Difference Between Prims and Kruskal Algorithm

One fundamental difference between Prims and Kruskal algorithm is how they build a graph's minimum spanning tree (MST). Prim's algorithm starts with a root vertex and gradually incorporates adjacent vertices. On the other hand, Kruskal's algorithm selects the smallest weighted edge to construct the MST. 

While both algorithms are used to find the MST or minimum weight spanning tree of a graph, they employ different steps or approaches to achieve that. In this article, we will explore the difference between Prim's algorithm and Kruskal's algorithm in great detail. But before we move ahead, let's go through the topics we will cover in this article. 

Table of Contents (TOC)

• Difference Between Prims and Kruskal Algorithm
• What is Prim's Algorithm?
• What is Kruskal's Algorithm?
• Key Differences Between Prims and Kruskal Algorithm

Difference Between Prims and Kruskal Algorithm

For better clarity, let's comprehend difference between Prims and Krushal algorithms in a tabular format.

Aspect	Prim's Algorithm	Kruskal's Algorithm
Starting Point	Begins with any vertex and expands the MST.	Starts with the least weighted edge in the graph.
Traversal of Nodes	May traverse a node more than once to find the minimum distance.	Traverses each node only once.
Time Complexity	O(V^2), can be improved to O(E log V) with Fibonacci heaps.	O(E log V), where V is the number of vertices.
Graph Type	Works only on connected graphs, giving a connected component.	Can work on disconnected components, potentially generating a forest.
Efficiency	Runs faster in dense graphs.	Runs faster in sparse graphs.
MST Starting Point	Generates the MST starting from the root vertex.	Generates the MST starting from the least weighted edge.
Applications	Used in the Travelling Salesman Problem, networks for roads, and rail tracks.	Used in LAN connections, TV networks, etc.
Preferred Data Structure	Prefers list data structures.	Prefers heap data structures.
Cycle Detection	Naturally avoids cycles by connecting only vertices outside the MST.	Requires explicit cycle detection to avoid creating cycles when adding edges.
Data Structure	Often uses a priority queue to manage edges.	Utilizes a disjoint-set (union-find) data structure to manage trees.
Approach	Greedy algorithm that grows a single tree.	Greedy algorithm that merges multiple trees into one.
Implementation Complexity	Generally simpler in dense graphs due to the effective use of data structures.	Simpler in sparse graphs due to efficient edge selection and cycle detection.

What is Prim's Algorithm?

A Prim's algorithm is a greedy algorithm that finds the minimum spanning tree of a connected, weighted, undirected graph. It begins constructing the shortest spanning tree from any vertex in the graph and traverses one node more than once to acquire the minimum distance.

It is widely used in network design, such as the layout of electrical grids, computer networks, and road networks, due to its ability to efficiently determine the least expensive way to connect a set of points.

What is Kruskal's Algorithm?

Kruskal's Algorithm is also a greedy algorithm that finds the MST for a connected, undirected graph. It begins constructing the shortest spanning tree from the vertex having the lowest weight in the graph and traverses one node only once to acquire the minimum distance.

It is widely used in various applications, including designing networks, such as telecommunication networks, road and rail networks, and in algorithms for clustering data.

For more information regarding the Kruskal's agorithm, please refer to the All About Kruskal’s Algorithm article. 

Key Differences Between Prims and Kruskal Algorithm

Here are the top five key differences:

• Kruskal's algorithm can work with disconnected components, whereas Prim's algorithm only operates on connected graphs. 
• Prim's algorithm prefers using list data structures for managing vertices and edges. In contrast, Kruskal's algorithm prefers heap data structures (or sometimes sorted arrays) for managing edges based on their weights. 
• Prim's Algorithm has a time complexity of O(V²), which can be improved to O(ElogV) (V is the number of vertices, and E is the number of edges.) Meanwhile, Kruskal's algorithm consistently has a time complexity of O(ElogV), leveraging sorting of edges and the disjoint-set (union-find) data structure for efficiency.
• Prim's algorithm may traverse nodes multiple times to find the minimum distance edge that connects a new vertex to the growing MST. In contrast, Kruskal's algorithm traverses each node only once, focusing on adding the smallest weighted edge available that does not form a cycle, regardless of its connection to a current tree.

You can also explore the Understanding Dijkstra’s Algorithm article.