1フロイドのアルゴリズム
1.1コード
/**
* 弗洛伊德(Floyd)算法:
1 解决问题: 多源最短路径问题
求每一对顶点之间的最短路径
Background: 带权有向图
2 算法思想: 动态规划(DP, Dynamic Programming)
3 时间复杂度: O(n^3)
*/
#include<stdio.h>
#include<iostream>
using namespace std;
// 1 定义图模型(邻接矩阵表示法)的基本存储结构体
# define MaxInt 32767 // 表示极大值 即 ∞ (无穷大)
# define MVNum 100 // 最大顶点数
typedef int VertexType; // 假设顶点的数据类型为整型
typedef int ArcType; // 假设Vi与Vj之边的权值类型为整型
typedef struct {
VertexType vexs[MVNum]; // 顶点表 (存储顶点信息)
ArcType arcs[MVNum][MVNum]; // 邻接矩阵
int vexnum,arcnum; // 图的当前顶点数与边数
}AMGraph; // Adjacent Matrix Graph 邻接矩阵
// 2 定义Floyd算法的辅助数据结构体
ArcType D[MVNum][MVNum]; // 记录顶点Vi和Vj之间的最短路径长度
int Path[MVNum][MVNum]; // 最短路径上顶点Vj的前一顶点的序号
/////////////////////// ↓算法区↓ ///////////////////////
void ShortestPath_Floyd(AMGraph G){
//step1 初始化各对结点的已知路径和距离
for(int i=0;i<G.vexnum;i++){
for(int j=0;j<G.vexnum;j++){
D[i][j] = G.arcs[i][j]; //D[i][j] 初始化
if(D[i][j]<MaxInt && i!=j){ // 【易漏】 i != j (防止产生自回环)
Path[i][j] = i; // 若 Vi与Vj之间存在弧(有序顶点对)： 将Vj的前驱置为 Vi
} else {
Path[i][j] = -1;
}
}
}
//step2 动态规划(DP)动态更新： <Vi,Vj>更短的最短路径的距离和路径
for(int k=0;k<G.vexnum;k++){ // 【易错】 中间结点Vk的循环 是在最外层
for(int i=0;i<G.vexnum;i++){
for(int j=0;j<G.vexnum;j++){
if(D[i][k] + D[k][j] < D[i][j]){ // 若从Vi【经Vk】到Vj的一条路径更短
D[i][j] = D[i][k] + D[k][j]; // 更新D[i][j]
Path[i][j] = Path[k][j]; // 【易错】 更改Vj的前驱为 Vk
}
}
}
}
}
// 初始化(邻接矩阵)带权有向图的图模型
void InitAMGraph(AMGraph &G){
cout<<"Please Input Vertexs Number:";
cin>>G.vexnum;
cout<<"\nPlease Directed Edge Number:";
cin>>G.arcnum;
for(int i=0;i<MVNum;i++){
for(int j=0;j<MVNum;j++){
if(i!=j){ // 【易错】 初始化<Vi, Vj>时: <Vi,Vj> 路径长度无穷大 (i!=j)
G.arcs[i][j] = MaxInt;
} else { // 【易错】 初始化<Vi, Vj>时: <Vi,Vi>【自回环】路径长度为0 (i==i)
G.arcs[i][j] = 0;
}
}
}
for(int i=0;i<G.vexnum;i++){
G.vexs[i] = i;
}
cout<<"\nPlease Input All Directed Edges and their Weight now.";
int i,j;
int weight;
for(int k=0;k<G.arcnum;k++){
cout<<"\n("<<(k+1)<<") Directed Edges(i,j,weight): ";
cin>>i;
cin>>j;
cin>>weight;
G.arcs[i][j] = weight;
}
cout<<endl;
}
void OutputD(AMGraph G){
for(int i=0;i<G.vexnum;i++){
for(int j=0;j<G.vexnum;j++){
cout<<"Shortest Distance Weight of the Pair of Directed Vertices ("<<i<<","<<j<<"): "<<D[i][j]<<endl;
}
}
}
void OutputPath(AMGraph G){
for(int i=0;i<G.vexnum;i++){
for(int j=0;j<G.vexnum;j++){
cout<<"Path("<<i<<","<<j<<"): "<<Path[i][j]<<endl;
}
}
// int fullPath[G.vexnum]; //逆序记录 <Vi,Vj>的最短路径的序号 ; 最大路径长度不会超过 G.vexnum
// for(int i=0;i<G.vexnum;i++){
// for(int j=0;j<G.vexnum;j++){
