Time Complexity of Kruskal Algorithm: Insights, Comparisons, and Applications
Updated on Mar 18, 2025 | 12 min read | 3.5k views
Share:
For working professionals
For fresh graduates
More
Updated on Mar 18, 2025 | 12 min read | 3.5k views
Share:
Table of Contents
Hey there, fellow curious minds! Welcome to our blog, where we embark on an exciting journey into the depths of Kruskal’s Algorithm, the rockstar of graph theory!
In this blog, we’ll explore the time complexity of Kruskal’s algorithm, real-life applications, and the importance of data structures used in Kruskal’s algorithm.
With crystal-clear explanations and captivating examples, we’ll equip you with the knowledge to tackle complex problems and design efficient networks like a pro! So, hop on board because this blog is your ticket to mastering Kruskal’s Algorithm and unlocking the secrets of data structures!
In the vast realm of data structures, Kruskal’s algorithm is an essential tool in graph theory. It plays a crucial role in solving problems like finding the Minimum Spanning Tree (MST) of a connected, undirected graph. This remarkable algorithm, named after Joseph Kruskal, was designed to find the minimum-weight spanning tree for a given graph, where the sum of all edge weights is minimized while ensuring the tree remains connected.
Kruskal’s algorithm utilizes a greedy approach to construct the MST. It starts by sorting the edges in ascending order of their weights and then gradually adds edges to the MST, ensuring no cycles are formed until all vertices are connected. The process is intuitive, and efficient, and offers a promising solution for many real-world applications. Gain detailed understanding of this via Data Analytics 360 Cornell Certificate Program.
Prim’s and Kruskal’s algorithms are popular approaches to finding the minimum spanning tree (MST) in a connected weighted graph. The MST is a subset of edges that connects all vertices with the least total weight possible. Prim’s algorithm starts with a single vertex and repeatedly adds the minimum-weight edge that connects the current MST to a new vertex until all vertices are included.
Learn data science courses online from the World’s top Universities. Earn Executive PG Programs, Advanced Certificate Programs, or Masters Programs to fast-track your career.
As students delve into Kruskal’s algorithm, understanding its time complexity becomes crucial to comprehend its computational efficiency. The time complexity of this algorithm mainly depends on three fundamental operations:
Check out our free courses to get an edge over the competition.
The Union-Find (Disjoint Set) data structure plays a vital role in Kruskal’s algorithm, making it imperative to explore the time complexity of its key operation, the Union function.
The Union operation merges two disjoint sets into one set.
When performing the Union operation, the algorithm must ensure that merging two sets does not create any cycles in the forest. The algorithm employs “union by rank” or “union by size” and path compression to achieve this.
With the “union by rank” and path compression techniques, the time complexity of the Union operation becomes very efficient, approximately O(1). Due to these optimizations, the overall time complexity of Kruskal’s algorithm becomes dominated by the sorting of edges, making it O(E log E), as previously discussed
In the context of Kruskal’s algorithm, the worst-case scenario arises when sorting the edges takes the longest time. The time complexity O(E+log E) for sorting is a solid guarantee of the algorithm’s efficiency in practice.
Let’s visualize the worst-case time complexity with a detailed example:
Suppose we have a connected graph G(V, E) with V vertices and E edges. Each edge has a unique weight such that all edge weights are distinct. In this scenario, the sorting of edges becomes the most time-consuming operation.
Consider the following graph G with five vertices and seven edges:
Hence, the overall worst-case time complexity of Kruskal’s algorithm for this example would be:
Total Time Complexity ≈ O(7) O(7) O(7) ≈ O(21) ≈ O(E)
In the best-case scenario, Kruskal’s Algorithm’s time complexity is primarily determined by two operations: sorting the edges and performing Union-Find operations. Let’s break down the complexities of these operations:
Kruskal’s Algorithm begins by sorting all the edges in the non-decreasing order of their weights.
The most commonly used sorting algorithms like Merge Sort, Quick Sort, or Heap Sort have a time complexity of O(E log E), where E is the number of edges in the graph.
Kruskal’s Algorithm performs a Union-Find operation for each edge to detect cycles efficiently.
