table of Contents
Compare the shortest path algorithm
Conditions
Fig may have negative weights, but can not have a negative circle (in the right side of the arc or circle and values less than 0)
Basic operation functions
- InitGraph (Graph & G) initialization function parameters: the role of G: vertex table of FIG initialization, the adjacency matrix, etc.
- InsertNode (Graph & G, VexType v) insertion point function parameters: graph G, vertex v effect: inserting a vertex v in the graph G, vertex i.e. change table
- InsertEdge (Graph & G, VexType v, VexType w) arc inserted Function parameters: Figure G, an arc end points v and w action: adding an arc between two points of G v, w, i.e. change the adjacency matrix
- Adjancent (Graph G, VexType v, VexType w) determines whether there arc (v, w) function parameters: Figure G, an arc end points v and w action: determining whether there is an arc (v, w)
- BFS (Graph G, int start) traversing the breadth of function parameters: Figure G, the start point node subscript start action: traversing width
- DFS (Graph G, int start) deep traversal function (recursive form) parameters: Figure G, the start point node subscript start action: deep traversals
- Dijkstra (Graph G, int v) Shortest Path - Dijkstra algorithm parameters: Figure G, the source v
- Bellman_Ford (Graph G, int v) Shortest Path - Bellman_Ford algorithm parameters: Figure G, v action source: negative free shortest path calculation graphs returns whether a ring
- Floyd_Wallshall (Graph G) Shortest Path - Floyd_Wallshall algorithm parameters: the role of G: computing the shortest path is free of negative returns whether there Graphs ring
Function realization function
- CreateGraph (Graph & G) implementing a function map creation function parameters: G InsertNode FIG action: Create FIG.
- BFSTraverse (Graph G) traversing the breadth of functions implemented function parameters: the role of G: traverse width
- DFSTraverse (Graph G) deep traversal functions implemented function parameters: the role of G: deep traversals
- Shortest_Dijkstra (Graph & G) call -Dijkstra shortest path algorithm parameters: Figure G, the source v
- Shortest_Bellman_Ford (Graph & G) call Shortest Path - - Bellman_Ford algorithm parameters: G in FIG.
- Shortest_Floyd_Wallshall (Graph & G) call Shortest Path - - Floyd_Wallshall algorithm parameters: G in FIG.
Testing using map
Algorithms to explain
initialization
Form behavior i, as j. Is the number of iterations in parentheses after the D and P.
Main diagonal matrix D is 0, the same as the rest of the adjacency matrix.
Deposit matrix P -1 , the shortest path is output as a recursive outlet.
Iteration
State transition equation D matrix is: D (m) [i] [j] = min {D (m-1) [i] [j], D (m-1) [i] [k] + D [k] [j]}, 0 << k << n-1, where, m is the iteration number, n is the number of nodes.
Ideas: adding a point Vk, Vk find into the arc Vi-> Vk, Vk to find out the arc, Vk-> Vj, Comparative D [i] [j] and D [i] [k] + D [k] [j] size.
If D matrix is updated, the value of the corresponding P matrix is the end of the shortest path of the first updated arcs .
D(1)
| 0 | 4 | -3 | ∞ |
| -3 | 0 | -7 | ∞ |
| ∞ | 10 | 0 | 3 |
| 5 | 6 | 6 | 0 |
P(1)
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
D(2)
| 0 | 4 | -3 | ∞ |
| -3 | 0 | -7 | ∞ |
| ∞ | 10 | 0 | 3 |
| 5 | 6 | 2 | 0 |
Addition points V0, V0 of the electric arc V1-> V0 and V3-> V0 , the electric arc V0-> V1 and V0-> V2 .
By comparison D (1) [3] [2]> D (1) [3] [0] + D (1) [0] [2], 6> 5-3 = 2, so the updated 6 2 .