// cout<<"Shortest Distance Path of the Pair of Directed Vertices ("<<i<<","<<j<<"): ";
// for(int p=0;i<G.vexnum;p++){ // 初始化记录最短路径的临时表
// fullPath[p]= -1; // -1 表示结束符
// }
// int m=i,n=j;
// int cusor=G.vexnum-1;
// while( (cusor>=0) && (Path[m][n]!=i || Path[m][n]!=MaxInt) ){
// fullPath[cusor] = Path[m][n]; // Vj的前驱
// n = fullPath[cusor]; // 【重难点】 源点m不变， 终点n更新为j的前驱
// cusor--; // 要保证 cusor>=0
// }
// //输出全路径
// cout<<" "<<i<<">";
// cusor++; // 当前cusor的位置是源点i的前一个空置空间，值为-1
// while(fullPath[cusor] != -1){
// cout<<fullPath[cusor]<<">"
// }
// }
// }
}
int main(){
AMGraph G;
InitAMGraph(G);//易错处
ShortestPath_Floyd(G); // 【重/难点】易错处
OutputD(G);
OutputPath(G);
return 0;
}
*/
1.2出力
Please Input Vertexs Number:4
Please Directed Edge Number:8
Please Input All Directed Edges and their Weight now.
(1) Directed Edges(i,j,weight): 0 1 1
(2) Directed Edges(i,j,weight): 1 3 2
(3) Directed Edges(i,j,weight): 2 0 3
(4) Directed Edges(i,j,weight): 0 3 4
(5) Directed Edges(i,j,weight): 2 1 5
(6) Directed Edges(i,j,weight): 3 2 6
(7) Directed Edges(i,j,weight): 2 3 8
(8) Directed Edges(i,j,weight): 1 2 9
Shortest Distance Weight of the Pair of Directed Vertices (0,0): 0
Shortest Distance Weight of the Pair of Directed Vertices (0,1): 1
Shortest Distance Weight of the Pair of Directed Vertices (0,2): 9
Shortest Distance Weight of the Pair of Directed Vertices (0,3): 3
Shortest Distance Weight of the Pair of Directed Vertices (1,0): 11
Shortest Distance Weight of the Pair of Directed Vertices (1,1): 0
Shortest Distance Weight of the Pair of Directed Vertices (1,2): 8
Shortest Distance Weight of the Pair of Directed Vertices (1,3): 2
Shortest Distance Weight of the Pair of Directed Vertices (2,0): 3
Shortest Distance Weight of the Pair of Directed Vertices (2,1): 4
Shortest Distance Weight of the Pair of Directed Vertices (2,2): 0
Shortest Distance Weight of the Pair of Directed Vertices (2,3): 6
Shortest Distance Weight of the Pair of Directed Vertices (3,0): 9
Shortest Distance Weight of the Pair of Directed Vertices (3,1): 10
Shortest Distance Weight of the Pair of Directed Vertices (3,2): 6
Shortest Distance Weight of the Pair of Directed Vertices (3,3): 0
Path(0,0): -1
Path(0,1): 0
Path(0,2): 3
Path(0,3): 1
Path(1,0): 2
Path(1,1): -1
Path(1,2): 3
Path(1,3): 1
Path(2,0): 2
Path(2,1): 0
Path(2,2): -1
Path(2,3): 1
Path(3,0): 2
Path(3,1): 0
Path(3,2): 3
Path(3,3): -1
参考2
- 「データ構造（C言語/ Dongmeiヤン魏ミン呉Weimin編）」
