Luogu4027 「NOI2007」货币兑换

Solution

推一下发现这个斜率优化不满足任何性质，那么平衡树维护凸包

Code

#include<bits/stdc++.h>
using namespace std;
#define int long long
#define reg register 
#define For(i,a,b) for(reg int i=a;i<=b;++i) 
#define Down(i,a,b) for(reg int i=a;i<=b;++i) 
namespace yspm{
    inline int read()
    {
        int res=0,f=1; char k;
        while(!isdigit(k=getchar())) if(k=='-') f=-1;
        while(isdigit(k)) res=res*10+k-'0',k=getchar();
        return res*f;
    }
    const double inf=1e18+10,eps=1e-9;
    const int N=1e5+10;
    int n,st,rt;
    int ls[N],rs[N],fa[N];
    double a[N],b[N],rat[N],l[N],r[N],x[N],y[N],f[N];
    inline void rotate(int x)
    {
        int y=fa[x],z=fa[y];
        if(ls[z]==y) ls[z]=x; else rs[z]=x;
        if(ls[y]==x) ls[y]=rs[x],fa[rs[x]]=y,rs[x]=y;
        else rs[y]=ls[x],fa[ls[x]]=y,ls[x]=y;
        fa[y]=x; fa[x]=z;
        return ;
    }
    inline void splay(int x,int g)
    {
        while(fa[x]!=g)
        {
            int y=fa[x],z=fa[y]; 
            if(z!=g) rotate((ls[z]==y)^(ls[y]==x)?x:y);
            rotate(x);
        } if(!g) rt=x;
        return ;
    }
    inline int find(int x,double num) 
    {
        if(!x) return 0;
        if(l[x]+eps>=num&&r[x]<=num+eps) return x;
        if(l[x]<num+eps) return find(ls[x],num);
        return find(rs[x],num);    
    }  
    inline double calc(int p,int q)
    {
        if(fabs(x[p]-x[q])<eps) return -inf;
        return (y[p]-y[q])/(x[p]-x[q]);
    }
    inline int pre(int x)
    {
        int now=ls[x],res=now; 
        while(now)
        {
            if(l[now]+eps>=calc(now,x)) res=now,now=rs[now];
            else now=ls[now];
        } return res;
    }
    inline int nxt(int x) 
    {   
        int now=rs[x],res=now;
        while(now) 
        {
            if(r[now]<=eps+calc(now,x)) res=now,now=ls[now];
            else now=rs[now]; 
        } return res;
    }
    inline void del(int x)
    {
        rt=ls[x]; fa[rt]=0; fa[rs[x]]=rt; rs[rt]=rs[x]; 
        r[rt]=l[rs[x]]=calc(rt,rs[x]);
        return ; 
    }
    inline void work(int x)
    {
        splay(x,0);
        if(ls[x])
        {
            int p=pre(x); splay(p,x); rs[p]=0;
            r[ls[x]]=l[x]=calc(ls[x],x);
        }else l[x]=inf;
        if(rs[x])
        {
            int tmp=nxt(x); splay(tmp,x); ls[tmp]=0;
            l[rs[x]]=r[x]=calc(rs[x],x);
        }else r[x]=-inf;
        if(l[x]<=eps+r[x]) del(x);
        return ;
    }
    inline void insert(int &now,int fat,int id)
    {
        if(!now){now=id; fa[now]=fat; return ;}
        if(x[id]<=x[now]+eps) insert(ls[now],now,id); else insert(rs[now],now,id);
        return ;
    }
    signed main()
    {
        n=read(); scanf("%lf",&f[0]); 
        For(i,1,n)
        {
            scanf("%lf%lf%lf",&a[i],&b[i],&rat[i]);
            int pos=find(rt,-1.0*a[i]/b[i]);
            f[i]=max(f[i-1],x[pos]*a[i]+y[pos]*b[i]);
            y[i]=f[i]*1.0/(rat[i]*a[i]+b[i]); x[i]=rat[i]*y[i];
            insert(rt,0,i); work(i);
        } 
        printf("%.6lf
",f[n]);
        return 0;
    }
}
signed main(){return yspm::main();}