Kruskal's algorithm

From Wikipedia Mirror

Jump to: navigation, search

Template:Graph search algorithm 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.

Other algorithms for this problem include Prim's algorithm, Reverse-Delete algorithm, and Borůvka's algorithm.

Contents

Performance

Where E is the number of edges in the graph and V is the number of 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 V2 and <math>\log V^2 = 2 \log V</math> is O(log V).
  • If we ignore isolated vertices, which will each be their own component of the minimum spanning forest, V ≤ E+1, so log V is 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 need to perform O(E) operations, two 'find' operations and possibly one union for each edge. 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 more sophisticated disjoint-set data structures to run in O(E α(V)) time, where α is the extremely slowly-growing inverse of the single-valued Ackermann function.

Example

Image Description
File:Prim Algorithm 0.svg This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted.
File:Kruskal Algorithm 1.svg AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted.
File:Kruskal Algorithm 2.svg CE is now the shortest arc that does not form a cycle, with length 5, so it is highlighted as the second arc.
File:Kruskal Algorithm 3.svg The next arc, DF with length 6, is highlighted using much the same method.
File:Kruskal Algorithm 4.svg The next-shortest arcs are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The arc BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen.
File:Kruskal Algorithm 5.svg The process continues to highlight the next-smallest arc, BE with length 7. Many more arcs 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.
File:Kruskal Algorithm 6.svg Finally, the process finishes with the arc 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 <math>P</math> be a connected, weighted graph and let <math>Y</math> be the subgraph of <math>P</math> produced by the algorithm. <math>Y</math> cannot have a cycle, since the last edge added to that cycle would have been within one subtree and not between two different trees. <math>Y</math> cannot be disconnected, since the first encountered edge that joins two components of <math>Y</math> would have been added by the algorithm. Thus, <math>Y</math> is a spanning tree of <math>P</math>.

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, there is some minimum spanning tree that contains F.

  • Clearly P is true at the beginning, when F is empty: any minimum spanning tree will do.
  • Now assume P is true for some non-final edge set F and let T be a minimum spanning tree that contains F. If the next chosen edge e is also in T, then P is true for F+e. Otherwise, T+e has a cycle C and there is another edge f that is in C but not F. Then T-f+e is a tree, and its weight is not more than the weight of T since otherwise the algorithm would choose f in preference to e. So T-f+e is a minimum spanning tree containing F+e and again P holds.
  • Therefore, by the principle of induction, P holds when F has become a spanning tree, which is only possible if F is a minimum spanning tree itself.

Pseudocode 

 1  function Kruskal(G)
 2    Define an elementary cluster C(v) ← {v}.
 3    Initialize a priority queue Q to contain all edges in G, using the weights as keys.
 4    Define a forest T ← Ø       //T will ultimately contain the edges of the MST
 5     // n is total number of vertices
 6    while T has fewer than n-1 edges do
 7      // edge u,v is the minimum weighted route from/to v
 8      (u,v) ← Q.removeMin()
 9      // prevent cycles in T. add u,v only if T does not already contain a path between u and v. 
10      // the vertices has been added to the tree.
11      Let C(v) be the cluster containing v, and let C(u) be the cluster containing u.
13      if C(v) ≠ C(u) then
14        Add edge (v,u) to T.
15        Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u).
16    return tree T

See also

References

External links

de:Algorithmus von Kruskal es:Algoritmo de Kruskal fa:الگوریتم کروسکال fr:Algorithme de Kruskal ko:크루스칼 알고리즘 it:Algoritmo di Kruskal he:האלגוריתם של קרוסקל ja:クラスカル法 nl:Kruskals algoritme pl:Algorytm Kruskala pt:Algoritmo de Kruskal ro:Algoritmul lui Kruskal ru:Алгоритм Краскала sk:Kruskalov algoritmus sr:Крускалов алгоритам sv:Kruskals algoritm zh:克鲁斯克尔演算法

Retrieved from "http://www.wiki-mirror.us/wiki/Kruskal%27s_algorithm"
Personal tools
Navigation