----- the next day -----
Small gray roadmap is as follows:
The first step, from the table using Dijkstra's algorithm, is determined from the apex A, the shortest distance to each other vertex:
The second step, to continue using Dijkstra's algorithm, obtained from the vertex B, the shortest distance to each of the other vertices.
The third step, starting from the vertex C, the shortest distance to each vertex.
The fourth step, starting from the vertex D ......
.......
Like this, it has been traversed to the vertex G.
The time complexity of this idea is how much?
If there are n vertices in the graph, the heap without considering optimization, a Dijkstra time complexity is O (n ^ 2). Therefore, to each vertex are calculated again, the overall time complexity of O (n ^ 3).
————————————
For a chestnut:
A vertex C and vertex edge graph is not directly connected to the direct distance between them is infinite.
If B as a "relay top", then the shortest path from A to C is ABC, the shortest distance is 2 + 3 = 5.
Take another chestnut:
Vertex A and vertex C is directly connected to the figure, the distance is 6. However, the presence of a "bypass" path ABC, the distance is 2 + 3 = 5 <6.
Therefore, after the relay vertex B, the shortest distance from A to C may be 5.
The following detailed steps Floyd algorithm we take a look.
1. To achieve Floyd algorithm, we first need to build a weighted graph adjacency matrix:
Among the adjacency matrix, each number represents a direct distance from one vertex to another vertex, the distance relay is not related to any vertex.
2. At this time, assuming that only allows vertex to vertex A as a relay, the distance between the apex of what will become of it?
The distance between B and C was originally infinity, in this case relay A, shorten the distance AB + AC distance = distance
5+2=7。
Updates the corresponding matrix element (orange represents the distance from the apex A to the other temporary vertex):
3. Next to the vertices A, B as a relay vertex, then the distance between the apex of what will become of it?
Distance between A and D are originally infinity, B is a relay case, to shorten the distance AB is the distance from the BD = + 5 + 1 = 6.
Distance between A and E originally infinity, B is a relay case, shorten the distance from the BE AB = distance + 5 + 6 = 11.
Updates the corresponding matrix element (orange represents the distance from the apex B to the other temporary vertex):
4. Next to the vertices A, B, C as a relay vertex, then the distance between the apex of what will become of it?
A和F之间的距离原本是无穷大，此时以C为中继，距离缩短为AC距离+CF距离=2+8=10。
更新对应矩阵元素（橙色区域代表顶点C到其他顶点的临时距离）：
.........
.........
以此类推，我们不断引入新的中继顶点，不断刷新矩阵中的临时距离。
最终，当所有顶点都可以作为中继顶点时，我们的距离矩阵更新如下：
此时，矩阵中每一个元素，都对应着某顶点到另一个顶点的最短距离。
为什么这么说呢？让我们回顾一下动态规划的两大要素：
问题的初始状态
问题的状态转移方程式
对于寻找图的所有顶点之间距离的问题，初始状态就是顶点之间的直接距离，也就是邻接矩阵。
而问题的状态转移方程式又是什么呢？
假设新引入的中继顶点是n，那么：
顶点i 到 顶点j 的新距离 = Min（顶点i 到 顶点j 的旧距离，顶点i 到 顶点n 的距离+顶点n 到 顶点j 的距离）
final static int INF = Integer.MAX_VALUE;
public static void floyd(int[][] matrix){
//循环更新矩阵的值
for(int k=0; k<matrix.length; k++){
for(int i=0; i<matrix.length; i++){
for(int j=0; j<matrix.length; j++){
if(matrix[i][k] == INF || matrix[k][j] == INF) {
continue;
}
matrix[i][j] = Math.min(matrix[i][j], matrix[i][k] + matrix[k][j]);
}
}
}
// 打印floyd最短路径的结果
System.out.printf("最短路径矩阵: \n");
for (int i = 0; i < matrix.length; i++) {
for (int j = 0; j < matrix.length; j++)
System.out.printf("%3d ", matrix[i][j]);
System.out.printf("\n");
}
}
public static void main(String[] args) {
int[][] matrix = {
{0, 5, 2, INF, INF, INF, INF},
{5, 0, INF, 1, 6, INF, INF},
{2, INF, 0, 6, INF, 8, INF},
{INF, 1, 6, 0, 1, 2, INF},
{INF, 6, INF, 1, 0, INF, 7},
{INF, INF, 8, 2, INF, 0, 3},
{INF, INF, INF, INF, 7, 3, 0}
};
floyd(matrix);
}
如何在短时间内成为Python工程师？
文章来自小灰，小灰的《漫画算法》上市后销量疯涨，出版社加印了好几次！占据各大电商图书板块第一位！
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小灰把两年多以来积累的漫画作品进行了筛选和优化，并加上了一些更为基础和系统的入门章节，最终完成了本书的六大篇章：
第一章 算法概述
介绍了算法和数据结构的相关概念，告诉大家算法是什么，数据结构又是什么，它们有哪些用途，如何分析时间复杂度，如何分析空间复杂度。
第二章 数据结构基础
介绍了最基本的数据结构，包括数组、链表、栈、队列、哈希表的概念和读写操作。
第三章 树
介绍了树和二叉树的概念、二叉树的各种遍历方式、二叉堆和优先队列的应用。
第四章 排序算法
介绍了几种典型的排序算法，包括冒泡排序、快速排序、堆排序、计数排序、桶排序。
第五章 面试中的算法
介绍了10余道职场上流行的算法面试题及详细的解题思路。例如怎样判断链表有环、怎样计算大整数相加等。
第六章 算法的实际应用
Some applications of the algorithm in the workplace, such as the use LRU algorithm to eliminate cold data, using statistical algorithms to Bitmap user characteristics and so on.
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