BZOJ 1597: [Usaco2008 Mar]土地购买

做法：斜率优化DP

斜率优化DP：Link

预处理:sort处理可合并土地，得到a[]严格上升，b[]严格下降的序列

朴素DP方程：F[i]=min f[j]+a[i]*b[j+1](0<j<i)

 朴素DP Code:

#include <cstdio>
#include <cstring>
#include <algorithm>
#define ll long long 
using namespace std;
inline int read()
{
    register int f=1,k=0;register char c=getchar();
    while (c<'0'||c>'9')c=='-'&&(f=-1),c=getchar();
    while (c>='0'&&c<='9')k=k*10+c-'0',c=getchar();
    return k*f;
}
const int maxn=50100;
struct qaq{
    ll a,b;
}x[maxn],y[maxn];
ll f[maxn];
inline bool cmp(const qaq&a,const qaq&b){return a.a<b.a||(a.a==b.a&&a.b<b.b);}
int main()
{
    memset(f,0x7f,sizeof(f));
    register int i,n=read(),cur=1;
    for (i=1;i<=n;i++)
    x[i].a=read(),x[i].b=read();
    sort(x+1,x+1+n,cmp);
    y[1]=x[1];
    for (register int i=2;i<=n;i++)
    {
        while (x[i].b>=y[cur].b&&cur>0)cur--; 
        y[++cur]=x[i];
    }
    f[1]=y[1].a*y[1].b;
    for (register int i=1;i<=cur;i++)
    for (register int j=1;j<i;j++)
    f[i]=min(f[i],f[j]+y[i].a*y[j+1].b);
    printf("%lld
",f[cur]);
}

 斜率优化DP：

#include <cstdio>
#include <cstring>
#include <algorithm>
#define ll long long 
using namespace std;
inline int read()
{
    register int f=1,k=0;register char c=getchar();
    while (c<'0'||c>'9')c=='-'&&(f=-1),c=getchar();
    while (c>='0'&&c<='9')k=k*10+c-'0',c=getchar();
    return k*f;
}
const int maxn=50100;
struct qaq{ll a,b;}x[maxn],y[maxn];
ll f[maxn],q[maxn];
inline int d(int a,int b){return (f[a]-f[b])/(y[b+1].b-y[a+1].b);}
inline bool cmp(const qaq&a,const qaq&b){return a.a<b.a||(a.a==b.a&&a.b<b.b);}
int main()
{
    register int i,n=read(),cur=1,head=0,tail=1;
    for (i=1;i<=n;i++)x[i].a=read(),x[i].b=read();
    sort(x+1,x+1+n,cmp);
    y[1]=x[1];
    for (register int i=2;i<=n;y[++cur]=x[i++])while (x[i].b>=y[cur].b&&cur>0)cur--;
    for (register int i=1;i<=cur;i++)
    {
        while (tail-head>1&&d(q[head],q[head+1])<y[i].a)head++;
        f[i]=f[q[head]]+1ll*y[i].a*y[q[head]+1].b;
        while (tail-head>1&&d(q[tail-1],q[tail-2])>d(q[tail-1],i))tail--;
        q[tail++]=i;
    }
    printf("%lld
",f[cur]); 
}