最短路径专题总结

Posted 2020-10-13

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一.模板：

1.dijsktra算法（矩阵）：

 1 int n,cost[1005][1005],dis[1005],vis[1005],dp[1005];  
 2 void dijkstra(int st)
 3 {
 4     memset(vis,0,sizeof(vis));
 5     for(int i = 1; i<=n; i++)
 6         dis[i] = (i==st?0:INF);
 7 
 8     for(int i = 1; i<=n; i++)
 9     {
10         int k, minn = INF;
11         for(int j = 1; j<=n; j++)
12             if(!vis[j]&&dis[j]<minn)
13                 minn = dis[k=j];
14 
15         vis[k] = 1;
16         for(int j = 1; j<=n; j++)
17             if(!vis[j] && cost[k][j]!=INF)        
18                 dis[j] = min(dis[j], dis[k] + cost[k][j]);
19     }
20 }

View Code

2.dijsktra算法（前向星）：

 1 struct edge
 2 {
 3     int to, w, next;
 4 }edge[maxn*maxn];
 5 int cnt, head[maxn];
 6 
 7 void addedge(int u, int v, int w)
 8 {
 9     edge[cnt].to = v;
10     edge[cnt].w = w;
11     edge[cnt].next = head[u];
12     head[u] = cnt++;
13 }
14 
15 void init()
16 {
17     cnt = 0;
18     memset(head, -1, sizeof(head));
19 }
20 
21 int dis[maxn], vis[maxn];
22 void dijkstra(int st)
23 {
24     memset(vis, 0, sizeof(vis));
25     for(int i = 1; i<=n; i++)
26         dis[i] = (i==st?0:INF);
27 
28     for(int i = 1; i<=n; i++)
29     {
30         int k, minn = INF;
31         for(int j = 1; j<=n; j++)
32             if(!vis[j] && dis[j]<minn)
33                 minn = dis[k=j];
34 
35         vis[k] = 1;
36         for(int j = head[k]; j!=-1; j = edge[j].next)
37             if(!vis[edge[j].to])
38                 dis[edge[j].to] = min(dis[edge[j].to], dis[k] + edge[j].w);
39     }
40 }

View Code

3.bellman-ford算法（前向星）：

 1 struct edge
 2 {
 3     int u, v, w;
 4 }edge[maxn*maxn];
 5 int cnt;
 6 
 7 void addedge(int u, int v, int w)
 8 {
 9     ++cnt;
10     edge[cnt].u = u;
11     edge[cnt].v = v;
12     edge[cnt].w = w;
13 }
14 
15 int dis[maxn];
16 bool bellman(int st)
17 {
18     for(int i = 1; i<=n; i++)
19         dis[i] = (i==st?0:INF);
20 
21     for(int i = 1; i<=n-1; i++)
22     {
23         for(int j = 1; j<=cnt; j++)
24         {
25             int u = edge[j].u;
26             int v = edge[j].v;
27             if(dis[u]!=INF)
28                 dis[v] = min(dis[v], dis[u]+edge[j].w);
29         }
30     }
31 
32     for(int i = 1; i<=cnt; i++)    //判断是否存在负环
33     {
34         int u = edge[i].u;
35         int v = edge[i].v;
36         if(dis[u]!=INF && dis[u]+edge[i].w<dis[v])
37             return false;
38     }
39     return true;
40 }

View Code

4.spfa算法（前向星）：

 1 struct edge
 2 {
 3     int to, w, next;
 4 }edge[maxn*maxn];
 5 int cnt, head[maxn];
 6 
 7 void addedge(int u, int v, int w)
 8 {
 9     edge[cnt].to = v;
10     edge[cnt].w = w;
11     edge[cnt].next = head[u];
12     head[u] = cnt++;
13 }
14 
15 void init()
16 {
17     cnt = 0;
18     memset(head, -1, sizeof(head));
19 }
20 
21 int dis[maxn], times[maxn], inq[maxn];
22 bool spfa(int st)
23 {
24     memset(inq, 0, sizeof(inq));
25     memset(times, 0, sizeof(times));
26     for(int i = 1; i<=n; i++)
27         dis[i] = (i==st?0:INF);
28 
29     queue<int>Q;
30     Q.push(st);
31     inq[st] = 1;
32     while(!Q.empty())
33     {
34         int u = Q.front();
35         Q.pop(); inq[u] = 0;
36         for(int i = head[u]; i!=-1; i = edge[i].next)
37         {
38             int v = edge[i].to;
39             if(dis[v]>dis[u]+edge[i].w)
40             {
41                 dis[v] = dis[u]+edge[i].w;
42                 if(!inq[v])
43                 {
44                     Q.push(v);
45                     inq[v] = 1;
46                     if(++times[v]>n) return false;
47                 }
48             }
49         }
50     }
51     return true;
52 }

View Code

5.floyd算法：

1 for(int i = 1; i<=n; i++)    
2     for(int j = 1; j<=n; j++)    
3         scanf("%d",&dis[i][j]);    
4 
5 for(int k = 1; k<=n; k++)   
6     for(int i = 1; i<=n; i++)    
7         for(int j = 1; j<=n; j++)    
8             dis[i][j] = min(dis[i][j], dis[i][k]+dis[k][j]);