最小生成树（minimum spanning tree）

Given a connected, undirected graph, a spanning tree of that graph is a subgraph which is a tree and connects all the vertices together. A single graph can have many different spanning trees. We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree.

 

A minimum spanning tree (MST) or minimum weight spanning tree is then a spanning tree with weight less than or equal to the weight of every other spanning tree. More generally, any undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components.

 

Prim's algorithm is known to be a good algorithm to find a minimum spanning tree.

1. Set i=0, S0= {u0=s}, L(u0)=0, and L(v)=infinity for v <> u0. If |V| = 1 then stop, otherwise go to step 2.
2. For each v in V/Si, replace L(v) by min{L(v), dvui}. If L(v) is replaced, put a label (L(v), ui) on v.
3. Find a vertex v which minimizes {L(v): v in V/Si}, say ui+1.
4. Let Si+1 = Si cup {ui+1}.
5. Replace i by i+1. If i=|V|-1 then stop, otherwise go to step 2.

 

The time required by Prim's algorithm is O(|V|2). It will be reduced to O(|E|log|V|) if heap is used to keep {v in V/Si : L(v) < infinity}.

 

资源链接

 

概念理解：
http://en.wikipedia.org/wiki/Minimum_spanning_tree
http://www.cs.man.ac.uk/~graham/cs2022/greedy/

demo演示：
http://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/dijkstra/Prim.shtml