Difference between revisions of "Kruskal's algorithm"
(Created page with ''''Kruskal's algorithm''' is a general-purpose algorithm for the minimum spanning tree problem, based on the disjoint sets data structure. The existence of very simple al…') |
|||
| Line 1: | Line 1: | ||
| − | '''Kruskal's algorithm''' is a general-purpose algorithm for the [[minimum spanning tree]] problem, based on the [[disjoint sets data structure]]. The existence of very simple algorithms to maintain disjoint sets in almost constant time gives rise to simple implementations of Kruskal's algorithm whose running times are close to | + | '''Kruskal's algorithm''' is a general-purpose algorithm for the [[minimum spanning tree]] problem, based on the [[disjoint sets data structure]]. The existence of very simple algorithms to maintain disjoint sets in almost [[Asymptotic analysis|constant time]] gives rise to simple implementations of Kruskal's algorithm whose running times are close to linear, usually outperforming [[Prim's algorithm]] in sparse graphs. |
=Theory of the algorithm= | =Theory of the algorithm= | ||
| − | Kruskal's may be characterized as a [[greedy algorithm]], which builds the MST one edge at a time. As befits a MST algorithm, the greedy strategy is to continually add the remaining edge of lowest weight. Unlike Prim's, however, Kruskal's adds edges without regard to the connectivity of the partially built MST; that is, it does not necessarily add an edge emanating from a vertex that is in the partially built MST. Indeed, it may be said that Kruskal's starts with <math>V</math> forests of one vertex each, and adds edges one by one, each one causing two trees in the forest to coalesce into one, until all vertices have been placed in the same connected component and the MST is complete. In doing so, one must be careful not to add an edge between two vertices that are ''already'' in the same component, for doing so would create a cycle. | + | Kruskal's may be characterized as a [[greedy algorithm]], which builds the MST one edge at a time. As befits a MST algorithm, the greedy strategy is to continually add the remaining edge of lowest weight. Unlike Prim's, however, Kruskal's adds edges without regard to the connectivity of the partially built MST; that is, it does not necessarily add an edge emanating from a vertex that is in the partially built MST. Indeed, it may be said that Kruskal's starts with <math>V</math> forests of one vertex each, and adds edges one by one, each one causing two trees in the forest to coalesce into one, until all vertices have been placed in the same connected component and the MST is complete. In doing so, one must be careful not to add an edge between two vertices that are ''already'' in the same component, for doing so would create a cycle. We shall assume that a spanning tree exists for the following sections. (If you find them too difficult, skip them.) |
| − | + | ==Lemma== | |
| + | <p>Suppose that a subset <math>S</math> of the edges of a graph <math>G</math> is known to be a subset of the edges of some spanning tree of <math>G</math>. Consider the set of edges <math>T</math> containing exactly those edges <math>\in G\S</math> which, when added to <math>S</math>, do not induce a cycle. If a minimal-weight edge <math>\in T</math> is added to <math>S</math>, the resulting set is also guaranteed to be a subset of the edges of some spanning tree of <math>G</math>. | ||
| − | + | ''Proof'': Consider the graph <math>G'</math> with vertex set <math>V(G)</math> and edge set <math>S</math>. Now, any edge in <math>G\S</math> connects either two vertices in different connected components in <math>G'</math> or two vertices in the same component. If it connects two vertices in the same component, adding it generates a cycle (since there is already a path from one component to the other that does not use that edge, and adding the edge generates a return path) and it is not in <math>T</math>. Otherwise, it does not generate a cycle because it is a bridge in the resulting single connected component, and it is in <math>T</math>. That is, <math>T</math> consists of exactly those edges linking different connected components in <math>G'</math>. It is clear that <!-- | |
| − | + | ||
| − | + | ||
| − | < | + | |
==Lemma 2== | ==Lemma 2== | ||
Revision as of 06:18, 22 December 2009
Kruskal's algorithm is a general-purpose algorithm for the minimum spanning tree problem, based on the disjoint sets data structure. The existence of very simple algorithms to maintain disjoint sets in almost constant time gives rise to simple implementations of Kruskal's algorithm whose running times are close to linear, usually outperforming Prim's algorithm in sparse graphs.
Theory of the algorithm
Kruskal's may be characterized as a greedy algorithm, which builds the MST one edge at a time. As befits a MST algorithm, the greedy strategy is to continually add the remaining edge of lowest weight. Unlike Prim's, however, Kruskal's adds edges without regard to the connectivity of the partially built MST; that is, it does not necessarily add an edge emanating from a vertex that is in the partially built MST. Indeed, it may be said that Kruskal's starts with 
forests of one vertex each, and adds edges one by one, each one causing two trees in the forest to coalesce into one, until all vertices have been placed in the same connected component and the MST is complete. In doing so, one must be careful not to add an edge between two vertices that are already in the same component, for doing so would create a cycle. We shall assume that a spanning tree exists for the following sections. (If you find them too difficult, skip them.)
Lemma
Suppose that a subset 
of the edges of a graph 
is known to be a subset of the edges of some spanning tree of . Consider the set of edges 
containing exactly those edges 
which, when added to , do not induce a cycle. If a minimal-weight edge 
is added to , the resulting set is also guaranteed to be a subset of the edges of some spanning tree of .
Proof: Consider the graph 
with vertex set 
and edge set . Now, any edge in 
connects either two vertices in different connected components in or two vertices in the same component. If it connects two vertices in the same component, adding it generates a cycle (since there is already a path from one component to the other that does not use that edge, and adding the edge generates a return path) and it is not in . Otherwise, it does not generate a cycle because it is a bridge in the resulting single connected component, and it is in . That is, consists of exactly those edges linking different connected components in . It is clear that
