Dijkstra算法
局限性：图中不能有负权边1
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74#include<bits/stdc++.h>
using namespace std;
const int max_point = 1000;
const int maxint = 999999999;
int dist[max_point];
int c[max_point][max_point];
void Dijkstra(int n , int v , int *dist , int c[max_point][max_point]){
int book[max_point];
for(int i = 0 ; i<n ; ++i){
book[i] = 0;
dist[i] = c[v][i];
}
book[v] = 1;
dist[v] = 0;
for(int i = 0 ; i<n-1 ; ++i){
int tmp = maxint;
int u = v;
for(int j = 0 ; j<n ; ++j){
if(!book[j] && dist[j] < tmp){
tmp = dist[j];
u = j;
}
}
book[u] = 1;
for(int j = 0 ; j<n ; ++j){
if(!book[j] && c[u][j] < maxint &&(dist[u] + c[u][j] < dist[j])){
dist[j] = dist[u] + c[u][j];
}
}
}
for(int i = 0 ; i<n ; ++i){
cout<<dist[i]<<' ';
}
}
int main()
{
freopen("input.txt","r",stdin);
//freopen("output.txt","w",stdout);
int n , m;
cin>>n>>m;
for(int i = 0 ; i<n ; ++i){
for(int j = 0 ; j<n ; ++j){
if(i == j){
c[i][j] = 0;
}
else c[i][j] = maxint;
}
}
for(int i = 0 ; i<m ; ++i){
int u1 , u2 , length;
cin>>u1>>u2>>length;
if(length < c[u1][u2]){
c[u1][u2] = length;
c[u2][u1] = length;
}
}
for(int i = 0 ; i<n ; ++i){
dist[i] = maxint;
}
Dijkstra(n , 0 , dist , c);
}
Dijkstra + dfs 遍历所有最短路径
Dijkstra算法有个好处，就是它拓展点只会拓展一次，而且只会拓展边一次 ，因此该点到源点的最短路径要修改与它相邻的点时不会有重复路径。
递归算法中回溯法很重要。1
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87#include<bits/stdc++.h>
using namespace std;
const int max_int = 1e8 , max_point = 1000; //距离最大值 最多点数
int mp[max_point][max_point]; //地图
vector<int> path[max_point]; //最短路径中，第i个点的前驱
int dist[max_point]; //最短距离
vector<int> p ; //储存最短 轨迹
int sum_path[max_point]; //第i个点到原点的最短路径个数
void Dijkstra(int n , int v , int *dist , int mp[max_point][max_point]){
int book[n];
for(int i = 0 ; i<n ; ++i){
book[i] = 0;
dist[i] = mp[v][i];
}
for(int i = 0 ; i<n ; ++i){
int u , tmp = max_int;
for(int j = 0 ; j<n ; ++j){
if(!book[j] && tmp > dist[j]){
tmp = dist[j];
u = j;
}
}
book[u] = 1;
for(int j = 0 ; j<n ; ++j){
if(!book[j] && dist[j] > dist[u] + mp[u][j]){
dist[j] = dist[u] + mp[u][j];
path[j].clear();
path[j].push_back(u);
sum_path[j] = sum_path[u];
}
else if(!book[j] && dist[j] == dist[u] + mp[u][j]){
path[j].push_back(u);
sum_path[j] += sum_path[u]; //由于算法拓展一个点时只会拓展与它相邻的点，且每条边只拓展一次，所以不会有重复路径
}
}
}
}
void find_path(int u , int v){ //u , v 是要求的最短路径的端点
if(v == u){
p.push_back(u);
for(int i = p.size()-1 ; i>=0 ; --i){
cout<<p[i]<<' ';
}
cout<<endl;
p.pop_back();
return;
}
p.push_back(v);
for(int i = 0 ; i<path[v].size() ; ++i){ //由于是遍历前驱，所以路是倒着的
find_path(u , path[v][i]);
}
p.pop_back(); //回溯！！
}
int main(){
freopen("input.txt","r",stdin);
//freopen("output.txt","w",stdout);
int n , m; //n个点，m条边
cin>>n>>m; //默认点从0到n-1
for(int i = 0 ; i<n ; ++i){
for(int j = 0 ; j<n ; ++j){
if(i == j) mp[i][j] = 0;
else mp[i][j] = max_int;
}
}
for(int i = 0 ; i<m ; ++i){
int c1 , c2 , length;
cin>>c1>>c2>>length;
mp[c1][c2] = length;
mp[c2][c1] = length;
}
sum_path[0]++; //必须把要求的源点的sum_path++;
Dijkstra(n , 0 , dist , mp);
find_path(0 , 4);
for(int i = 0 ; i<n ; ++i){
cout<<sum_path[i]<<' ';
}
}