P(2)
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
| -1 | -1 | 0 | -1 |
P(2)[3][2]由P(1)[3][2]改为0，因为最短路径为V3->V0->V2，第一条弧的终点为V0。
D(3)
| 0 | 4 | -3 | ∞ |
| -3 | 0 | -7 | ∞ |
| 7 | 10 | 0 | 3 |
| 3 | 6 | -1 | 0 |
加入点V1，V1入弧有V0->V1，V2->V1以及V3->V1，出弧有V1->V2,V1->V0。
经比较,D(2)[2][0]>D(2)[2][1]+D(2)[1][0]，∞>10-3=7，所以，将∞更新为7。
D(2)[3][0]>D(1)[3][1]+D(2)[1][0]，5>6-3=3，所以，将5更新为3。
D(2)[3][2]>D(1)[3][1]+D(2)[1][2]，2>6-7=-1，所以，将2更新为-1。
P(3)
| -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 |
| 1 | -1 | -1 | -1 |
| 1 | -1 | 1 | -1 |
P(3)[2][0]改为1，因为最短路径为V2->V1->V0，第一条弧的终点为V1。
P(3)[3][0]改为1，因为最短路径为V3->V1->V0，第一条弧的终点为V1。
P(3)[3][2]改为1，因为最短路径为V3->V1->V2，第一条弧的终点为V1。
下面的由读者根据原理及矩阵自己补充，加深印象。
D(4)
| 0 | 4 | -3 | 0 |
| -3 | 0 | -7 | -4 |
| 7 | 10 | 0 | 3 |
| 3 | 6 | -1 | 0 |
P(4)
| -1 | -1 | -1 | 2 |
| -1 | -1 | -1 | 2 |
| 1 | -1 | -1 | -1 |
| 1 | -1 | 1 | -1 |
D(5)
| 0 | 4 | -3 | 0 |
| -3 | 0 | -7 | -4 |
| 6 | 9 | 0 | 3 |
| 3 | 6 | -1 | 0 |
P(5)
| -1 | -1 | -1 | 2 |
| -1 | -1 | -1 | 2 |
| 3 | 3 | -1 | -1 |
| 1 | -1 | 1 | -1 |
注意：弗洛伊德算法的最短路径在输出时不是倒着的，我们记录的是第一条弧的终点。例如，p[2][0]=3，P[3][0]=1,P[1][0]=-1，
则V[2]到V[0]的最短路径为2->3->1->0，值为6。也就是看P矩阵的列，这是与前面两篇最短路径算法不同的地方，需注意。
弗洛伊德算法代码
//最短路径 - Floyd_Wallshall算法 参数：图G 作用：计算不含负圈图的最短路径 返回是否有圈
bool Floyd_Wallshall(Graph G)
{
//初始化
for (int i = 0; i<G.vexnum; i++)
for (int j = 0; j < G.vexnum; j++)
{
if (i == j)F_D[i][j] = 0;
else F_D[i][j] = G.Edge[i][j];
P[i][j] = -1;
}
//初始化结束，开始迭代
for(int k=0;k<G.vexnum;k++)
for (int i = 0; i<G.vexnum; i++)
for (int j = 0; j<G.vexnum; j++)
if (F_D[i][j] > F_D[i][k] + F_D[k][j])
{
F_D[i][j] = F_D[i][k] + F_D[k][j];
P[i][j] = k;
}
bool flag = true;
for (int i = 0; i < G.vexnum; i++)
for (int j = 0; j < G.vexnum; j++)
if (i==j&&F_D[i][j] < 0)
{
flag = false;
break;
}
return flag;
}全部代码
/*
Project: 图-最短路径-Bellman-Ford算法（可含有负权弧）
Date: 2019/10/24
Author: Frank Yu
基本操作函数:
InitGraph(Graph &G) 初始化函数 参数：图G 作用：初始化图的顶点表，邻接矩阵等
InsertNode(Graph &G,VexType v) 插入点函数 参数：图G,顶点v 作用：在图G中插入顶点v,即改变顶点表
InsertEdge(Graph &G,VexType v,VexType w) 插入弧函数 参数：图G,某弧两端点v和w 作用：在图G两点v,w之间加入弧，即改变邻接矩阵
Adjancent(Graph G,VexType v,VexType w) 判断是否存在弧(v,w)函数 参数：图G，某弧两端点v和w 作用：判断是否存在弧(v,w)
BFS(Graph G, int start) 广度遍历函数 参数：图G,开始结点下标start 作用：宽度遍历
DFS(Graph G, int start) 深度遍历函数（递归形式）参数：图G,开始结点下标start 作用：深度遍历
