Kruskal's algorithm

• create a forest F (a set of trees), where each vertex in the graph is a separate tree
• create a 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.

A demo for Kruskal's algorithm on a complete graph with weights based on Euclidean distance.

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)}
            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. These running times are equivalent because:

• E is at most ${\displaystyle V^{2}}$ and ${\displaystyle \log V^{2}=2\log V\in O(\log V)}$.
• Each isolated vertex is a separate component of the minimum spanning forest. If we ignore isolated vertices we obtain V ≤ 2E, so log V is ${\displaystyle O(\log E)}$.

We can achieve this bound as follows: first sort the edges by weight using a comparison sort in O(E log E) time; this allows the step "remove an edge with minimum weight from S" to operate in constant time. Next, we use a disjoint-set data structure to keep track of which vertices are in which components. We place each vertex into its own disjoint set, which takes O(V) operations. Finally, in worst case, we need to iterate through all edges, and for each edge we need to do two 'find' operations and possibly one union. Even a simple disjoint-set data structure such as disjoint-set forests with union by rank can perform O(E) operations in O(E log V) time. Thus the total time is O(E log E) = O(E log V).

Provided that the edges are either already sorted or can be sorted in linear time (for example with counting sort or radix sort), the algorithm can use a more sophisticated disjoint-set data structure to run in O(E α(V)) time, where α is the extremely slowly growing inverse of the single-valued Ackermann function.

The proof consists of two parts. First, it is proved that the algorithm produces a spanning tree. Second, it is proved that the constructed spanning tree is of minimal weight.

Kruskal's algorithm is inherently sequential and hard to parallelize. It is, however, possible to perform the initial sorting of the edges in parallel or, alternatively, to use a parallel implementation of a binary heap to extract the minimum-weight edge in every iteration.[3] As parallel sorting is possible in time ${\displaystyle O(n)}$ on ${\displaystyle O(\log n)}$ processors,[4] the runtime of Kruskal's algorithm can be reduced to O(E α(V)), where α again is the inverse of the single-valued Ackermann function.

A variant of Kruskal's algorithm, named Filter-Kruskal, has been described by Osipov et al.[5] and is better suited for parallelization. The basic idea behind Filter-Kruskal is to partition the edges in a similar way to quicksort and filter out edges that connect vertices of the same tree to reduce the cost of sorting. The following pseudocode demonstrates this.

function filter_kruskal(G) is
    if |G.E| < kruskal_threshold:
        return kruskal(G)
    pivot = choose_random(G.E)
    ${\displaystyle E_{<=}}$, ${\displaystyle E_{>}}$ = partition(G.E, pivot)
    A = filter_kruskal(${\displaystyle E_{<=}}$)
    ${\displaystyle E_{>}}$ = filter(${\displaystyle E_{>}}$)
    A = A ∪ filter_kruskal(${\displaystyle E_{>}}$)
    return A

function partition(E, pivot) is
    ${\displaystyle E_{<=}}$ = ∅, ${\displaystyle E_{>}}$ = ∅
    foreach (u, v) in E do
        if weight(u, v) <= pivot then
            ${\displaystyle E_{<=}}$ = ${\displaystyle E_{<=}}$ ∪ {(u, v)}
        else
            ${\displaystyle E_{>}}$ = ${\displaystyle E_{>}}$ ∪ {(u, v)}
    return ${\displaystyle E_{<=}}$, ${\displaystyle E_{>}}$

function filter(E) is
    ${\displaystyle E_{filtered}}$ = ∅
    foreach (u, v) in E do
        if find_set(u) ≠ find_set(v) then
            ${\displaystyle E_{filtered}}$ = ${\displaystyle E_{filtered}}$ ∪ {(u, v)}
    return ${\displaystyle E_{filtered}}$

Filter-Kruskal lends itself better for parallelization as sorting, filtering, and partitioning can easily be performed in parallel by distributing the edges between the processors.[5]

Finally, other variants of a parallel implementation of Kruskal's algorithm have been explored. Examples include a scheme that uses helper threads to remove edges that are definitely not part of the MST in the background,[6] and a variant which runs the sequential algorithm on p subgraphs, then merges those subgraphs until only one, the final MST, remains.[7]

• Prim's algorithm
• Dijkstra's algorithm
• Borůvka's algorithm
• Reverse-delete algorithm
• Single-linkage clustering
• Greedy geometric spanner

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 23.2: The algorithms of Kruskal and Prim, pp. 567–574.
• Michael T. Goodrich and Roberto Tamassia. Data Structures and Algorithms in Java, Fourth Edition. John Wiley & Sons, Inc., 2006. ISBN 0-471-73884-0. Section 13.7.1: Kruskal's Algorithm, pp. 632..

• Data for the article's example.
• Gephi Plugin For Calculating a Minimum Spanning Tree source code.
• Kruskal's Algorithm with example and program in c++
• Kruskal's Algorithm code in C++ as applied to random numbers