Kruskal's Algorithm Calculator

The Ultimate Guide to Kruskal's Algorithm

Welcome to the definitive guide on Kruskal's algorithm, a cornerstone of graph theory and computer science. This greedy algorithm is the classic method for finding a Minimum Spanning Tree (MST) in a connected, undirected graph. Our interactive Kruskal's Algorithm Calculator above provides a visual, step-by-step demonstration to help you master this essential concept.

What is a Minimum Spanning Tree (MST)?

Before diving into the algorithm, let's understand its goal. A **Minimum Spanning Tree** of a weighted, connected graph is a subgraph that connects all the vertices (nodes) together, without any cycles, and with the minimum possible total edge weight. Imagine connecting a group of cities with fiber optic cables; the MST would represent the layout that connects all cities with the least amount of cable.

How Kruskal's Algorithm Works

Kruskal's algorithm for minimum spanning tree is elegant and intuitive. It builds the MST by adding one edge at a time, always choosing the cheapest possible option that doesn't create a problem.

1. 🔀 Sort All Edges: Create a list of all edges in the graph and sort them in non-decreasing order of their weights.
2. 🌳 Initialize a Forest: Start by considering each vertex as its own separate tree (this is a "forest" of trees).
3. 🔁 Iterate and Select: Go through the sorted list of edges, one by one, from the smallest weight to the largest. For each edge:
   • Check if adding this edge would form a cycle with the edges already selected for the MST.
   • If it **does not** form a cycle, add the edge to the MST.
   • If it **does** form a cycle, discard the edge and move to the next one.
4. 🏁 Stop: The algorithm terminates when the MST has (V-1) edges, where V is the number of vertices. At this point, all vertices are connected.

The key challenge is efficiently detecting cycles. This is where a data structure called Disjoint Set Union (DSU) or Union-Find is used. It keeps track of which vertices belong to which "tree" in the forest, allowing for near-instant cycle checks. This is how our Kruskal's algorithm calculator works internally.

Kruskal's Algorithm Example

Let's apply Kruskal's algorithm to find a minimum spanning tree for the default graph in the calculator. The first step is sorting the edges:

• (A, D, 5)
• (C, E, 5)
• (D, F, 6)
• (A, B, 7)
• (B, E, 7)
• ... and so on.

The algorithm will proceed as follows:

1. Consider (A, D, 5): Accepted. No cycle. MST: {(A,D)}.
2. Consider (C, E, 5): Accepted. No cycle. MST: {(A,D), (C,E)}.
3. Consider (D, F, 6): Accepted. No cycle. MST: {(A,D), (C,E), (D,F)}.
4. Consider (A, B, 7): Accepted. No cycle. MST: {(A,D), (C,E), (D,F), (A,B)}.
5. Consider (B, E, 7): Accepted. Joins the two separate tree structures. No cycle.
6. Consider (B, D, 9): Rejected! Adding this edge would form a cycle A-B-D-A.

This process continues until a full tree connecting all nodes is formed. The visualization above will illustrate this step-by-step.

Kruskal's Algorithm Time Complexity

Understanding the efficiency is crucial. The time complexity of Kruskal's algorithm is dominated by the initial sorting of edges. If E is the number of edges and V is the number of vertices:

• Sorting Edges: O(E log E)
• Iterating and Union-Find Operations: For each edge, we perform two `find` operations and one `union` operation. With path compression and union by rank/size optimizations, these operations are nearly constant time, close to O(α(V)), where α is the inverse Ackermann function (which is practically a small constant). So, this part is approximately O(E).

Therefore, the overall Kruskal's algorithm time complexity is O(E log E) or O(E log V), as in a connected graph E can be up to V².

Kruskal's vs. Prim's Algorithm

Kruskal's algorithm is often compared to Prim's algorithm, another greedy algorithm for finding MSTs.

• Kruskal's: Builds the MST by adding the cheapest edges from anywhere in the graph, as long as they don't form a cycle. It maintains a "forest" of trees that eventually merge.
• Prim's: Builds the MST by starting from a single vertex and growing the tree outwards, always adding the cheapest edge that connects a vertex in the tree to a vertex outside the tree. It always maintains a single, connected tree.

While some argue that "Prim's algorithm is simpler than Kruskal's algorithm" conceptually, Kruskal's can be more efficient for sparse graphs (where E is much smaller than V²). A key feature is that Kruskal's algorithm can also run on disconnected graphs, where it will find a Minimum Spanning Forest (a collection of MSTs, one for each connected component).

Finding a Maximum Spanning Tree

How can you explain how Prim's and/or Kruskal's algorithm can be used to find a maximum spanning tree? It's surprisingly simple! To find a Maximum Spanning Tree (a tree that connects all vertices with the largest possible total weight), you can:

1. Negate all edge weights (e.g., a weight of 10 becomes -10).
2. Run Kruskal's or Prim's algorithm as usual to find the MST on this new graph with negative weights.
3. The resulting tree is the Maximum Spanning Tree of the original graph.

Alternatively, when using Kruskal's, you can simply sort the edges in descending order of weight and proceed with the algorithm.

Conclusion: A Fundamental Algorithm for Networks

Kruskal's algorithm is more than just an academic exercise; it's a foundational concept for network design, circuit layout, and clustering problems. This interactive calculator is designed to provide a deep, intuitive understanding of its mechanics. By visualizing how the algorithm greedily selects the best edges while intelligently avoiding cycles, you can master a technique that is essential for computer science, operations research, and beyond.