Kruskal's algorithm
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Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted 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. If the graph is not connected, then it finds a minimum spanning forest (a minimum spanning tree for each connected component). Kruskal's algorithm is an example of a greedy algorithm.
It works as follows:
- 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
- remove an edge with minimum weight from S
- if that edge connects two different trees, then add it to the forest, combining two trees into a single tree
- otherwise discard that edge
At the termination of the algorithm, the forest has only one component and forms a minimum spanning tree of the graph.
This algorithm first appeared in Proceedings of the American Mathematical Society, pp. 48–50 in 1956, and was written by Joseph Kruskal.
With the use of a suitable data structure, Kruskal's algorithm can be shown to run in O (m log n) time, where m is the number of edges in the graph and n the number of vertices. The complexity bound given that the edges are already sorted (observe that the algorithm requires knowing the edge with minimum weight) is O(m <math>\alpha<math>(n)), using a disjoint sets data structure called Union-Find, where <math>\alpha<math> is the inverse of the Ackermann Function.
Proof
Let P be a connected, weighted graph and let Y be the subgraph of P produced by the algorithm. Y cannot have a cycle, since the last edge added to that cycle would have been within one subtree and not between two different trees. Y cannot be disconnected, since the first encountered edge that joins two components of Y would have been added by the algorithm. Thus, Y is a spanning tree of P.
Let Y1 be a minimum spanning tree for P which has the greatest number of edges in common with Y. If Y1=Y then Y is a minimum spanning tree. Otherwise, let e be the first edge considered by the algorithm that is in Y but not in Y1. Let C1 and C2 be the components of F which e lies between at the stage when e is considered. Since Y1 is a tree, Y1+e has a cycle and there is some different edge f of that cycle that also lies between C1 and C2. Then Y2=Y1+e-f is also a spanning tree. Since e is considered by the algorithm before f, the weight of e is at most equal to the weight of f, and since Y1 is a minimum spanning tree the weights of these two edges must in fact be equal. Therefore, Y2 is a minimum spanning tree having more edges in common with Y than Y1 does, contradicting our assumption about Y1. This proves that Y must be a minimum spanning tree.
Other algorithms for this problem include Prim's algorithm, and Boruvka's algorithm.
External links
- Animation of Kruskal's algorithm (http://students.ceid.upatras.gr/~papagel/project/kruskal.htm)
- Create and Solve Mazes by Kruskal's and Prim's algorithms (http://www.cut-the-knot.org/Curriculum/Games/Mazes.shtml)de:Algorithmus von Kruskal
fr:Algorithme de Kruskal he:אלגוריתם קרוסקל pl:Algorytm Kruskala sv:Kruskals algoritm