【SDOI2014】向量集

题目描述

我们分析一波：

假设我们询问\((A,B)\)，\(x_i>x_j\)若
\[ A\cdot x_i+B\cdot y_i>A\cdot x_j+B\cdot y_j\\ A\cdot(x_i-x_j)>B\cdot(y_j-y_i) \]
当\(B>0\)
\[ -\frac{A}{B}<\frac{y_i-y_j}{x_i-x_j} \]
否则
\[ -\frac{A}{B}>\frac{y_i-y_j}{x_i-x_j} \]
所以，对于\(B>0\)，我们在上凸壳上三分，否则在下凸壳上三分。\(B=0\)随意。

三分细节：条件是\(r-l\geq 3\)，然后两个端点是\(\frac{l+l+r}{3}\)以及\(\frac{l+r+r}{3}\)。

代码：

#include<bits/stdc++.h>
#define ll long long
#define inf 1e16
#define eps 1e-7
#define N 400005

using namespace std;
inline int Get() {int x=0,f=1;char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') f=-1;ch=getchar();}while('0'<=ch&&ch<='9') {x=(x<<1)+(x<<3)+ch-'0';ch=getchar();}return x*f;}

int n;
ll ans;
char type;
int decode(int x) {return (type!='E')?x^(ans&0x7fffffff):x;}

struct point {
    double x,y;
    bool operator <(const point &a)const {return x<a.x;}
};

double slope(const point &a,const point &b) {
    if(fabs(a.x-b.x)<eps) return a.y<b.y?inf:-inf;
    return (a.y-b.y)/(a.x-b.x);
}
double cdot(const point &a,const point &b) {return a.x*b.x+a.y*b.y;}

vector<point>tem;
struct Convex {
    vector<point>st;
    int size() {return st.size();}
    void Insert(point a) {st.push_back(a);}
    void build(int flag) {
        sort(st.begin(),st.end());
        tem.clear();
        if(flag) {
            //上凸壳 
            for(int i=0;i<st.size();i++) {
                while(tem.size()>1&&slope(tem[tem.size()-2],tem[tem.size()-1])<slope(tem[tem.size()-1],st[i])+eps) tem.pop_back();
                tem.push_back(st[i]);
            }
        } else {
            //下凸壳 
            for(int i=0;i<st.size();i++) {
                while(tem.size()>1&&slope(tem[tem.size()-2],tem[tem.size()-1])+eps>slope(tem[tem.size()-1],st[i])) tem.pop_back();
                tem.push_back(st[i]);
            }
        }
        st=tem;
    }
    ll query(point a) {
        int l=0,r=st.size()-1;
        while(r-l>=3) {
            int lmid=(l+l+r)/3;
            int rmid=(l+r+r)/3;
            if(cdot(st[lmid],a)>cdot(st[rmid],a)) r=rmid;
            else l=lmid;
        }
        double ans=-inf;
        for(;l<=r;l++) ans=max(ans,cdot(st[l],a));
        return (ll)ans;
    }
};
struct tree {
    int l,r;
    Convex u,d;
}tr[N<<2];
void build(int v,int l,int r) {
    tr[v].l=l,tr[v].r=r;
    if(l==r) return ;
    int mid=l+r>>1;
    build(v<<1,l,mid),build(v<<1|1,mid+1,r);
}
void Insert(int v,int p,point a) {
    tr[v].u.Insert(a);
    tr[v].d.Insert(a);
    if(tr[v].u.size()==tr[v].r-tr[v].l+1) {
        tr[v].u.build(1);
        tr[v].d.build(0);
    }
    if(tr[v].l==tr[v].r) return ;
    int mid=tr[v].l+tr[v].r>>1;
    if(p<=mid) Insert(v<<1,p,a);
    else Insert(v<<1|1,p,a);
}
ll query(int v,int l,int r,point a) {
    if(tr[v].l>r||tr[v].r<l) return -inf;
    if(l<=tr[v].l&&tr[v].r<=r) return a.y>0?tr[v].u.query(a):tr[v].d.query(a);
    return max(query(v<<1,l,r,a),query(v<<1|1,l,r,a));
}
int main() {
    n=Get();
    scanf("%c",&type);
    build(1,1,n);
    char op;
    point a;
    int l,r;
    int now=0;
    while(n--) {
        while(op=getchar(),!isalpha(op));
        if(op=='A') {
            now++;
            a.x=decode(Get()),a.y=decode(Get());
            Insert(1,now,a);
        } else {
            a.x=decode(Get()),a.y=decode(Get());
            l=decode(Get()),r=decode(Get());
            cout<<(ans=query(1,l,r,a))<<"\n";
        }
    }
    return 0;
}