# Prim’s Algorithm vs. Kruskal’s Algorithm: Know the Difference

By Shumaila Saeed || Published on February 4, 2024

**Prim’s Algorithm builds a minimum spanning tree by adding the cheapest edge from a vertex, while Kruskal’s Algorithm does so by adding the cheapest overall edge.**

## Key Differences

Prim’s Algorithm and Kruskal’s Algorithm both are used to find the minimum spanning tree in a graph. Prim's Algorithm starts from a single vertex and grows the spanning tree one edge at a time, always choosing the smallest edge that connects a vertex in the tree to a vertex outside. Kruskal's Algorithm, in contrast, sorts all the edges in the graph by weight and adds them one by one, but only if they don't form a cycle.

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Feb 04, 2024

Prim’s Algorithm and Kruskal’s Algorithm differ in their approach to cycle detection and edge selection. Prim's Algorithm inherently avoids cycles by connecting new edges to a growing tree. Kruskal's Algorithm requires a method like Union-Find to detect and avoid cycles as it builds the spanning tree from a collection of disjoint sets.

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Feb 04, 2024

In Prim’s Algorithm, each step involves finding the edge with the minimum weight that connects a vertex in the already built tree to any vertex outside the tree. Kruskal’s Algorithm, however, initially treats each vertex as a separate tree and combines them by repeatedly choosing the smallest edge that connects two different trees.

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Feb 04, 2024

The efficiency of Prim’s Algorithm is often better in dense graphs where the number of edges is high compared to the number of vertices. Kruskal’s Algorithm can be more efficient in sparse graphs, where the number of edges is much lower compared to the number of vertices.

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Feb 04, 2024

Implementation-wise, Prim’s Algorithm can be optimized using priority queues which can result in better performance for dense graphs. Kruskal’s Algorithm is generally implemented using a disjoint-set data structure which is efficient for cycle detection in graphs.

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Feb 04, 2024

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## Comparison Chart

### Starting Point

Begins at a single vertex and expands.

Treats each vertex as an individual tree.

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### Edge Selection

Chooses the smallest edge from a vertex to the tree.

Chooses the smallest edge overall.

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### Cycle Prevention

Inherently avoids cycles.

Requires a disjoint-set or similar structure.

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### Graph Type

More efficient for dense graphs.

More efficient for sparse graphs.

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### Implementation

Often uses priority queues.

Typically uses a disjoint-set data structure.

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## Prim's Algorithm and Kruskal's Algorithm Definitions

#### Prim's Algorithm

Prim’s Algorithm starts from a chosen vertex and expands the tree edge by edge.

The algorithm began at the central hub, expanding outward using Prim’s Algorithm.

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#### Kruskal's Algorithm

Kruskal’s Algorithm ensures the least total weight for the spanning tree.

To minimize wiring length, the technician used Kruskal’s Algorithm for the network layout.

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#### Prim's Algorithm

It incrementally builds the spanning tree by selecting the cheapest edge at each step.

Prim’s Algorithm selected the shortest bridge to add, keeping the overall construction under budget.

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#### Kruskal's Algorithm

It treats each node as a separate component and merges them without forming cycles.

Kruskal’s Algorithm systematically connected the isolated villages, avoiding redundant paths.

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#### Prim's Algorithm

It's particularly efficient for dense graphs in network optimization.

For the dense city grid, Prim’s Algorithm was ideal for laying out electrical lines.

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#### Kruskal's Algorithm

Kruskal’s Algorithm is ideal for sparse graphs with fewer edges.

In the sparse rural area, Kruskal’s Algorithm optimized the water pipeline network.

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#### Prim's Algorithm

Prim’s Algorithm continuously connects the nearest unconnected vertex.

Prim’s Algorithm connected the outlying areas to the main network, ensuring minimal distance.

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Jan 23, 2024

#### Kruskal's Algorithm

Kruskal’s Algorithm forms a minimum spanning tree by adding edges in order of increasing weight.

Kruskal’s Algorithm was used to efficiently layout roads between towns.

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#### Prim's Algorithm

Prim’s Algorithm is a greedy method to construct a minimum spanning tree for a weighted graph.

Using Prim’s Algorithm, the network design minimized the total cost of cabling.

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Jan 23, 2024

#### Kruskal's Algorithm

It requires a disjoint-set data structure for cycle detection and union operations.

Implementing Kruskal’s Algorithm, the engineer used a disjoint-set to manage the railway sections.

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Jan 23, 2024

## Repeatedly Asked Queries

#### How does Kruskal’s Algorithm work?

It constructs the minimum spanning tree by sorting all edges and adding them to the tree if they don't form a cycle.

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Feb 04, 2024

#### When should I use Kruskal’s Algorithm?

It's ideal for sparse graphs with fewer edges compared to the number of vertices.

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#### What is Prim’s Algorithm?

It's a method to find the minimum spanning tree for a weighted undirected graph by building it one edge at a time.

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#### Does Prim’s Algorithm require a starting vertex?

Yes, it starts from a specific vertex and expands the tree from there.

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#### Can Kruskal’s Algorithm work with disconnected graphs?

It can, but it will only find the minimum spanning forest, not a single tree.

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#### How does Kruskal’s Algorithm handle equal weight edges?

It can choose any of the equal weight edges as long as they don't form a cycle.

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#### What data structure is commonly used in Prim’s Algorithm?

Priority queues are often used for efficient edge selection.

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#### Does Kruskal’s Algorithm require sorting of edges?

Yes, it sorts all edges by weight before constructing the tree.

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#### Is Prim’s Algorithm suitable for all graph types?

It's best for dense graphs where the number of edges is much larger than the number of vertices.

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Feb 04, 2024

#### How does Kruskal’s Algorithm ensure no cycles are formed?

It typically uses a disjoint-set data structure to detect and prevent cycles.

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#### Is Prim’s Algorithm a greedy algorithm?

Yes, it selects the smallest edge at each step, making it a greedy approach.

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#### How does Kruskal’s Algorithm compare in efficiency to Prim’s?

It's generally more efficient for sparse graphs, while Prim's is better for dense graphs.

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#### Can Prim’s Algorithm be used for directed graphs?

It's designed for undirected graphs, and adaptations for directed graphs are non-trivial.

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#### What is a minimum spanning tree in the context of Prim’s Algorithm?

It's a subset of edges forming a tree that connects all vertices with the minimum total edge weight.

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#### Can Prim’s Algorithm produce different trees for different start vertices?

Yes, the resulting tree can vary based on the chosen starting vertex.

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#### What is the time complexity of Kruskal’s Algorithm?

It's O(E log E) or O(E log V), since sorting the edges dominates the time complexity.

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#### What is the time complexity of Prim’s Algorithm?

With a priority queue, it's typically O(E log V), where E is edges and V is vertices.

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Feb 04, 2024

#### Is Prim’s Algorithm deterministic?

Yes, given the same starting vertex, it will always produce the same result.

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#### What kind of graph is best suited for Kruskal’s Algorithm?

A graph where the edge to vertex ratio is low, meaning it's not densely connected.

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#### How does cycle prevention in Kruskal’s Algorithm benefit its process?

It ensures that the algorithm always produces a tree, not a graph with cycles.

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Feb 04, 2024

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About Author

Written by

Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.