Dijkstra(Graph G, int v) 最短路径 - Dijkstra算法 参数：图G、源点v
Bellman_Ford(Graph G, int v) 最短路径 - Bellman_Ford算法 参数：图G、源点v 作用：计算不含负圈图的最短路径 返回是否有圈
Floyd_Wallshall(Graph G) 最短路径 - Floyd_Wallshall算法 参数：图G 作用：计算不含负圈图的最短路径 返回是否有圈
功能实现函数：
CreateGraph(Graph &G) 创建图功能实现函数 参数：图G InsertNode 作用：创建图
BFSTraverse(Graph G) 广度遍历功能实现函数 参数：图G 作用：宽度遍历
DFSTraverse(Graph G) 深度遍历功能实现函数 参数：图G 作用：深度遍历
Shortest_Dijkstra(Graph &G) 调用最短路径-Dijkstra算法 参数：图G、源点v
Shortest_Bellman_Ford(Graph &G) 调用最短路径- - Bellman_Ford算法 参数：图G
Shortest_Floyd_Wallshall(Graph &G) 调用最短路径- - Floyd_Wallshall算法 参数：图G
*/
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<cmath>
#include<string>
#include<set>
#include<list>
#include<queue>
#include<vector>
#include<map>
#include<iterator>
#include<algorithm>
#include<iostream>
#define MaxVerNum 100 //顶点最大数目值
#define VexType char //顶点数据类型
#define EdgeType int //弧数据类型,有向图时邻接矩阵不对称，有权值时表示权值，没有时1连0不连
#define INF 0x3f3f3f3f//作为最大值
using namespace std;
//图的数据结构
typedef struct Graph
{
VexType Vex[MaxVerNum];//顶点表
EdgeType Edge[MaxVerNum][MaxVerNum];//弧表
int vexnum, arcnum;//顶点数、弧数
}Graph;
//迪杰斯特拉算法全局变量
bool S[MaxVerNum]; //顶点集
int D[MaxVerNum]; //到各个顶点的最短路径
int F_D[MaxVerNum][MaxVerNum];//Floyd的D矩阵 记录最短路径
int Pr[MaxVerNum]; //记录前驱
//*********************************************基本操作函数*****************************************//
//初始化函数 参数：图G 作用：初始化图的顶点表，邻接矩阵等
int P[MaxVerNum][MaxVerNum];//最短路径记录矩阵
void InitGraph(Graph &G)
{
memset(G.Vex, '#', sizeof(G.Vex));//初始化顶点表
//初始化弧表
for (int i = 0; i < MaxVerNum; i++)
for (int j = 0; j < MaxVerNum; j++)
G.Edge[i][j] = INF;
G.arcnum = G.vexnum = 0; //初始化顶点数、弧数
}
//插入点函数 参数：图G,顶点v 作用：在图G中插入顶点v,即改变顶点表
bool InsertNode(Graph &G, VexType v)
{
if (G.vexnum < MaxVerNum)
{
G.Vex[G.vexnum++] = v;
return true;
}
return false;
}
//插入弧函数 参数：图G,某弧两端点v和w 作用：在图G两点v,w之间加入弧，即改变邻接矩阵
bool InsertEdge(Graph &G, VexType v, VexType w, int weight)
{
int p1, p2;//v,w两点下标
p1 = p2 = -1;//初始化
for (int i = 0; i<G.vexnum; i++)//寻找顶点下标
{
if (G.Vex[i] == v)p1 = i;
if (G.Vex[i] == w)p2 = i;
}
if (-1 != p1&&-1 != p2)//两点均可在图中找到
{
G.Edge[p1][p2] = weight;//有向图邻接矩阵不对称
G.arcnum++;
return true;
}
return false;
}
//判断是否存在弧(v,w)函数 参数：图G，某弧两端点v和w 作用：判断是否存在弧(v,w)
bool Adjancent(Graph G, VexType v, VexType w)
{
int p1, p2;//v,w两点下标
p1 = p2 = -1;//初始化
