我的理解：
给定一个变量如X0（类似于物理的参照物），求所有Xk-X0的最大值Dk。
如：
Xa-X0<=Dk，
如果Xb-Xa<=L，则有
Xb-X0<=Dk+L，
对于Xb-Xa<=L，
变形为Xa+L>=Xb，
(Xa-X0)+L>=(Xb-X0)
Da+L>=Db
相当于点a到点b有一条长度为L的有向边，当Da+L>=b，点b的最短路修改为从起点到点a的最短路+点a到点b。
求最短路后，相当于求出Xk-X0<=V中，V的最小值(其它V都无意义，Xk-X0<=V<=any V')。这就是Xk-X0的最大值（它不能取大于V的值，否则式子Xk-X0<=V不成立）。
study from：
https://blog.csdn.net/my_sunshine26/article/details/72849441
https://blog.csdn.net/lyy289065406/article/details/6648679
题目：
1.
http://poj.org/problem?id=1716
study from：https://blog.csdn.net/lyy289065406/article/details/6648679
贪心
1 #include <cstdio> 2 #include <cstdlib> 3 #include <cmath> 4 #include <cstring> 5 #include <time.h> 6 #include <string> 7 #include <set> 8 #include <map> 9 #include <list> 10 #include <stack> 11 #include <queue> 12 #include <vector> 13 #include <bitset> 14 #include <algorithm> 15 #include <iostream> 16 using namespace std; 17 #define ll long long 18 #define minv 1e-6 19 #define inf 1e9 20 #define pi 3.1415926536 21 #define nl 2.7182818284 22 const ll mod=1e9+7;//998244353 23 const int maxn=1e4+10; 24 25 struct node 26 { 27 int a,b; 28 }f[maxn]; 29 30 bool vis[maxn]={0}; 31 32 int cmp(node x,node y) 33 { 34 if (x.b==y.b) 35 return x.a<y.a; 36 else 37 return x.b<y.b; 38 } 39 40 int main() 41 { 42 int n,i,sum=0,x,y; 43 scanf("%d",&n); 44 for (i=1;i<=n;i++) 45 scanf("%d%d",&f[i].a,&f[i].b); 46 sort(f+1,f+n+1,cmp); 47 x=-1,y=-1; 48 for (i=1;i<=n;i++) 49 { 50 if (y<f[i].a) 51 y=f[i].b,sum++; 52 if (x<f[i].a) 53 { 54 sum++; 55 if (y!=f[i].b) 56 { 57 x=y; 58 y=f[i].b; 59 } 60 else 61 x=f[i].b-1; 62 } 63 } 64 cout<<sum; 65 return 0; 66 }
sum[b]-sum[a-1]>=2
sum[b]-2>=sum[a-1]
b -> a-1 len=-2
sum[x+1]+0>=sum[x]
x+1 -> x len=0
sum[x]+1>=sum[x+1]
x -> x+1 len=1
第三个式子容易忘
1 #include <cstdio> 2 #include <cstdlib> 3 #include <cmath> 4 #include <cstring> 5 #include <time.h> 6 #include <string> 7 #include <set> 8 #include <map> 9 #include <list> 10 #include <stack> 11 #include <queue> 12 #include <vector> 13 #include <bitset> 14 #include <algorithm> 15 #include <iostream> 16 using namespace std; 17 #define ll long long 18 #define minv 1e-6 19 #define inf 1e9 20 #define pi 3.1415926536 21 #define nl 2.7182818284 22 const ll mod=1e9+7;//998244353 23 const int maxn=1e4+10; 24 25 /* 26 sum[b]-sum[a-1]>=2 27 sum[b]-2>=sum[a-1] 28 b -> a-1 len=-2 29 30 sum[x+1]+0>=sum[x] 31 x+1 -> x len=0 32 33 sum[x]+1>=sum[x+1] 34 x -> x+1 len=1 35 */ 36 37 struct node 38 { 39 int d,len; 40 node *next; 41 }*prenode[10],*e[maxn]; 42 43 int preint[10],dist[maxn],q[maxn]; 44 bool vis[maxn]={0}; 45 46 int main() 47 { 48 node *p; 49 int n,i,a,b,head,tail,d,dd,value=10000; 50 scanf("%d",&n); 51 for (i=1;i<=n;i++) 52 { 53 scanf("%d%d",&a,&b); 54 p=(node*) malloc (sizeof(node)); 55 p->d=a-1; 56 p->len=-2; 57 p->next=e[b]; 58 e[b]=p; 59 } 60 for (i=0;i<=value;i++) 61 { 62 p=(node*) malloc (sizeof(node)); 63 p->d=i-1; 64 p->len=0; 65 p->next=e[i]; 66 e[i]=p; 67 } 68 for (i=-1;i<=value-1;i++) 69 { 70 p=(node*) malloc (sizeof(node)); 71 p->d=i+1; 72 p->len=1; 73 p->next=e[i]; 74 e[i]=p; 75 } 76 fill(dist-1,dist+value,inf); 77 dist[value]=0; 78 fill(vis-1,vis+value,0); 79 vis[value]=1; 80 head=0,tail=1; 81 q[1]=value; 82 while (head!=tail) 83 { 84 head=(head+1)%10001; 85 d=q[head]; 86 p=e[d]; 87 while (p) 88 { 89 dd=p->d; 90 if (dist[dd]>dist[d]+p->len) 91 { 92 dist[dd]=dist[d]+p->len; 93 if (!vis[dd]) 94 { 95 vis[dd]=1; 96 tail=(tail+1)%10001; 97 q[tail]=dd; 98 } 99 } 100 p=p->next; 101 } 102 vis[d]=0; 103 } 104 printf("%d",-dist[-1]); 105 return 0; 106 }
2.
