Minimum Spanning Tree

Definition

A Minimum Spanning Tree (MST) is a fundamental concept in graph theory and network design.

“It refers to a tree-like subgraph of a given connected graph that connects all its vertices with the minimum possible total edge weight.”

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A Minimum Spanning Tree is a subgraph of a connected graph that
1. Contains all the vertices of the graph.
2. Is a tree, which means it has no cycles or loops.
3. Has the minimum possible total sum of edge weights among all such subgraphs.

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Suppose ABCD is a graph.

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G(V, E) represents the above graph, where ‘V‘ is the number of vertices and ‘E‘ is the number of edges.

The number of spanning trees in a complete graph with n vertices, often denoted as K_n, can be calculated using Cayley’s formula. Cayley’s formula states that the number of spanning trees in a complete graph with n vertices is equal to n^(n-2).

Cayley’s formula K₄ = 4^(4-2) = 16

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There are 16 spanning trees and the weights decide the minimum spanning tree.
The tree with minimum sum of weights is called the minimum spanning tree (MST).

• The above graph’s spanning tree would be stated as ST(V’, E’).
• In this situation, V’ = V denotes that the number of vertices in the spanning tree is equal to the number of vertices in the graph, but the number of edges is different.
• The number of edges must be E’ = V – 1, because edges is one less than the vertices.
• Also, if you create a spanning tree in a complete graph, then you must remove some edges. So, the maximum number of edges can be removed is equal to E – V + 1.
  In our example, the edges is 6, the vertices are 4, so we can remove 6 – 4 + 1 edges = 3 edges.

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Different Algorithms of Minimum Spanning Tree

1. Prim’s Algorithm – It starts with an arbitrary vertex and grows the minimum spanning tree one vertex at a time by adding the shortest edge that connects a vertex in the tree to a vertex outside the tree.

2. Kruskal’s Algorithm – It starts with an empty set of edges and adds the edges with the smallest weights while avoiding the creation of cycles. It’s based on sorting edges by weight.

Prim’s & Kruskal’s algorithm is greedy in nature.

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Time Complexity of Minimum Spanning Tree (MST)

The time complexity of finding the minimum spanning tree (MST) depends on the algorithm used. Two commonly used algorithms for finding the minimum spanning tree are Kruskal’s algorithm and Prim’s algorithm.

In both the algorithm, the time complexity is O(E log E) or O(E log V), where E is the number of edges and V is the number of vertices.

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Advantages of Minimum Spanning Tree (MST)

1. MST helps find the most cost-effective way to connect points. It’s like figuring out the cheapest and most efficient routes between different locations.
2. It ensures that all points (like cities or nodes in a network) are connected in the most efficient way possible, minimizing the total distance or cost of connections.
3. MST avoids creating unnecessary loops or redundant paths.
4. In computer networks or communication systems, MST helps design efficient connections, reducing delays and ensuring smooth communication between devices.
5. By selecting the minimum necessary connections, MST helps conserve resources such as time, energy, and materials.

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Disadvantages of Minimum Spanning Tree (MST)

1. There can be multiple MSTs for the same graph if there are edges with the same weight. This lack of uniqueness might complicate decision-making in certain applications.
2. MSTs are typically calculated based on static data, assuming fixed edge weights (travel times in the case of Google Maps). However, real-world conditions are dynamic and subject to change (e.g., accidents, road closures, traffic congestion).
3. For large graphs, finding the MST can become computationally expensive.
4. MST algorithms assume that the distance (or weight) between two points is the same in both directions. For example, if the distance from point A to point B is X, then the distance from point B to point A is also X. But in some real-world scenarios, distances may not be symmetric.

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Applications of Minimum Spanning Tree (MST)

1. Network Design

In communication networks, such as telephone or computer networks, MSTs help design the most efficient and cost-effective way to connect different locations or devices.

2. Cluster Analysis

MSTs can be used in clustering analysis to identify natural groupings or clusters within a set of data points.

3. Circuit Design

In electronic circuit design, MSTs assist in connecting components with minimal wire length, reducing the overall cost and improving the efficiency of the circuit.

4. Transportation Planning

MSTs can model transportation networks, helping in planning routes for vehicles or designing efficient delivery networks.

6. Resource Management

In resource management, such as water or power distribution systems, MSTs can help determine the most efficient way to connect various distribution points.

7. Image Segmentation

In image processing, MSTs are used for segmentation to identify distinct regions or objects within an image.

8. Robotics and Sensor Networks

MSTs are applied in robotics and sensor networks for efficient data transmission and communication. They help optimize the connectivity of sensors or robots in a network.

9. Approximation Algorithms

MSTs are fundamental in designing approximation algorithms for solving complex optimization problems. For example, the famous Traveling Salesman Problem (TSP) can be approximated using MSTs.

10. VLSI Design

Very Large Scale Integration (VLSI) design involves placing and connecting components on a chip. MSTs can be used to minimize the total wire length, improving the performance and reliability of the design.

11. Spanning Tree Protocol 

The Spanning Tree Protocol (STP) used in computer networks is a distributed algorithm that creates a loop-free topology.