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Encyclopedia > Kruskal's 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. In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... The minimum spanning tree of a planar graph. ... This article just presents the basic definitions. ... This article just presents the basic definitions. ... In an undirected graph, a connected component or component is a maximal connected subgraph. ... Greedy Algorithms are algorithms which follow the problem solving meta-heuristic of making the locally optimum choice at each stage with the hope of finding the global optimum. ...

This graph is a single well-connected component; it is provided as a sample subject for the algorithm.
This graph is a single well-connected component; it is provided as a sample subject for the algorithm.

It works as follows: Image File history File links Complete_network. ... Image File history File links Complete_network. ...

  • 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. A labeled tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ...


This algorithm first appeared in Proceedings of the American Mathematical Society, pp. 48–50 in 1956, and was written by Joseph Kruskal. Proceedings of the AMS, October 2005 issue. ... Year 1956 (MCMLVI) was a leap year starting on Sunday of the Gregorian calendar. ... Joseph Bernard Kruskal (b. ...


Other algorithms for this problem include Prim's algorithm, and Borůvka's algorithm. Prims algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. ... Borůvkas algorithm is an algorithm for finding minimum spanning trees. ...

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: The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ... Plot of log2 x In mathematics, the binary logarithm (log2 n) is the logarithm for base 2. ...

  • E is at most V2 and log V2 = 2 × logV is O(log V).
  • If we ignore isolated vertices, which will each be their own component of the minimum spanning tree anyway, V ≤ 2E, 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). A comparison sort is a particular type of sorting algorithm; a number of well-known algorithms are comparison sorts. ... Given a set of elements, it is often useful to break them up or partition them into a number of separate, nonoverlapping groups. ...


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. Counting sort is a sorting algorithm which (like bucket sort) takes advantage of knowing the range of the numbers in the array to be sorted (array A). ... To meet Wikipedias quality standards, this article may require cleanup. ... In computability theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. ...


Example

This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted.
AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted.
However, CE is now the shortest arc that does not form a loop, with length 5, so it is highlighted as the second arc.
The next arc, DF with length 6, is highlighted using much the same method.
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 it would form a loop ABD if it were chosen.
The process continutes 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.
Finally, the process finishes with the arc EG of length 9, and the minimum spanning tree is found.

Image File history File links Kruskal_Algorithm_0. ... Image File history File links Kruskal_Algorithm_1. ... Image File history File links Kruskal_Algorithm_2. ... Image File history File links Kruskal_Algorithm_3. ... Image File history File links Kruskal_Algorithm_4. ... Image File history File links Kruskal_Algorithm_5. ... Image File history File links Kruskal_Algorithm_6. ...

Proof of correctness

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.


For simplicity, assume that all edges have different weights. Let Y1 be a minimum spanning tree. 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. Y1+e has a cycle, because you cannot add an edge to a spanning tree and still have a tree. This cycle contains another edge f which at the stage of the algorithm where e is added to Y, has not been considered. This is because otherwise e would not connect different trees but two branches of the same tree. Then Y2=Y1+e-f is also a spanning tree. Its total weight is less than the total weight of Y1. This is because the algorithm visits e before f. It follows that Y1 is no minimum spanning tree, and the assumption that there exists an edge in Y, but not in Y1, was incorrect. This proves that Y=Y1, i.e., Y is a minimum spanning tree.


Pseudocode

 1 function Kruskal(G) 2 for each vertex v in G do 3 Define an elementary cluster C(v) ← {v}. 4 Initialize a priority queue Q to contain all edges in G, using the weights as keys. 5 Define a tree T ← Ø //T will ultimately contain the edges of the MST 6 while T has fewer than n-1 edges do 7 (u,v) ← Q.removeMin() 8 Let C(v) be the cluster containing v, and let C(u) be the cluster containing u. 9 if C(v) ≠ C(u) then 10 Add edge (v,u) to T. 11 Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u). 12 return tree T 

References

  • J. B. Kruskal: On the shortest spanning subtree and the traveling salesman problem. In: Proceedings of the American Mathematical Society. 7 (1956), pp. 48–50
  • 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.

Thomas H. Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. ... Charles E. Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof; as part of this effort, he developed the Cilk multithreaded language. ... Professor Ron Rivest Professor Ronald Linn Rivest (born 1947, Schenectady, New York) is a cryptographer, and is the Viterbi Professor of Computer Science at MITs Department of Electrical Engineering and Computer Science. ... Clifford Stein is a computer scientist, currently working as a professor at Columbia University in New York, NY. He earned his BSE from Princeton University in 1987, a MS from Massachusetts Institute of Technology in 1989, and a PhD from Massachusetts Institute of Technology in 1992. ... Cover of the second edition Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. ...

See also

Prims algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. ... Borůvkas algorithm is an algorithm for finding minimum spanning trees. ... Dijkstras algorithm, named after its discoverer, Dutch computer scientist Edsger Dijkstra, is an algorithm that solves the single-source shortest path problem for a directed graph with nonnegative edge weights. ...

External links

  • Animation of Kruskal's algorithm (Requires Java plugin)
  • Create and Solve Mazes by Kruskal's and Prim's algorithms at cut-the-knot
  • Minimum Spanning Tree Problem: Kruskal's Algorithm

  Results from FactBites:
 
summary (677 words)
Algorithm engineering views design, analysis, implementation, and experimental analysis of algorithms as an integrated process in the tradition of Popper's scientific method.
Several aspects of algorithms for miminum spanning trees most of which are related to data structures.
Our algorithm can be viewed as a positive answer to this question albeit using a different (better) algorithm.
Take a BrainSip (509 words)
'''Kruskal's algorithm''' is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph.
Kruskal's algorithm is an example of a greedy algorithm.
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.
  More results at FactBites »

 
 

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