Difference Between Prim’s Algorithm and Kruskal’s Algorithm
By Mukesh Kumar
Updated on Apr 22, 2025 | 8 min read | 1.64K+ views
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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
Feature/Parameter |
Prim’s Algorithm |
Kruskal’s Algorithm |
| Approach | Grows the MST from a starting vertex | Adds the shortest edges regardless of starting point |
| Graph Type Preference | Works best for dense graphs | More efficient for sparse graphs |
| Edge Sorting Required? | Not required | Yes, mandatory |
| Cycle Detection | Not explicitly needed | Required (via Union-Find / DSU) |
| Data Structure Used | Min-Heap (Priority Queue) and Adjacency List/Matrix | Disjoint Set (Union-Find) |
| Time Complexity | O(E log V) with Min-Heap & Adjacency List | O(E log E) with efficient DSU |
| Implementation | Comparatively complex due to priority queue operations | Conceptually simpler and easier to implement |
| MST Construction Style | Node-based (grows one node at a time) | Edge-based (adds the smallest edge that connects trees) |
| Graph Connectivity Needed | Requires graph to be connected | Can work even with disconnected graphs (builds forest) |
| Edge Count Handled | Handles dense graphs efficiently | Handles fewer edges efficiently |
| Start Node Required | Yes | No |
| Typical Use Case | Network design with live updates (e.g., real-time mapping) | Offline edge list processing (e.g., pre-computed routing) |
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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:
Data Structure |
Time Complexity |
Notes |
| Adjacency Matrix | O(V²) | Good for dense graphs |
| Adjacency List + Min-Heap (Binary Heap) | O(E log V) | Efficient and widely used |
| Adjacency List + Fibonacci Heap | O(E + log V) | Theoretically optimal, complex to implement |
- 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):
From |
To |
Weight |
| A | B | 2 |
| A | C | 3 |
| B | C | 1 |
| B | D | 4 |
| C | D | 5 |
| C | E | 6 |
| D | E | 7 |
Step-by-Step Execution (starting from A):
Step |
Current MST Nodes |
Edges Considered |
Edge Added |
MST Weight |
| 1 | A | A-B (2), A-C (3) | A-B (2) | 2 |
| 2 | A, B | A-C (3), B-C (1), B-D (4) | B-C (1) | 3 |
| 3 | A, B, C | C-D (5), C-E (6), B-D (4) | B-D (4) | 7 |
| 4 | A, B, C, D | D-E (7), C-E (6) | C-E (6) | 13 |
| 5 | A, B, C, D, E | — | — | 13 |
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:
From |
To |
Weight |
| A | B | 2 |
| A | C | 3 |
| B | C | 1 |
| B | D | 4 |
| C | D | 5 |
| C | E | 6 |
| D | E | 7 |
Step-by-Step Execution:
- 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) - Initialize disjoint sets: {A}, {B}, {C}, {D}, {E}
- Add edges (skip cycles):
Step |
Edge Added |
Action |
Disjoint Sets After Step |
| 1 | B-C (1) | No cycle, add to MST | {A}, {BC}, {D}, {E} |
| 2 | A-B (2) | No cycle, add to MST | {ABC}, {D}, {E} |
| 3 | A-C (3) | Forms a cycle (skip) | — |
| 4 | B-D (4) | No cycle, add to MST | {ABCD}, {E} |
| 5 | C-D (5) | Forms a cycle (skip) | — |
| 6 | C-E (6) | No cycle, add to MST | {ABCDE} |
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)
Implementation Element |
Time Complexity |
| Sorting edges | O(E log E) |
| Union-Find (efficient DSU) | O(E × α(V)) |
| Overall | O(E log E) |
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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Frequently Asked Questions
1. What is the primary difference between Prim’s and Kruskal’s algorithms?
Prim’s algorithm builds the MST by expanding from a starting vertex, adding the smallest edge connecting the tree to a new vertex. In contrast, Kruskal’s algorithm sorts all edges and adds them one by one, ensuring no cycles are formed, regardless of the starting point.
2. When should I prefer Prim’s algorithm over Kruskal’s?
Prim’s is more efficient for dense graphs where the number of edges is high, especially when the graph is represented using an adjacency matrix. It’s also suitable when you need to start from a specific node.
3. In which scenarios is Kruskal’s algorithm more efficient?
Kruskal’s excels in sparse graphs with fewer edges and when the graph is represented as an edge list. It’s also beneficial when the entire list of edges is known beforehand.
4. Do Prim’s and Kruskal’s algorithms always produce the same MST?
If all edge weights are unique, both algorithms will yield the same MST. However, with equal edge weights, multiple valid MSTs may exist, and the algorithms might produce different ones.
5. Can Prim’s and Kruskal’s algorithms handle disconnected graphs?
Prim’s algorithm requires a connected graph to function correctly. Kruskal’s algorithm can handle disconnected graphs by producing a Minimum Spanning Forest, which is a union of MSTs for each connected component.
6. What data structures are commonly used in these algorithms?
Prim’s algorithm typically uses a priority queue (often implemented with a min-heap) to select the next edge with the smallest weight. Kruskal’s algorithm uses a Disjoint Set Union (DSU) or Union-Find structure to detect cycles efficiently.
7. How do the time complexities of Prim’s and Kruskal’s algorithms compare?
Prim’s algorithm has a time complexity of O(E + V log V) when implemented with a min-heap and adjacency list. Kruskal’s algorithm has a time complexity of O(E log E) due to the initial sorting of edges.
8. Are there real-world applications where one algorithm is preferred over the other?
Yes. Prim’s algorithm is often used in network routing protocols and designing circuit connections where starting from a specific node is essential. Kruskal’s algorithm is preferred in clustering algorithms and when dealing with sparse graphs.
9. Can these algorithms be parallelized for faster computation?
Kruskal’s algorithm is more amenable to parallelization since sorting edges and performing union operations can be distributed. Prim’s algorithm is inherently sequential due to its dependency on the growing MST.
10. How do these algorithms handle negative edge weights?
Both algorithms can handle negative edge weights as long as the graph doesn’t contain negative cycles. Since MSTs are acyclic by definition, negative cycles are not a concern.
11. Is there a scenario where neither Prim’s nor Kruskal’s algorithm is ideal?
In dynamic graphs where edges are frequently added or removed, both algorithms may not be efficient due to the need to recompute the MST. In such cases, dynamic MST algorithms or data structures like dynamic trees are more suitable.
310 articles published
Mukesh Kumar is a Senior Engineering Manager with over 10 years of experience in software development, product management, and product testing. He holds an MCA from ABES Engineering College and has l...
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