**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Kruskal's algorithm

Summary

Kruskal's algorithm (also known as Kruskal's method) finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.) It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest.
This algorithm first appeared in Proceedings of the American Mathematical Society, pp. 48–50 in 1956, and was written by Joseph Kruskal. It was rediscovered by .
Other algorithms for this problem include Prim's algorithm, the reverse-delete algorithm, and Borůvka's algorithm.
create a forest F (a set of trees), where each vertex in the graph is a separate tree
create a sorted set S containing all the edges in the graph
while S is nonempty and F is not yet spanning
remove an edge with minimum weight from S
if the removed edge connects two different trees then add it to the forest F, combining two trees into a single tree
At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.
The following code is implemented with a disjoint-set data structure. Here, we represent our forest F as a set of edges, and use the disjoint-set data structure to efficiently determine whether two vertices are part of the same tree.
algorithm Kruskal(G) is
F:= ∅
for each v ∈ G.V do
MAKE-SET(v)
for each (u, v) in G.E ordered by weight(u, v), increasing do
if FIND-SET(u) ≠ FIND-SET(v) then
F:= F ∪ {(u, v)} ∪ {(v, u)}
UNION(FIND-SET(u), FIND-SET(v))
return F
For a graph with E edges and V vertices, Kruskal's algorithm can be shown to run in O(E log E) time, or equivalently, O(E log V) time, all with simple data structures.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (14)

Kruskal's algorithm

Kruskal's algorithm (also known as Kruskal's method) finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.

Prim's algorithm

In computer science, Prim's algorithm (also known as Jarník's algorithm) is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. 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. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.

Minimum spanning tree

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected 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.

Related courses (20)

Related lectures (122)

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

CS-119(c): Information, Computation, Communication

L'objectif de ce cours est d'introduire les étudiants à la pensée algorithmique, de les familiariser avec les fondamentaux de l'Informatique et de développer une première compétence en programmation (

CS-250: Algorithms

The students learn the theory and practice of basic concepts and techniques in algorithms. The course covers mathematical induction, techniques for analyzing algorithms, elementary data structures, ma

Linear Algebra: Bases and Change of Basis

Covers the concept of bases in R² and the process of changing bases.

Optimisation with Constraints: Interior Point Algorithm

Explores optimization with constraints using KKT conditions and interior point algorithm on two examples of quadratic programming.

Graph Algorithms II: Traversal and Paths

Explores graph traversal methods, spanning trees, and shortest paths using BFS and DFS.