hoj第三场G-manhattanp ositioning system

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一、题意

在二维坐标系内，给定若干个点和目标点距离该点的曼哈顿距离，求是否存在唯一解使得目标点坐标为整数。

 

 

二、题解

重新思考题目模型，考虑曼哈顿距离一定时，几何含义为，以给定点为中心，以二倍曼哈顿距离为对角线的矩形的轮廓线。则原题可以理解为，寻找唯一的被所有矩形覆盖的整数点。

于是，考虑扫描线思路，首先将所有涉及到的坐标离散化并标记出来，其次建立线段树，用于统计某个区间出现的最多的点的次数。则，nodes[0]中保存着当前各个节点最大覆盖次数。当覆盖次数为n时，必然存在实数解，但是不一定存在整数解。

考虑模型实际上在y轴上推进，x轴上做统计。考虑到几何意义，则有2种边——横着的和竖着的。横边在扫描线扫描过程中，应当在切换下一坐标之前消除。但是竖边应当用一个点来表示，考虑几何意义，应当在下一个横边之前消除。另外题目的特殊情况包括了r为0的点，要把他当做横边特殊判断一下。

#include<iostream>
#include <stdio.h>
#include<stdlib.h>
#include <assert.h>
#include <algorithm>
#include <cmath>
#include <set>
#include <vector>

using namespace std;

#define ll long long
#define ld long double
#define ppd pair<ld,ld>
#define equal(x,y) (abs(x-y)<=EXP)
#define idx(x) (lower_bound(mapp,mapp+mapp_num,x)-mapp)

#define init_coor(x) (equal(x.first,-2333666)&&equal(x.second,-2333666))

const int MAXN = 100233;
const ld EXP = 1e-2;
const ld SQRT_2 = sqrt(2);

ppd rotate(ppd p){
    ld dis = sqrt(p.first*p.first+p.second*p.second);
    if(equal(dis,0))return p;
    ld coss = (p.first - p.second)/(dis*SQRT_2);
    ld sins = (p.first + p.second)/(dis*SQRT_2);
    return make_pair(dis*coss,dis*sins);
}

ppd anti_rotate(ppd p){

    ld dis = sqrt(p.first*p.first+p.second*p.second);
    if(equal(dis,0))return p;
    ld coss = (p.first + p.second)/(dis*SQRT_2);
    ld sins = (-p.first + p.second)/(dis*SQRT_2);
    return make_pair(dis*coss,dis*sins);
}

class Order{
    public:
        ld x1,x2,y;int key;
        int tt;
        Order(){}
        Order(ld x,ld xx,ld yy,int kk){
            this->x1 = x;
            this->x2 = xx;
            this->y = yy;
            this->key = kk;
            this->tt = 1;
        }
};

inline bool is_negative_point(Order o){
    return(equal(o.x1,o.x2)&&o.key<0&&o.tt);
}
inline bool is_positive_point(Order o){
    return(equal(o.x1,o.x2)&&o.key>0&&o.tt);
}
inline bool is_point(Order o){
    return equal(o.x1,o.x2)&&o.tt;
}

bool cmp1(Order o1,Order o2){
    if(equal(o1.y,o2.y)){
        if(is_negative_point(o1) && !is_negative_point(o2))return true;
        if(!is_negative_point(o1) && is_negative_point(o2))return false;
        if(is_positive_point(o1) && !is_positive_point(o2))return false;
        if(!is_positive_point(o1) && is_positive_point(o2))return true;



        return o1.key > o2.key;
    }
    return o1.y<o2.y;
}

Order orders[MAXN];
int orders_num;

ld mapp[MAXN];
int mapp_num;

bool cmp(ld l1,ld l2){
    if(equal(l1,l2))return l1>l2;
    else return l1<l2;
}

void show(Order o){
    cout<<"check_order: "<<o.x1<<" "<<o.x2<<" at: "<<o.y<<" key: "<<o.key<<endl;
}

class Node{
    public:
        int l,r,lc,rc,num,lazy;
};
Node nodes[MAXN];
int nodes_num;

void tree_init(int a,int b){
    int now = nodes_num++;
    nodes[now].l = a;
    nodes[now].r = b;
    nodes[now].num = nodes[now].lazy = 0;
    if(a == b-1)return ;
    int mid = (a+b)/2;

    nodes[now].lc = nodes_num;
    tree_init(a,mid);
    nodes[now].rc = nodes_num;
    tree_init(mid,b);
}

void push_down(int now){
    int lazy = nodes[now].lazy;
    nodes[now].lazy = 0;
    int l = nodes[now].l;
    int r = nodes[now].r;
    if(l == r-1)return ;
    int lc = nodes[now].lc;
    int rc = nodes[now].rc;
    nodes[lc].lazy += lazy;
    nodes[lc].num += lazy;
    nodes[rc].lazy += lazy;
    nodes[rc].num += lazy;
}

void tree_insert(int now,int a,int b,int key){
    int l = nodes[now].l;
    int r = nodes[now].r;
    push_down(now);
    if(a == l && b == r){
        nodes[now].num += key;
        nodes[now].lazy += key;
        return ;
    }
    int mid = (l+r)/2;
    int lc = nodes[now].lc;
    int rc = nodes[now].rc;
    if(a<mid){
        tree_insert(lc,a,min(b,mid),key);
        if(b>mid) tree_insert(rc,mid,b,key);
    }else tree_insert(rc,a,b,key);
    nodes[now].num = max(nodes[nodes[now].lc].num,nodes[nodes[now].rc].num);
}

ld fitness_x,lasty;

void show(ppd p)
{
    // cout<<"check_coor: "<<p.first<<" "<<p.second<<endl;
    printf("check_coor: %.10lf %.10lf\n",(double)p.first,(double)p.second);
}

bool check_int(ppd p){
    p.first += EXP;
    p.second += EXP;

