A Minimum Spanning Tree (MST) is a subset of edges in a connected, undirected graph that connects all the nodes without cycles and with the minimum possible total edge weight. It's widely used in network design, clustering, and optimization problems.

Key Concepts

  • Spanning Tree: A tree that includes all nodes of a graph with the minimum number of edges (no cycles, fully connected).
  • Minimum Weight: The sum of all edge weights in the spanning tree is minimized.
  • Applications:
    • Designing efficient networks (e.g., telecommunications, transportation).
    • Approximating solutions for NP-hard problems like the Traveling Salesman Problem.
    • Image segmentation in computer vision.

Common Algorithms

  1. Kruskal's Algorithm

    • Sorts all edges by weight and adds them one by one, avoiding cycles.
    • Uses a Union-Find data structure for cycle detection.
    Kruskal_algorithm

  2. Prim's Algorithm

    • Starts with a node and greedily adds the cheapest edge to expand the tree.
    • Efficient for dense graphs (more edges than nodes).
    Prim_algorithm

  3. Boruvka's Algorithm

    • A greedy approach that works well for graphs with many nodes.
    • Often used in distributed systems due to its parallel nature.

Practical Example

For a graph with nodes A, B, C, D and edge weights:

  • A-B: 1
  • B-C: 2
  • C-D: 3
  • A-D: 4
    The MST would include edges A-B (1), B-C (2), and C-D (3), totaling 6.

Extend Your Knowledge

Explore our Graph Theory Basics tutorial to understand how MSTs relate to other graph concepts.

💡 Tip: MSTs are always acyclic and contain exactly n-1 edges for n nodes. Ensure your graph is connected before applying these algorithms!