s use the same graph:

Step-by-Step Execution:

1. Sort edges by weight:
   B-C (1), A-B (2), A-C (3), B-D (4), C-D (5), C-E (6), D-E (7)
2. Initialize disjoint sets: {A}, {B}, {C}, {D}, {E}
3. Add edges (skip cycles):

Final MST Weight = 1 + 2 + 4 + 6 = 13

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Time Complexity and Implementation of Kruskal’s Algorithm

Kruskal’s efficiency depends on how you sort edges and how you manage disjoint sets.

Key Components:

• Edge Sorting: O(E log E)
• Union-Find (Disjoint Set Union) with path compression and union by rank: nearly constant time per operation → O(α(N)), where α is the inverse Ackermann function (very slow-growing)

Best suited for: Sparse graphs (fewer edges)

Kruskal’s is modular, intuitive, and highly efficient in scenarios where you can pre-process all edges and use smart cycle detection.

Which Algorithm Should You Use? Use Case Analysis

Choosing between Prim’s Algorithm and Kruskal’s Algorithm depends on the structure of your graph, the way your data is stored, and the nature of your application.

When to Use Prim’s Algorithm:

• You’re working with a dense graph (lots of edges).
• Your graph is represented using an adjacency matrix or list.
• You need to build the MST incrementally, especially from a known starting point.
• Ideal for real-time applications like GPS routing, network expansion planning, or dynamic map construction.

Example: Expanding an existing internet or road network where you start from a central hub and want to grow outwards efficiently.

When to Use Kruskal’s Algorithm:

• You’re dealing with a sparse graph (fewer edges than vertices).
• You have an edge list readily available.
• You want a simple and modular algorithm to apply offline.
• Useful when cycle detection is crucial or when working with disconnected graphs (will return a forest of MSTs).

Example: Offline processing of railway lines or circuit layout planning where all edges are known in advance.

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