各种蕴含算法思想的DP

study from:

https://www.cnblogs.com/flashhu/p/9480669.html

3.斜率dp

study from:http://www.cnblogs.com/MashiroSky/p/6009685.html

或单减

https://www.luogu.org/problemnew/show/P3195

study from:http://www.cnblogs.com/MashiroSky/p/5968118.html

 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 1e9
21 #define pi 3.1415926536
22 #define E  2.7182818284
23 const ll mod=1e9+7;//998244353
24 const int maxn=5e4+10;
25 
26 ll sum[maxn],a[maxn],f[maxn],L;
27 int q[maxn];
28 
29 double slope(int j,int k)
30 {
31     return (f[k]-f[j]+(a[k]+L+1)*(a[k]+L+1)-(a[j]+L+1)*(a[j]+L+1))
32         /2.0/(a[k]-a[j]);
33 }
34 
35 int main()
36 {
37     int n,l,r,i;
38     ll s;
39     scanf("%d%lld",&n,&L);
40     sum[0]=0;
41     l=1,r=1;
42     f[0]=0;
43     q[1]=0;
44     for (i=1;i<=n;i++)
45     {
46         scanf("%lld",&s);
47         sum[i]=sum[i-1]+s;
48         a[i]=sum[i]+i;
49         while (r>l && slope(q[l],q[l+1])<=a[i]) //have =
50             l++;
51         f[i]=f[q[l]]+pow(i-q[l]-1+sum[i]-sum[q[l]]-L,2);
52         while (r>l && slope(q[r-1],q[r])>slope(q[r],i))//have no =
53             r--;
54         q[++r]=i;
55     }
56     printf("%lld",f[n]);
57     return 0;
58 }

https://www.luogu.org/problemnew/show/P2900

 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 1e9
21 #define pi 3.1415926536
22 #define nl 2.7182818284
23 const ll mod=1e9+7;//998244353
24 const int maxn=5e4+10;
25 
26 struct node
27 {
28     ll x,y;
29 }s[maxn],d[maxn];
30 
31 ll f[maxn];
32 int q[maxn];
33 
34 int cmp(node a,node b)
35 {
36     if (a.y==b.y)
37         return a.x>b.x;
38     else
39         return a.y>b.y;
40 }
41 
42 double slope(int j,int k)
43 {
44     return 1.0*(f[k]-f[j])/(d[j+1].y-d[k+1].y);
45 }
46 
47 int main()
48 {
49     int n,i,m,k,l,r;
50     scanf("%d",&n);
51     for (i=1;i<=n;i++)
52         scanf("%lld%lld",&s[i].x,&s[i].y);
53     sort(s+1,s+n+1,cmp);
54     m=0;
55     k=0;
56     for (i=1;i<=n;i++)
57         if (s[i].x>k)
58         {
59             d[++m]=s[i];
60             k=s[i].x;
61         }
62 
63     f[0]=0;
64     q[1]=0;
65     l=1,r=1;
66     for (i=1;i<=m;i++)
67     {
68         while (l<r && slope(q[l],q[l+1])<=d[i].x)
69             l++;
70         f[i]=f[q[l]]+d[q[l]+1].y*d[i].x;
71         while (l<r && slope(q[r-1],q[r])>slope(q[r],i))
72             r--;
73         q[++r]=i;
74     }
75     printf("%lld",f[m]);
76     return 0;77 }