# Kruskal's Minimum Spanning Tree Algorithm

October 30, 2019 • ☕️ 1 min read

Kruskal’s algorithm is a minimum-spanning-tree algorithm which
finds an edge of the least possible weight that connects any two
trees in the forest. It is a greedy algorithm in graph theory
as it finds a minimum spanning tree for a connected weighted
graph adding increasing cost arcs at each step. This means it
finds a subset of the edges that forms a tree that includes every
vertex, where the total weight of all the edges in the tree is
minimized. If the graph is not connected, then it finds a
minimum spanning forest (a minimum spanning tree for each
connected component).

A demo for Kruskal’s algorithm based on Euclidean distance.

## Minimum Spanning Tree

A **minimum spanning tree** (MST) or minimum weight spanning tree
is a subset of the edges of a connected, edge-weighted
(un)directed graph that connects all the vertices together,
without any cycles and with the minimum possible total edge
weight. That is, it is a spanning tree whose sum of edge weights
is as small as possible. More generally, any edge-weighted
undirected graph (not necessarily connected) has a minimum
spanning forest, which is a union of the minimum spanning
trees for its connected components.

A planar graph and its minimum spanning tree. Each edge is
labeled with its weight, which here is roughly proportional
to its length.

This figure shows there may be more than one minimum spanning
tree in a graph. In the figure, the two trees below the graph
are two possibilities of minimum spanning tree of the given graph.