for (int i = 0; i<G.vexnum; i++)//寻找顶点下标
{
if (G.Vex[i] == v)p1 = i;
if (G.Vex[i] == w)p2 = i;
}
if (-1 != p1&&-1 != p2)//两点均可在图中找到
{
if (G.Edge[p1][p2] == 1)//存在弧
{
return true;
}
return false;
}
return false;
}
bool visited[MaxVerNum];//访问标记数组，用于遍历时的标记
//广度遍历函数 参数：图G,开始结点下标start 作用：宽度遍历
void BFS(Graph G, int start)
{
queue<int> Q;//辅助队列
cout << G.Vex[start];//访问结点
visited[start] = true;
Q.push(start);//入队
while (!Q.empty())//队列非空
{
int v = Q.front();//得到队头元素
Q.pop();//出队
for (int j = 0; j<G.vexnum; j++)//邻接点
{
if (G.Edge[v][j] < INF && !visited[j])//是邻接点且未访问
{
cout << "->";
cout << G.Vex[j];//访问结点
visited[j] = true;
Q.push(j);//入队
}
}
}//while
cout << endl;
}
//深度遍历函数（递归形式）参数：图G,开始结点下标start 作用：深度遍历
void DFS(Graph G, int start)
{
cout << G.Vex[start];//访问
visited[start] = true;
for (int j = 0; j < G.vexnum; j++)
{
if (G.Edge[start][j] <INF && !visited[j])//是邻接点且未访问
{
cout << "->";
DFS(G, j);//递归深度遍历
}
}
}
//最短路径 - Dijkstra算法 参数：图G、源点v3
void Dijkstra(Graph G, int v)
{
//初始化
int n = G.vexnum;//n为图的顶点个数
for (int i = 0; i < n; i++)
{
S[i] = false;
D[i] = G.Edge[v][i];
if (D[i] < INF)Pr[i] = v; //v与i连接，v为前驱
else Pr[i] = -1;
}
S[v] = true;
D[v] = 0;
//初始化结束,求最短路径，并加入S集
for (int i = 1; i < n; i++)
{
int min = INF;
int temp;
for (int w = 0; w < n; w++)
if (!S[w] && D[w] < min) //某点temp未加入s集，且为当前最短路径
{
temp = w;
min = D[w];
}
S[temp] = true;
//更新从源点出发至其余点的最短路径 通过temp
for (int w = 0; w < n; w++)
if (!S[w] && D[temp] + G.Edge[temp][w] < D[w])
{
D[w] = D[temp] + G.Edge[temp][w];
Pr[w] = temp;
}
}
}
//最短路径 - Bellman_Ford算法 参数：图G、源点v 作用：计算不含负圈图的最短路径 返回是否有圈
bool Bellman_Ford(Graph G, int v)
{
//初始化
int n = G.vexnum;//n为图的顶点个数
for (int i = 0; i < n; i++)
{
D[i] = G.Edge[v][i];
if (D[i] < INF)Pr[i] = v; //v与i连接，v为前驱
else Pr[i] = -1;
}
D[v] = 0;
//初始化结束，开始双重循环
for (int i = 2; i<G.vexnum - 1; i++)
for (int j = 0; j<G.vexnum; j++) //j为源点
for (int k = 0; k<G.vexnum; k++) //k为终点
if (D[k] > D[j] + G.Edge[j][k])
{
D[k] = D[j] + G.Edge[j][k];
Pr[k] = j;
}
//判断是否含有负圈
bool flag = true;
for (int j = 0; j<G.vexnum - 1; j++) //j为源点
for (int k = 0; k<G.vexnum - 1; k++) //k为终点
if (D[k] > D[j] + G.Edge[j][k])
{
flag = false;
break;
}
return flag;
}
//最短路径 - Floyd_Wallshall算法 参数：图G 作用：计算不含负圈图的最短路径 返回是否有圈
bool Floyd_Wallshall(Graph G)
{
//初始化