In the best case, the Union-Find operations have a time complexity of approximately O(log V), where V is the number of vertices in the graph. Considering both the sorting and Union-Find operations, the best-case time complexity of Kruskal’s Algorithm is approximately O(E+log E E log V).
Here are a few real-world applications where the Kruskals algorithm proves useful in the scenarios:
In some situations, it is uncommon for a graph to be structured as a forest. However, in worst-case scenarios, the Kruskals algorithm maintains an efficient time complexity of O(E log E).
Now let’s explore some instances where the Kruskals algorithm does not achieve its best-case time complexity:
In the average-case scenario, the time complexity analysis involves the probabilities of edge selections during the algorithm’s execution. To understand this better, let’s consider an example:
Example:
Suppose we have a connected graph with V vertices and E edges, where each edge has a unique weight. The edges are sorted in non-decreasing order. When we start adding edges to the MST, we can classify them into three categories based on the result of the Union-Find operation:
Now, let’s analyze the probabilities associated with these edge categories:
When adding edges to the MST, the probability of selecting a safe edge is approximately 1/V.
This is because there is only one way to form the MST for each edge added to the MST, and it contains one more vertex.
Thus, the probability of selecting a safe edge is 1/V.
The probability of choosing an unsafe edge is approximately 1/V.
Since there can be, at most, V-1 edges in the MST, the number of unsafe edges is V – (V-1) = 1.
Critical edges are those where adding them to the MST will increase its edge count by 1 without creating a cycle.
The probability of selecting a critical edge is approximately (V-2)/V.
Now, let’s calculate the average time complexity of Kruskal’s Algorithm using these probabilities:
Average Time Complexity = Ps * Ts Pu * Tu Pc * Tc
Ts, Tu, and Tc are the time complexities of safe, unsafe, and critical edge selection.
As safe edges and unsafe edges require O(log V) time for Union-Find operations, and critical edges require O(log V) time as well, the average case time complexity can be approximated as O(E log V).
In addition to time complexity, understanding the space complexity of Kruskal’s Algorithm is essential to evaluate its efficiency in memory usage. The primary space-consuming factor in the algorithm is the Kruskal algorithm in the data structure used for Union-Find operations.
The Union-Find Data Structure typically requires O(V) space to store each vertex’s parent and rank information. Sorting the edges can be done in place without requiring additional space. Hence, the space complexity of Kruskal’s Algorithm is O(V) for the Union-Find Data Structure.
Kruskal algorithm Python uses a Union-Find data structure to detect cycles and create the MST efficiently. The steps involve sorting the edges in ascending order of their weights and then iteratively adding edges to the MST while ensuring that no cycles are formed. The algorithm continues until all vertices are included in the MST.
Kruskal’s algorithm in C follows the same logic as in Python, but it uses arrays and loops to handle data structures. The algorithm efficiently selects edges while avoiding cycles, ultimately forming the MST.
Kruskal’s Algorithm in (DAA) Design and Analysis of Algorithms: Design and Analysis often involves a theoretical explanation of Kruskal’s algorithm. It emphasizes its greedy nature and proves its correctness and optimality regarding the MST.
Kruskal’s algorithm offers a powerful and efficient solution to graph theory’s Minimum Spanning Tree problem. The time complexity of Kruskal’s algorithm is O(E log E), making it an appealing choice for real-world applications such as network design, clustering, and transportation planning.
By understanding the time complexity of Kruskal’s algorithm and its inner workings, students can appreciate its efficiency in solving complex problems. So, embrace the power of Kruskal’s algorithm, and let it guide you through the intriguing world of data structures! Learn these concepts via
Unlock the power of data with our popular Data Science courses, designed to make you proficient in analytics, machine learning, and big data!
Elevate your career by learning essential Data Science skills such as statistical modeling, big data processing, predictive analytics, and SQL!
Stay informed and inspired with our popular Data Science articles, offering expert insights, trends, and practical tips for aspiring data professionals!
Get Free Consultation
By submitting, I accept the T&C and
Privacy Policy
Start Your Career in Data Science Today
Top Resources