Algorithm 为什么Floyd Warshall中3个循环的顺序很重要？_Algorithm_Graph Algorithm

Algorithm 为什么Floyd Warshall中3个循环的顺序很重要？

algorithm

Algorithm 为什么Floyd Warshall中3个循环的顺序很重要？,algorithm,graph-algorithm,Algorithm,Graph Algorithm,在popular实现中，我们使用三个循环，循环变量i，j，k。此处i&j分别用于指示两个顶点的源和目标，因此“k”表示“中间”顶点。如果我把循环变量“k”放在第三位而不是第一位，我会得到错误的答案。为什么？以下是弗洛伊德·沃沙尔算法背后的想法： if: i is connected to k and k is connected to j then if i and j are not connected, create a connection between i and j 另一种解释：

在popular实现中，我们使用三个循环，循环变量i，j，k。此处i&j分别用于指示两个顶点的源和目标，因此“k”表示“中间”顶点。如果我把循环变量“k”放在第三位而不是第一位，我会得到错误的答案。为什么？

以下是弗洛伊德·沃沙尔算法背后的想法：

if:
i is connected to k and
k is connected to j
then if i and j are not connected,
create a connection between i and j

另一种解释：

if:
i -> k and
k -> j
then if not i -> j
create i -> j

我想说的是，K将处于逻辑连接的中间。中间顶点k负责为N个相邻顶点提供直接连接的权限

上述算法必须应用于| V |顶点，因此k上的循环必须是最外层的for循环

例如：

DiGraph:
Vertices: 0,1,2
Edges: (0,1), (1,2)
Representation: 0 -> 1 -> 2

算法：

k = 0
i = 0 ignore because k = i
i = 1 false because 1 -> 0 is wrong
i = 2 false because 2 -> 0 is wrong

k = 1
i = 0 true because 0 -> 1
j = 0 ignore because i = j
j = 1 ignore because j = k
j = 2 true because 1 -> 2
Since 0 -> 2 does not exist, create this edge
i = 1 ignore because i = k
i = 2 false because 2 -> 1 is wrong

k = 2
i = 0 true because we created this edge in the previous iteration
j = 0 ignore because i = j
j = 1 false because 2 -> 1 is wrong
j = 2 ignore because j = k
i = 1 true because 1 -> 2
j = 0 false because 2 -> 0 is wrong
j = 1 ignore because i = j
j = 2 ignore because j = k
i = 2 ignore because i = k

总之，
k需要执行| V |次
我需要为每个k执行| V |次
j需要为i->k上的每个成功i执行| V |次

时间复杂度：O（n3）

注意：如果您想应用或使用该算法，以便更好地了解其工作原理，请检查my，方法名称为

transitiveClosure（）

。该方法在。

中实现，以下是Floyd Warshall算法背后的思想：

if:
i is connected to k and
k is connected to j
then if i and j are not connected,
create a connection between i and j