2011-02-26 CLRS Chapter23 Minimum Spanning Trees 最小生成树

Minimum Spanning Trees

General Solution：
GENERIC-MST(G, w)
1  A ← Ø
2  while A does not form a spanning tree
3      do find an edge (u, v) that is safe for A
4          A ← A ∪ {(u, v)}
5  return A
Kruskal's algorithm：
MST-KRUSKAL(G, w)
1  A ← Ø
2  for each vertex v ∈ V[G]
3       do MAKE-SET(v)
4  sort the edges of E into nondecreasing order by weight w
5  for each edge (u, v) ∈ E, taken in nondecreasing order by weight
6       do if FIND-SET(u) ≠ FIND-SET(v)
7             then A ← A ∪ {(u, v)}
8                  UNION(u, v)        //将两棵树中的root点进行合并
9  return A
Prim's algorithm：
MST-PRIM(G, w, r)
 1  for each u ∈ V [G]
 2       do key[u] ← ∞
 3          π[u] ← NIL
 4  key
← 0
 5   Q ← V [G]
 6   while Q ≠ Ø
 7       do u ← EXTRACT-MIN(Q)             //从Q中取出key值最小的点，加入到A中
 8          for each v ∈ Adj[u]
 9              do if v ∈ Q and w(u, v) < key[v]
10                    then π[v] ← u
11                         key[v] ← w(u, v)
//MST生成后，树中的点是通过π关系联结起来的，因此不用显示的“加入”到A中