The basic idea of Dijkstra's algorithm for solving the shortest path:
(1) Set the auxiliary array Dist, where each component Dist[k] represents the shortest path currently obtained from the source point to the remaining vertices k.
(2) In general, Dist[k] = Min{<the weight of the arc from the source point to vertex k>, <the shortest path length from the source point to other vertices> + <the arc on the arc from other vertices to vertex k> Weight>};
(3) First, clear the labels of all points;
(4) Set d[0]=0, other d[i]=inf;
(5) Loop n times, among all unlabeled nodes, select the node x with the smallest d value, and mark the node x. For all edges (x, y) starting from x, update d[y]= min{d[y], d[x]+w(x,y)}.
Algorithm implementation
-
Build a map
-
In the console, abstract the map as
-
The macros defined according to design requirements and actual needs are as follows
#define M 8//地点数
#define N 12//路径数
#define mapRowNum 10//地图行数
#define mapColNum 10//地图列数
- Define a two-dimensional array of maps
//地图二维数组
static char map[mapRowNum][mapRowNum] = {
{
' ', ' ', ' ', ' ', '0', '*', ' ', ' ', ' ', ' '},
{
' ', ' ', ' ', '*', '*', ' ', '1', ' ', ' ', ' '},
{
' ', ' ', '*', ' ', '2', '*', ' ', '*', ' ', ' '},
{
' ', '*', ' ', ' ', '*', ' ', ' ', ' ', '*', ' '},
{
'*', ' ', ' ', '*', '3', '*', ' ', ' ', ' ', '*'},
{
'*', ' ', '*', ' ', '*', ' ', '*', ' ', ' ', '*'},
{
'4', '*', ' ', ' ', '*', ' ', ' ', '*', '*', '6'},
{
' ', ' ', '*', '*', '5', '*', '*', ' ', ' ', ' '},
{
' ', ' ', ' ', ' ', '*', ' ', ' ', ' ', ' ', ' '},
{
' ', ' ', ' ', ' ', '7', ' ', ' ', ' ', ' ', ' '},
};
- Define an array of distances between two adjacent points
//相邻两点间的距离数组
static int dist[3][12] = {
{
0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 5, 5},
{
1, 2, 4, 2, 6, 3, 4, 5, 6, 5, 6, 7},
{
3, 2, 12, 3, 10, 2, 9, 4, 11, 7, 9, 2}
};
- Define an array of place names
//地点名
static char loca[8][8] = {
"图书馆", "行远楼", "信南", "四区", "东操", "餐厅", "北操", "宿舍区"};
- Define other arrays
int edge[N][N], weight[N], disp[N];
bool visited[N] = {
false};
const int inf = 999999;
- Functions for printing maps and legends
//打印地图和图例
void printMap() {
printf(" ");
for(int i = 0; i < mapRowNum; i++) {
for(int j = 0; j < mapColNum; j++) {
printf("%c ", map[i][j]);
}
printf("\n ");
}
printf("\n");
printf("图例：(共8个地点)\n 0-图书馆 1-行远楼 2-信南 3-四区");
printf(" 4-东操 5-餐厅 6-北操 7-宿舍区\n\n");
printf("各地点间距离：(任意两点间共12条路径)\n ");
printf("0-1：3; 0-2：2; 0-4：12; 1-2：3; 1-6：10; 2-3：2; ");
printf("3-4：9; 3-5：4; 3-6：11; 4-5：7; 5-6：9; 5-7：2 ");
}
- Function to solve the shortest path
//求解最短路径
void calculatePath(){
int prev[N];//用来标注当前结点的上一个结点，以便输出两点间最短路线
fill(edge[0], edge[0] + N * N, inf);//注意这里要是N*N，要不然不是全部
fill(disp, disp + N, inf);
int start_point;
cout<<"\n\n请输入起点:";
cin>>start_point;
//建立二维距离矩阵
for(int i=0; i < N; i++) {
edge[dist[0][i]][dist[1][i]] = edge[dist[1][i]][dist[0][i]] = dist[2][i];
}
for(int i = 0; i < N; i++) {
prev[i] = i;//初始状态设每个点的前驱为自身
edge[i][i] = 0;
}
//DijkStra算法求解最短路径
disp[start_point] = 0;
for(int i = 0; i < M; i++) {
int u = -1, min = inf;
for(int j = 0; j < M; j++) {
if(visited[j] == false && disp[j] <= min) {
u = j;
min = disp[j];
}
}
if(u == -1) break;
visited[u] = true;
for(int v = 0; v < M; v++) {
if(visited[v] == false && edge[u][v] != inf) {
if(disp[u] + edge[u][v] < disp[v]) {
disp[v] = disp[u] + edge[u][v];
prev[v] = u;//prev保存当前结点的上一个节点，以便输出最短路线
}
}
}
}
int end_point;//终点
cout<<"请输入终点:";
cin>>end_point;
stack<int> myStack;//最短路径栈
//构建最短路径栈
int temp = end_point;
myStack.push(end_point);//先加终点
while(start_point != temp) {
temp = prev[temp];
myStack.push(temp);//注意这里是把当前点的上一个结点放进去，此时换成了temp
}
cout<<"\n"<<loca[start_point]<<"("<<start_point<<")"<<" -> "<<loca[end_point];
cout<<"("<<end_point<<")"<<"的最短距离为: "<<disp[end_point]<< "\n对应的路径为: ";
//输出最短路径
while(!myStack.empty()) {
cout<<loca[myStack.top()]<<"("<<myStack.top()<<")";
myStack.pop();
if(!myStack.empty())
cout<<"->";
}
}
- Main function
#include <iostream>
#include <algorithm>
#include <stack>
using namespace std;
int main() {
printMap();
calculatePath();
return 0;
}
operation result