for (int i = 0; i<G.vexnum; i++)
for (int j = 0; j < G.vexnum; j++)
{
if (i == j)F_D[i][j] = 0;
else F_D[i][j] = G.Edge[i][j];
P[i][j] = -1;
}
//初始化结束，开始迭代
for(int k=0;k<G.vexnum;k++)
for (int i = 0; i<G.vexnum; i++)
for (int j = 0; j<G.vexnum; j++)
if (F_D[i][j] > F_D[i][k] + F_D[k][j])
{
F_D[i][j] = F_D[i][k] + F_D[k][j];
P[i][j] = k;
}
bool flag = true;
for (int i = 0; i < G.vexnum; i++)
for (int j = 0; j < G.vexnum; j++)
if (i==j&&F_D[i][j] < 0)
{
flag = false;
break;
}
return flag;
}
//输出最短路径
void Path(Graph G, int v)
{
if (Pr[v] == -1)
return;
Path(G, Pr[v]);
cout << G.Vex[Pr[v]] << "->";
}
// 输出Floyd最短路径 v是终点
void F_Path(Graph G, int v, int w)
{
cout << "->"<< G.Vex[P[v][w]] ;
if (P[v][w] == -1)
return;
F_Path(G, v,P[v][w]);
}
//**********************************************功能实现函数*****************************************//
//打印图的顶点表
void PrintVex(Graph G)
{
for (int i = 0; i < G.vexnum; i++)
{
cout << G.Vex[i] << " ";
}
cout << endl;
}
//打印图的弧矩阵
void PrintEdge(Graph G)
{
for (int i = 0; i < G.vexnum; i++)
{
for (int j = 0; j < G.vexnum; j++)
{
if (G.Edge[i][j] == INF)cout << "∞ ";
else cout << G.Edge[i][j] << " ";
}
cout << endl;
}
}
//创建图功能实现函数 参数：图G InsertNode 作用：创建图
void CreateGraph(Graph &G)
{
VexType v, w;
int vn, an;//顶点数，弧数
cout << "请输入顶点数目:" << endl;
cin >> vn;
cout << "请输入弧数目:" << endl;
cin >> an;
cout << "请输入所有顶点名称:" << endl;
for (int i = 0; i<vn; i++)
{
cin >> v;
if (InsertNode(G, v)) continue;//插入点
else {
cout << "输入错误！" << endl; break;
}
}
cout << "请输入所有弧（每行输入起点，终点及权值）:" << endl;
for (int j = 0; j<an; j++)
{
int weight;
cin >> v >> w >> weight;
if (InsertEdge(G, v, w, weight)) continue;//插入弧
else {
cout << "输入错误！" << endl; break;
}
}
cout << "图的顶点及邻接矩阵：" << endl;
PrintVex(G);
PrintEdge(G);
}
//广度遍历功能实现函数 参数：图G 作用：宽度遍历
void BFSTraverse(Graph G)
{
for (int i = 0; i<MaxVerNum; i++)//初始化访问标记数组
{
visited[i] = false;
}
for (int i = 0; i < G.vexnum; i++)//对每个连通分量进行遍历
{
if (!visited[i])BFS(G, i);
}
}
//深度遍历功能实现函数 参数：图G 作用：深度遍历
void DFSTraverse(Graph G)
{
for (int i = 0; i<MaxVerNum; i++)//初始化访问标记数组
{
visited[i] = false;
}
for (int i = 0; i < G.vexnum; i++)//对每个连通分量进行遍历
{
if (!visited[i])
{
DFS(G, i); cout << endl;
}
}
}
//调用最短路径-Dijkstra算法 参数：图G
void Shortest_Dijkstra(Graph &G)
{
char vname;
int v = -1;
