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Minimum Spanning Tree (Kruskal's Algorithm)

Updated on ・2 min read


This post requires knowledge of graphs and union-find (covered in earlier posts).

  1. Kruskal's algorithm video explanation
  2. Another video explanation
  3. OO implementation of Kruskal's
  4. Wikipedia article on Minimum Spanning Tree


  • A Minimum Spanning Tree (MST) is a subset of edges of an undirected, connected, weighted graph.
    • This means a MST connects all vertices together in a path that has the smallest total edge weight.
  • One algorithm for finding the MST of a graph is Kruskal's Algorithm.
  • Kruskal's algorithm is a greedy algorithm - this means it will make locally optimum choices, with an intent of finding the overall optimal solution.
  • Kruskal's algorithm relies on the union-find data structure.
    • First the algorithm sorts the graph's edges in ascending order (by weight).
    • Then for every edge, if it's vertices have different root vertices (determined by union-find's Find()), it will add the edge to a list & Union() it's vertices within the union-find data structure.
    • If roots are the same, it will skip the edge.
    • The final list represents the MST of the graph.
  • Another common algorithm for finding the MST of a graph is Prim's Algorithm. Commonly, Prim's uses a heap or priority queue in it's implementation.
  • Time complexity for Kruskal's algorithm is O(e log v) where e is the number of edges and v is the number of vertices in the graph. Space is O(e + v).

Below is my implementation of Kruskal's algorithm:

As always, if you found any errors in this post please let me know!

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