http://poj.org/problem?id=1201
1 #include <cstdio> 2 #include <cstdlib> 3 #include <cmath> 4 #include <cstring> 5 #include <time.h> 6 #include <string> 7 #include <set> 8 #include <map> 9 #include <list> 10 #include <stack> 11 #include <queue> 12 #include <vector> 13 #include <bitset> 14 #include <algorithm> 15 #include <iostream> 16 using namespace std; 17 #define ll long long 18 #define minv 1e-6 19 #define inf 1e9 20 #define pi 3.1415926536 21 #define nl 2.7182818284 22 const ll mod=1e9+7;//998244353 23 const int maxn=5e4+10; 24 25 /* 26 sum[b]-sum[a-1]>=2 27 sum[b]-2>=sum[a-1] 28 b -> a-1 len=-2 29 30 sum[x+1]+0>=sum[x] 31 x+1 -> x len=0 32 33 sum[x]+1>=sum[x+1] 34 x -> x+1 len=1 35 */ 36 37 struct node 38 { 39 int d,len; 40 node *next; 41 }*prenode[10],*e[maxn]; 42 43 int preint[10],dist[maxn],q[maxn]; 44 bool vis[maxn]={0}; 45 46 int main() 47 { 48 node *p; 49 int n,i,a,b,c,head,tail,d,dd,value=50000; 50 scanf("%d",&n); 51 for (i=1;i<=n;i++) 52 { 53 scanf("%d%d%d",&a,&b,&c); 54 p=(node*) malloc (sizeof(node)); 55 p->d=a-1; 56 p->len=-c; 57 p->next=e[b]; 58 e[b]=p; 59 } 60 for (i=0;i<=value;i++) 61 { 62 p=(node*) malloc (sizeof(node)); 63 p->d=i-1; 64 p->len=0; 65 p->next=e[i]; 66 e[i]=p; 67 } 68 for (i=-1;i<=value-1;i++) 69 { 70 p=(node*) malloc (sizeof(node)); 71 p->d=i+1; 72 p->len=1; 73 p->next=e[i]; 74 e[i]=p; 75 } 76 fill(dist-1,dist+value,inf); 77 dist[value]=0; 78 fill(vis-1,vis+value,0); 79 vis[value]=1; 80 81 head=0,tail=1; 82 q[1]=value; 83 while (head!=tail) 84 { 85 head=(head+1)%50001; 86 d=q[head]; 87 p=e[d]; 88 while (p) 89 { 90 dd=p->d; 91 if (dist[dd]>dist[d]+p->len) 92 { 93 dist[dd]=dist[d]+p->len; 94 if (!vis[dd]) 95 { 96 vis[dd]=1; 97 tail=(tail+1)%50001; 98 q[tail]=dd; 99 } 100 } 101 p=p->next; 102 } 103 vis[d]=0; 104 } 105 printf("%d",-dist[-1]); 106 return 0; 107 }
3.
