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Advanced Topic 38 of 51 Algorithms

Minimum Spanning Tree

Cheapest set of edges connecting all vertices.

How it works

A minimum spanning tree of a connected, weighted, undirected graph is a subset of edges that connects all vertices with no cycles and has the smallest possible total weight. It has V minus 1 edges. The cut property and cycle property justify greedy algorithms like Kruskal's and Prim's, and if all weights are distinct the tree is unique.

Where it is used

Designing low-cost networks of cables, pipelines or roads, clustering data, approximating the traveling salesman problem, image segmentation and laying out circuits use minimum spanning trees.

Common pitfall

It applies to undirected graphs; directed graphs need minimum arborescences instead. A disconnected graph has a spanning forest, not a tree. Tied weights can give several valid trees. A minimum spanning tree does not give shortest paths between vertices, a common confusion with Dijkstra.

Where it fits in the course

Minimum Spanning Tree is topic 38 of 51 in Algorithms, rated Advanced. It follows A* Search. The next topic is Kruskal's Algorithm.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Algorithms quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Minimum Spanning Tree?
Cheapest set of edges connecting all vertices. It belongs to Algorithms and is an advanced-level topic.
Where do people use Minimum Spanning Tree?
Designing low-cost networks of cables, pipelines or roads, clustering data, approximating the traveling salesman problem, image segmentation and laying out circuits use minimum spanning trees.
What is the most common mistake with Minimum Spanning Tree?
It applies to undirected graphs; directed graphs need minimum arborescences instead.
What should I learn before this topic?
Start with A* Search, the topic just before it in Algorithms.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Algorithms quiz. You can also explore Graph Traversal in the Visual Lab.
Is Minimum Spanning Tree hard to learn?
It is an Advanced topic. Expect to read the explanation more than once and to connect it with earlier topics. Working through a small example by hand helps most.
How long does it take to learn Minimum Spanning Tree?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Minimum Spanning Tree?
A* Search, Kruskal's Algorithm, Floyd-Warshall Algorithm are closely connected. The full list is in the related topics section of this page.
What should I study after Minimum Spanning Tree?
Continue with Kruskal's Algorithm, the next topic in Algorithms.

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