Kruskal's algorithm


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.
This algorithm first appeared in Proceedings of the American Mathematical Society, pp. 48-50 in 1956, and was written by Joseph Kruskal.
Other algorithms for this problem include Prim's algorithm, Reverse-delete algorithm, and Borůvka's algorithm.

Algorithm

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

Pseudocode

The following code is implemented with disjoint-set data structure:
algorithm Kruskal is
A := ∅
for each v ∈ G.V do
MAKE-SET
for each in G.E ordered by weight, increasing do
if FIND-SET ≠ FIND-SET then
A := A ∪
UNION, FIND-SET)
return A

Complexity

Kruskal's algorithm can be shown to run in O time, or equivalently, O time, where E is the number of edges in the graph and V is the number of vertices, all with simple data structures. These running times are equivalent because:
We can achieve this bound as follows: first sort the edges by weight using a comparison sort in O 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 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 operations in O time. Thus the total time is O = O.
Provided that the edges are either already sorted or can be sorted in linear time, the algorithm can use a more sophisticated disjoint-set data structure to run in O time, where α is the extremely slowly growing inverse of the single-valued Ackermann function.

Example

ImageDescription
AD and CE are the shortest edges, with length 5, and AD has been arbitrarily chosen, so it is highlighted.
CE is now the shortest edge that does not form a cycle, with length 5, so it is highlighted as the second edge.
The next edge, DF with length 6, is highlighted using much the same method.
The next-shortest edges are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The edge BD has been highlighted in red, because there already exists a path between B and D, so it would form a cycle if it were chosen.
The process continues to highlight the next-smallest edge, BE with length 7. Many more edges are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD.
Finally, the process finishes with the edge EG of length 9, and the minimum spanning tree is found.

Proof of correctness

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.

Spanning tree

Let be a connected, weighted graph and let be the subgraph of produced by the algorithm. cannot have a cycle, being within one subtree and not between two different trees. cannot be disconnected, since the first encountered edge that joins two components of would have been added by the algorithm. Thus, is a spanning tree of.

Minimality

We show that the following proposition P is true by induction: If F is the set of edges chosen at any stage of the algorithm, then there is some minimum spanning tree that contains F and none of the edges rejected by the algorithm.
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.
As parallel sorting is possible in time on processors, the runtime of Kruskal's algorithm can be reduced to O, 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. 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 is
if |G.E| < kruskal_threshold:
return kruskal
pivot = choose_random
, = partition
A = filter_kruskal
= filter
A = A ∪ filter_kruskal
return A
function partition is
= ∅, = ∅
foreach in E do
if weight <= pivot then
= ∪
else
= ∪
return,
function filter is
= ∅
foreach in E do
if find_set ≠ find_set then
= ∪
return
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.
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, and a variant which runs the sequential algorithm on p subgraphs, then merges those subgraphs until only one, the final MST, remains.