http://poj.org/problem?id=3169
BellmanFord
在这里，神奇地速度竟然比SPFA快
1 #include <cstdio> 2 #include <cstdlib> 3 #include <cmath> 4 #include <cstring> 5 #include <time.h> 6 #include <string> 7 #include <set> 8 #include <map> 9 #include <list> 10 #include <stack> 11 #include <queue> 12 #include <vector> 13 #include <bitset> 14 //#include <ext/rope> 15 #include <algorithm> 16 #include <iostream> 17 using namespace std; 18 #define ll long long 19 #define minv 1e-6 20 #define inf 2e9 21 #define pi 3.1415926536 22 #define nl 2.7182818284 23 const ll mod=1e9+7;//998244353 24 const int maxn=1e4+10; 25 26 /** 27 a[y]-a[x]<=z 28 a[x]+z>=a[y] 29 x->y z 30 31 a[y]-a[x]>=z 32 a[y]-z>=a[x] 33 y->x -z 34 35 a[x+1]-a[x]>=0 36 a[x+1]>=a[x] 37 x+1->x 0 38 **/ 39 40 struct rec 41 { 42 int x,y,z; 43 }f[30000];///ml+md+n 44 45 int dist[maxn],g=0; 46 47 bool bell() 48 { 49 bool vis=0; 50 int i; 51 for (i=1;i<=g;i++) 52 if (dist[f[i].y]>dist[f[i].x]+f[i].z) ///don't need to judge ==inf !=inf 53 { 54 dist[f[i].y]=dist[f[i].x]+f[i].z; 55 vis=1; 56 } 57 return vis; 58 } 59 60 int main() 61 { 62 int n,ml,md,i,x,y,z; 63 scanf("%d%d%d",&n,&ml,&md); 64 for (i=1;i<=ml;i++) 65 { 66 scanf("%d%d%d",&x,&y,&z); 67 f[++g]={x,y,z}; 68 } 69 for (i=1;i<=md;i++) 70 { 71 scanf("%d%d%d",&x,&y,&z); 72 f[++g]={y,x,-z}; 73 } 74 75 for (i=1;i<n;i++) 76 f[++g]={i+1,i,0}; 77 78 fill(dist,dist+n+1,inf); 79 dist[1]=0; 80 for (i=1;i<=n;i++) 81 if (!bell()) 82 break; 83 if (i==n+1) 84 printf("-1"); 85 else if (dist[n]==inf) 86 printf("-2"); 87 else 88 printf("%d",dist[n]); 89 return 0; 90 } 91 /* 92 4 2 0 93 1 2 10 94 3 4 20 95 96 4 1 1 97 1 2 8 98 1 2 12 99 100 3 1 1 101 1 3 10 102 2 3 12 103 104 4 1 1 105 2 3 8 106 2 3 12 107 */
SPFA
有些样例不对，因为如果使用SPFA，有些点不能到达，但代码通过了这题
如：
4 1 1
2 3 8
2 3 12
wrong!
1 #include <cstdio> 2 #include <cstdlib> 3 #include <cmath> 4 #include <cstring> 5 #include <time.h> 6 #include <string> 7 #include <set> 8 #include <map> 9 #include <list> 10 #include <stack> 11 #include <queue> 12 #include <vector> 13 #include <bitset> 14 #include <ext/rope> 15 #include <algorithm> 16 #include <iostream> 17 using namespace std; 18 #define ll long long 19 #define minv 1e-6 20 #define inf 2e9 21 #define pi 3.1415926536 22 #define nl 2.7182818284 23 const ll mod=1e9+7;//998244353 24 const int maxn=1e3+10; 25 26 /** 27 a[y]-a[x]<=z 28 a[x]+z>=a[y] 29 x->y z 30 31 a[y]-a[x]>=z 32 a[y]-z>=a[x] 33 y->x -z 34 35 a[x+1]-a[x]>=0 36 a[x+1]>=a[x] 37 x+1->x 0 38 **/ 39 40 struct node 41 { 42 int d,len; 43 node *next; 44 }*e[maxn]; 45 46 int dist[maxn],q[maxn],use[maxn]; 47 bool vis[maxn]; 48 49 int main() 50 { 51 node *p; 52 int n,ml,md,i,x,y,z,d,dd,head,tail; 53 scanf("%d%d%d",&n,&ml,&md); 54 for (i=1;i<=ml;i++) 55 { 56 scanf("%d%d%d",&x,&y,&z); 57 p=(node*) malloc (sizeof(node)); 58 p->d=y; 59 p->len=z; 60 p->next=e[x]; 61 e[x]=p; 62 } 63 for (i=1;i<=md;i++) 64 { 65 scanf("%d%d%d",&x,&y,&z); 66 p=(node*) malloc (sizeof(node)); 67 p->d=x; 68 p->len=-z; 69 p->next=e[y]; 70 e[y]=p; 71 } 72 for (i=1;i<n;i++) 73 { 74 p=(node*) malloc (sizeof(node)); 75 p->d=i; 76 p->len=0; 77 p->next=e[i+1]; 78 e[i+1]=p; 79 } 80 81 fill(dist,dist+n+1,inf); 82 dist[1]=0; 83 memset(vis,0,sizeof(vis)); 84 vis[1]=1; 85 memset(use,0,sizeof(use)); 86 q[1]=1; 87 head=0; 88 tail=1; 89 while (head!=tail) 90 { 91 head=(head+1)%n; 92 d=q[head]; 93 p=e[d]; 94 while (p) 95 { 96 dd=p->d; 97 if (dist[dd]>dist[d]+p->len) 98 { 99 dist[dd]=dist[d]+p->len; 100 use[dd]++; 101 if (use[dd]>=n) 102 { 103 printf("-1"); 104 return 0; 105 } 106 if (!vis[dd]) 107 { 108 vis[dd]=1; 109 tail=(tail+1)%n; 110 q[tail]=dd; 111 } 112 } 113 p=p->next; 114 } 115 vis[d]=0; 116 } 117 if (dist[n]==inf) 118 printf("-2"); 119 else 120 printf("%d",dist[n]); 121 return 0; 122 } 123 /* 124 4 2 0 125 1 2 10 126 3 4 20 127 128 4 1 1 129 1 2 8 130 1 2 12 131 132 3 1 1 133 1 3 10 134 2 3 12 135 136 4 1 1 137 2 3 8 138 2 3 12 139 wrong! 140 */