MST Construction Examples

Graph Theory · Axiom Academy

EXAMPLE MST Construction Examples Step-by-step walkthrough of Prim's and Kruskal's algorithms Graph Details: 5 vertices, 8 edges with weights shown Key Takeaways: MST Construction Algorithms Excellent work! You've mastered both Prim's and Kruskal's algorithms. Here's what we learned: Prim's Algorithm: Grows the MST from a starting vertex by always adding the minimum-weight edge that connects a new vertex to the existing MST. Uses a vertex-centric approach. Kruskal's Algorithm: Sorts all edges by weight and adds them in order, skipping any edge that would create a cycle. Uses an edge-centric approach with union-find data structure. Correctness Guarantee: Both algorithms always produce a minimum spanning tree. If edge weights are unique, the MST is unique. With duplicate weights, multiple valid MSTs may exist with the same total weight. Time Complexity: Prim's runs in O(E log V) with a priority queue. Kruskal's runs in O(E log E) due to edge sorting, where E is edges and V is vertices. When to Use Which: Prim's is better for dense graphs (many edges). Kruskal's is better for sparse graphs (few edges) and when edges are already sorted. Non-unique MSTs: When multiple edges have identical weights, different selection orders can produce different MST edge sets, but all valid MSTs have the same total weight. Practice both algorithms on different graphs to build intuition about when each approach is most efficient. Remember: both always find an optimal MST!

This is the written version of the interactive lesson above. See the full Graph Theory course.