cout << "请输入源点名称:" << endl;
cin >> vname;
for (int i = 0; i < G.vexnum; i++)
if (G.Vex[i] == vname)v = i;
if (v == -1)
{
cout << "没有找到输入点！" << endl;
return;
}
Dijkstra(G, v);
cout << "目标点" << "\t" << "最短路径值" << "\t" << "最短路径" << endl;
for (int i = 0; i < G.vexnum; i++)
{
if (i != v)
{
cout << " " << G.Vex[i] << "\t" << " " << D[i] << "\t";
Path(G, i);
cout << G.Vex[i] << endl;
}
}
}
//调用最短路径- - Bellman_Ford算法 参数：图G
void Shortest_Bellman_Ford(Graph &G)
{
char vname;
int v = -1;
cout << "请输入源点名称:" << endl;
cin >> vname;
for (int i = 0; i < G.vexnum; i++)
if (G.Vex[i] == vname)v = i;
if (v == -1)
{
cout << "没有找到输入点！" << endl;
return;
}
if (Bellman_Ford(G, v))
{
cout << "目标点" << "\t" << "最短路径值" << "\t" << "最短路径" << endl;
for (int i = 0; i < G.vexnum; i++)
{
cout << " " << G.Vex[i] << "\t" << " " << D[i] << "\t";
Path(G, i);
cout << G.Vex[i] << endl;
}
}
else
{
cout << "输入的图中含有负圈，不能使用该算法！" << endl;
}
}
//调用最短路径- - Floyd_Wallshall算法 参数：图G
void Shortest_Floyd_Wallshall(Graph &G)
{
if (Floyd_Wallshall(G))
{
cout << "最短路径值" << "\t" << "最短路径" << endl;
for (int i = 0; i < G.vexnum; i++)
for (int j = 0; j < G.vexnum; j++)
{
cout << " "<<F_D[i][j] << " \t";
cout << G.Vex[i];
F_Path(G, i,j);
cout << G.Vex[j] << endl;
}
}
else
{
cout << "输入的图中含有负圈，不能使用该算法！" << endl;
}
}
//菜单
void menu()
{
cout << "************1.创建图 2.广度遍历******************" << endl;
cout << "************3.深度遍历 4.迪杰斯特拉****************" << endl;
cout << "************5.贝尔曼福特 6.弗洛伊德******************" << endl;
cout << "************7.退出*************************************" << endl;
}
//主函数
int main()
{
int choice = 0;
Graph G;
InitGraph(G);
while (1)
{
menu();
printf("请输入菜单序号：\n");
scanf("%d", &choice);
if (7 == choice) break;
switch (choice)
{
case 1:CreateGraph(G); break;
case 2:BFSTraverse(G); break;
case 3:DFSTraverse(G); break;
case 4:Shortest_Dijkstra(G); break;
case 5:Shortest_Bellman_Ford(G); break;
case 6:Shortest_Floyd_Wallshall(G); break;
default:printf("输入错误！！！\n"); break;
}
}
return 0;
}实验结果
最短路径算法比较
| 算法\比较内容 | 适用条件 | 算法思想 | 时间复杂度 |
| Dijkstra | 无负权的图，单源或多源 | 贪心 | O(v^2)、O(v^3) |
| Bellman-Ford | 可以有负权但无负圈的图 | 动态规划 | O(v^3)、O(ve) |
| Floyd-Warshall | 无负权的图，多源 | 动态规划 | O(v^3) |
更多数据结构与算法实现：数据结构（严蔚敏版）与算法的实现（含全部代码）
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