Minimum Spanning Tree (MST)

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🌐 What is a Spanning Tree?

Take a connected graph (a set of points joined by lines).
A spanning tree is:

• A set of edges
• That connects all vertices
• With no cycles
• And uses only the edges of the original graph

Think of it like placing just enough wires between rooms of a house so that electricity reaches everywhere, but you don’t waste wire on loops.

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💰 So What Makes It “Minimum”?

A Minimum Spanning Tree (MST) is a spanning tree where the total weight (sum of edge costs) is as small as possible.

If weights represent:

• distance → shortest cable length
• cost → cheapest network
• time → fastest connection

then MST helps you build the most efficient system.

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🎯 A Simple Weighted Graph

Let’s take a small graph as an example:

      (2)
   A ------- B
   | \       |
 (3|  \5)   (1)
   |    \    |
   C ---- D
      (4)

Each number is a weight — think of it as kilometers, cost, or effort.

Your goal:
Pick a set of edges that connects A, B, C, and D, avoids loops, and keeps the total weight as low as possible.

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🌳 An MST of the Above Graph

One possible MST for the graph looks like this:

      (2)
   A ------- B
   |
 (3|
   |
   C ------- D
        (4)

Selected edges: 2, 3, 4
Total weight = 2 + 3 + 4 = 9

Notice:

• All nodes are connected
• No cycles are formed
• Total weight is minimal

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🧠 How Do We Find an MST?

There are two famous methods, and both are surprisingly intuitive.

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⭐ 1. Kruskal’s Algorithm — “Pick the cheapest first”

Kruskal feels like shopping during a sale.
You sort all edges from cheapest to costliest, and then:

1. Pick the smallest edge.
2. Add the next smallest edge.
3. Skip an edge if it creates a cycle.
4. Keep going until all vertices are connected.

Super simple idea: Buy the cheap stuff first, but don’t take duplicates.

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⭐ 2. Prim’s Algorithm — “Grow like a tree”

Prim works like you’re slowly spreading out from one town.

1. Start at any vertex.
2. Choose the smallest edge that connects to a new vertex.
3. Repeat — always attach the next nearest vertex.
4. Stop when all vertices are included.

It’s like expanding a network step-by-step, always choosing the most affordable connection.

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✨ Why Minimum Spanning Trees Matter

MSTs appear in many real-world problems:

• Designing computer networks
• Planning telecom towers
• Laying railway tracks or roads
• Electricity grid design
• Clustering algorithms in machine learning
• Social network analysis

Anywhere you want everything connected efficiently, MSTs are extremely helpful.

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