## Resources:

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

- Kruskal's algorithm video explanation
- Another video explanation
- OO implementation of Kruskal's
- Wikipedia article on Minimum Spanning Tree

Takeaways:

- 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!

## Discussion