    // show(p);
    // printf("check_coor: %.5f %.5f %.5f %.5f\n",(double)p.first,(double)ceil(p.first),(double)p.second,(double)ceil(p.second));

    return equal(p.first,floor(p.first)) || equal(p.second,floor(p.second));
}

int find_max_num(int now){
    if(now == 0)fitness_x = -2333666;
    int maxx = nodes[now].num;
    int l = nodes[now].l;
    int r = nodes[now].r;
    push_down(now);
    if(l == r-1){
        ld xx = mapp[l];
        ld yy = lasty;
        ppd coor = make_pair(xx,yy);
        ppd std_coor = anti_rotate(coor);
        // show(coor);
        // show(std_coor);
        if(check_int(std_coor)){
            fitness_x = mapp[l];
            return 1;
        }
        return 0;
    }
    int lc = nodes[now].lc;
    int rc = nodes[now].rc;
    int ret = 0;
    if(nodes[lc].num == maxx)ret += find_max_num(lc);
    if(nodes[rc].num == maxx)ret += find_max_num(rc);
    return ret;
}

int n;

ppd succ_coor;

void init(){
    nodes_num = 0;
    orders_num = 0;
    mapp_num = 0;

    succ_coor = make_pair(-2333666,-2333666);

    mapp[mapp_num++] = 1e8;
    mapp[mapp_num++] = -1e8;

    for(int i=0;i<n;++i){
        ld x,y,r;
        cin>>x>>y>>r;
        ppd p1 = make_pair(x-r,y);
        ppd p2 = make_pair(x,y-r);
        ppd p3 = make_pair(x+r,y);

if(equal(r,0)){

    ppd coor = rotate(make_pair(x,y));
    mapp[mapp_num++] = coor.first;
    orders[orders_num++] = Order(coor.first,coor.first,coor.second,1);
    orders[orders_num-1].tt = 0;
    orders[orders_num++] = Order(coor.first,coor.first,coor.second,-1);
    orders[orders_num-1].tt = 0;
    continue;
}
        // show(p1);
        // show(p2);
        // show(p3);

        p1 = rotate(p1);
        p2 = rotate(p2);
        p3 = rotate(p3);

        // show(anti_rotate(p1));
        // show(anti_rotate(p2));
        // show(anti_rotate(p3));

        orders[orders_num++] = Order(p1.first,p2.first,p1.second,1);
        orders[orders_num++] = Order(p1.first,p2.first,p3.second,1);
        orders[orders_num++] = Order(p1.first,p2.first,p1.second,-1);
        orders[orders_num++] = Order(p1.first,p2.first,p3.second,-1);

        orders[orders_num++] = Order(p1.first,p1.first,p1.second,1);
        orders[orders_num++] = Order(p2.first,p2.first,p1.second,1);       
        orders[orders_num++] = Order(p1.first,p1.first,p3.second,-1);
        orders[orders_num++] = Order(p2.first,p2.first,p3.second,-1);   

        mapp[mapp_num++] = p1.first;
        mapp[mapp_num++] = p2.first;
        // cout<<p1.first<<" "<<p2.first<<endl;
    }
    sort(orders,orders+orders_num,cmp1);
    sort(mapp,mapp+mapp_num,cmp);
    
    int len = mapp_num;
    mapp_num = 0;
    for(int i=1;i<len;++i){

                    // cout<<mapp[i]<<endl;
        if(!(equal(mapp[i],mapp[mapp_num]))){
            mapp_num++;
            mapp[mapp_num] = mapp[i];

        }
    }mapp_num++;

    tree_init(0,mapp_num);

    int succ = 0;

    lasty = orders[0].y;

    for(int i=0;i<orders_num;++i){


        // cout<<"circle: "<<i<<" "<<orders_num<<endl;
            // cout<<"check_num: "<<nodes[0].num<<endl;
            // show(orders[i]);
            // cout<<"check_mapp: "<<mapp[idx(orders[i].x1)]<<" "<<orders[i].x1<<endl;
            // printf("check_mapp: %.5f %.5f\n",(double)mapp[idx(orders[i].x1)],(double)orders[i].x1);
            // printf("check_mapp: %.5f %.5f\n",(double)mapp[idx(orders[i].x2)],(double)orders[i].x2);

        if(nodes[0].num == n){
            int maxx_num = find_max_num(0);
            if(maxx_num >1){
                succ = -2;
                break;
            }
            if(maxx_num == 0)continue;
            
            if(init_coor(succ_coor))succ_coor = make_pair(fitness_x,lasty);
            else{
                succ = -2;
                break;
            }
        }
        tree_insert(0,idx(orders[i].x1),idx(orders[i].x2)+1,orders[i].key);
        lasty = orders[i].y;
    }

    if(succ == -2){
        puts("uncertain");
        return;
    }
    if(init_coor(succ_coor)){
        puts("impossible");
        return;
    }else{
        succ_coor = anti_rotate(succ_coor);
        // succ_coor.first += EXP;
        // succ_coor.second += EXP;
        printf("%.0f %.0f\n",(double)succ_coor.first,(double)succ_coor.second);
    }

}

int main(){

    while(cin>>n)init();

    return 0;
}