Now that we understand the Cut Property guarantees we can safely add the 'lightest' crossing edge, let's explore an algorithm that directly applies this principle at every step: Prim's algorithm.

  • Prim's algorithm builds a Minimum Spanning Tree by greedily growing it from a single starting point. Think of it like a crystal forming, expanding one connection at a time.
  • The core idea is to start with an MST containing just one arbitrary starting vertex.
  • In a loop, the algorithm finds the minimum-weight edge that connects a vertex already in the MST to a vertex outside the MST.
  • It adds this cheapest "crossing" edge and its corresponding vertex to the MST, repeating until all vertices are included.
  • At each step, the algorithm implicitly defines a cut (MST vs. non-MST vertices) and leverages the Cut Property by always picking the lightest crossing edge, guaranteeing an optimal solution.

Prim's Algorithm (High-Level)

An iterative algorithm that finds an MST for a connected, weighted, undirected graph. It initializes the MST with an arbitrary vertex, then "grows" it by repeatedly adding the cheapest edge that connects a vertex in the growing tree to a vertex not yet in the tree, until all vertices from the original graph are included.