Minimum Spanning Tree (Prim)

Lesson notes

Overview

Grow a tree by always adding the cheapest edge leaving it. This topic is rated beginner and is number 15 of 117 in Computer Science. Core ideas and first intuitions. No prior knowledge needed.

What you are seeing

The panel above shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

A minimum spanning tree connects all nodes of a weighted graph using the least total edge weight and no cycles. Prim’s algorithm starts at any node and repeatedly adds the cheapest edge that links the tree to a new node. It is greedy, yet provably optimal. Applications include laying cable or pipes at minimum cost, designing networks, and clustering. Kruskal’s algorithm is an alternative that sorts the edges.

Reading the relationship

The expression "connect all nodes at least cost" is a compact summary of the idea. It uses the symbols c, o, n, e, t, a; name what each one stands for before you start. Notice that it contains a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.

Try this

Start by only moving one slider at a time. Say aloud what you expect, then check.

Common pitfalls

Test small inputs by hand and then the largest ones. Off-by-one errors and forgetting how cost grows with input size are the most common traps.

How it connects

It builds on “Dijkstra’s Shortest Path”, the topic before it in Computer Science. Once you are comfortable, continue with “PageRank”.

Why it matters

Computing ideas decide how fast, how safely and how well software can solve problems.