Prim 算法
Prim算法的执行非常类似于寻找图的最短通路的Dijkstra算法。Prim算法的特点是集合A中的边总是只形成单棵树。如图5所示,阴影覆盖的边属于正在生成的树,树中的结点为黑色。在算法的每一步,树中的结点确定了图的一个割,并且通过该割的轻边被加进树中。树从任意根结点r开始形成并逐渐生长直至该树跨越了V中的所有结点。在每一步,连接A中某结点到V-A中某结点的轻边被加入到树中,由推论2,该规则仅加大对A安全的边,因此当算法终止时,A中的边就成为一棵最小生成树。因为每次添加到树中的边都是使树的权尽可能小的边,因此上述策略也是贪心的。
有效实现Prim算法的关键是设法较容易地选择一条新的边添加到由A的边所形成的树中,在下面的伪代码中,算法的输入是连通图G和将生成的最小生成树的根r。在算法执行过程中,不在树中的所有结点都驻留于优先级基于key域的队列Q中。对每个结点v,key[v]是连接v到树中结点的边所具有的最小权值;按常规,若不存在这样的边则key[v]=∞。域p[v]说明树中v的“父母”。在算法执行中,GENERIC-MST的集合A隐含地满足:
A={(v,p[v])|vÎV-{r}-Q}
当算法终止时,优先队列Q为空,因此G的最小生成树A满足:
A={(v,p[v])|vÎ V-{r}}
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图5 Prim算法在图1所示的图上的执行流程
MST-PRIM(G,w,r)
1. Q←V[G]
2. for 每个uÎQ
3. do key[u]←∞
4. key[r]←0
5. p[r]←NIL
6. while Q≠Æ
7. do u←EXTRACT-MIN(Q)
8. for 每个vÎAdj[u]
9. do if vÎQ and w(u,v)<key[v]
10. then p[v]←u
11. key[v]←w(u,v)
Prim算法的工作流程如图5所示。第1-4行初始化优先队列Q使其包含所有结点,置每个结点的key域为∞(除根r以外),r的key域被置为0。第5行初始化p[r]的值为NIL,这是由于r没有父母。在整个算法中,集合V-Q包含正在生长的树中的结点。第7行识别出与通过割(V-Q,Q)的一条轻边相关联的结点uÎQ(第一次迭代例外,根据第4行这时u=r)。从集合Q中去掉u后把它加入到树的结点集合V-Q中。第8-11行对与u邻接且不在树中的每个结点v的key域和p域进行更新,这样的更新保证key[v]=w(v,p[v])且(v,p[v])是连接v到树中某结点的一条轻边。
Prim算法的性能取决于我们如何实现优先队列Q。若用二叉堆来实现Q,我们可以使用过程BUILD-HEAP来实现第1-4行的初始化部分,其运行时间为O(V)。循环需执行|V|次,且由于每次EXTRACT-MIN操作需要O(㏒V)的时间,所以对EXTRACT-MIN的全部调用所占用的时间为O(V㏒V)。第8-11行的for循环总共要执行O(E)次,这是因为所有邻接表的长度和为2|E|。在for循环内部,第9行对队列Q的成员条件进行测试可以在常数时间内完成,这是由于我们可以为每个结点空出1位(bit)的空间来记录该结点是否在队列Q中,并在该结点被移出队列时随时对该位进行更新。第11行的赋值语句隐含一个对堆进行的DECREASE-KEY操作,该操作在堆上可用0(㏒V)的时间完成。因此,Prim算法的整个运行时间为O(V㏒V+E㏒V)=O(E㏒V),从渐近意义上来说,它和实现Kruskal算法的运行时间相同。
通过使用Fibonacci堆,Prim算法的渐近意义上的运行时间可得到改进。在Fibonacci堆中我们己经说明,如果|V|个元素被组织成Fibonacci堆,可以在O(㏒V)的平摊时间内完成EXTRACT-MIN操作,在O(1)的平摊时间里完成DECREASE-KEY操作(为实现第11行的代码),因此,若我们用Fibonacci堆来实现优先队列Q,Prim算法的运行时间可以改进为O(E+V㏒V)。
算法实现如下:
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测试数据:
9 14
0 1 4
1 2 8
2 3 7
3 4 9
4 5 10
5 6 2
6 7 1
7 8 7
8 2 2
2 5 4
1 7 11
0 7 8
3 5 14
6 8 6