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.
Shumaila Saeed
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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Feb 04, 2024
Edge Selection
Chooses the smallest edge from a vertex to the tree.
Chooses the smallest edge overall.
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Feb 04, 2024
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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Feb 04, 2024
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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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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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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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
What data structure is commonly used in Prim’s Algorithm?
Priority queues are often used for efficient edge selection.
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Feb 04, 2024
Does Kruskal’s Algorithm require sorting of edges?
Yes, it sorts all edges by weight before constructing the tree.
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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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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Feb 04, 2024
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.
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