[Sdoi2010]Hide and Seek【K-D Tree】

题目链接 BZOJ 1941

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  题意：给N个点，选其中一个点为起点，使得它到其他所有点的最远曼哈顿距离和最近曼哈顿距离的差值最小。

  所以，题目就变成了对每个点进行查询，去查询它们对应的二维空间的最远到达的点和最近到达的点，最近到达的就是直接用临近算法来解决。

  最远到达的话，我们于最近到达的“球面半径”一样，进行估值函数来传递信息，我们估值最远的可能点来进行优化。

#include <iostream>
#include <cstdio>
#include <cmath>
#include <string>
#include <cstring>
#include <algorithm>
#include <limits>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <bitset>
//#include <unordered_map>
//#include <unordered_set>
#define lowbit(x) ( x&(-x) )
#define pi 3.141592653589793
#define e 2.718281828459045
#define INF 0x3f3f3f3f3f3f3f3f
#define eps 1e-8
#define HalF (l + r)>>1
#define lsn rt<<1
#define rsn rt<<1|1
#define Lson lsn, l, mid
#define Rson rsn, mid+1, r
#define QL Lson, ql, qr
#define QR Rson, ql, qr
#define myself rt, l, r
#define MP(a, b) make_pair(a, b)
#define MAX_3(a, b, c) max(a, max(b, c))
#define Rabc(x) x > 0 ? x : -x
using namespace std;
typedef unsigned long long ull;
typedef unsigned int uit;
typedef long long ll;
const int K = 2, maxN = 1e5 + 7;
int N, Kth;
ll I_want;
struct node
{
    ll d[K]; int id;
    node(ll a=0, ll b=0, int c=0):d{a, b}, id(c) {}
    void In() { scanf("%lld%lld", &d[0], &d[1]); }
} a[maxN], b[maxN];
int op;
inline bool cmp(node e1, node e2) { return e1.d[op] < e2.d[op]; }
node q, point;
int key[maxN << 2];
double var[maxN << 2];
ll dis(node a, node b)
{
    ll sum = 0;
    for(int i=0; i<K; i++) sum += abs(a.d[i] - b.d[i]);
    return sum;
}
struct KD_Close
{
    node tree[maxN << 2];
    ll ans;
    void build(int rt, int l, int r)
    {
        if(l > r) return;
        op = 0; key[rt] = 0;
        for(int i=0; i<K; i++)
        {
            double ave = 0.;
            var[i] = 0;
            for(int j=l; j<=r; j++) ave += a[j].d[i];
            ave /= (r - l + 1.);
            for(int j=l; j<=r; j++) var[i] += abs(ave - a[j].d[i]);
            var[i] /= (r - l + 1.);
            if(var[i] > var[key[rt]])
            {
                key[rt] = i;
                op = i;
            }
        }
        int mid = HalF;
        nth_element(a + l, a + mid, a + r + 1, cmp);
        tree[rt] = a[mid];
        build(rt << 1, l, mid - 1); build(Rson);
    }
    void query(int rt, int l, int r)
    {
        if(l > r) return;
        int mid = HalF;
        ll dist = dis(q, tree[rt]);
        if(q.id ^ tree[rt].id)
        {
            if(dist < ans)
            {
                ans = dist;
                point = tree[rt];
            }
        }
        int k_key = key[rt];
        ll ra = abs(q.d[k_key] - tree[rt].d[k_key]);
        if(q.d[k_key] < tree[rt].d[k_key])
        {
            query(rt << 1, l, mid - 1);
            if(ra < ans) query(Rson);
        }
        else
        {
            query(Rson);
            if(ra < ans) query(rt << 1, l, mid - 1);
        }
    }
} close_Tree;
struct KD_far
{
    node tree[maxN << 2];
    ll ans, mx[maxN << 2][K], mn[maxN << 2][K];
    void pushup(int rt, int l, int r)
    {
        int mid = HalF;
        if(l < mid && mid < r)
        {
            for(int i=0; i<K; i++)
            {
                mx[rt][i] = max(tree[rt].d[i], max(mx[lsn][i], mx[rsn][i]));
                mn[rt][i] = min(tree[rt].d[i], min(mn[lsn][i], mn[rsn][i]));
            }
        }
        else if(l < mid)
        {
            for(int i=0; i<K; i++)
            {
                mx[rt][i] = max(tree[rt].d[i], mx[lsn][i]);
                mn[rt][i] = min(tree[rt].d[i], mn[lsn][i]);
            }
        }
        else if(mid < r)
        {
            for(int i=0; i<K; i++)
            {
                mx[rt][i] = max(tree[rt].d[i], mx[rsn][i]);
                mn[rt][i] = min(tree[rt].d[i], mn[rsn][i]);
            }
        }
        else
        {
            for(int i=0; i<K; i++) mx[rt][i] = mn[rt][i] = tree[rt].d[i];
        }
    }
    void build(int rt, int l, int r)
    {
        if(l > r) return;
        op = 0; key[rt] = 0;
        for(int i=0; i<K; i++)
        {
            double ave = 0.;
            var[i] = 0;
            for(int j=l; j<=r; j++) ave += a[j].d[i];
            ave /= (r - l + 1.);
            for(int j=l; j<=r; j++) var[i] += abs(ave - a[j].d[i]);
            var[i] /= (r - l + 1.);
            if(var[i] > var[key[rt]])
            {
                key[rt] = i;
                op = i;
            }
        }
        int mid = HalF;
        nth_element(a + l, a + mid, a + r + 1, cmp);
        tree[rt] = a[mid];
        build(rt << 1, l, mid - 1); build(Rson);
        pushup(myself);
    }
    ll _Get(int rt)
    {
        ll sum = 0;
        for(int i=0; i<K; i++) sum += max(abs(q.d[i] - mx[rt][i]), abs(q.d[i] - mn[rt][i]));
        return sum;
    }
    void query(int rt, int l, int r)
    {
        if(l > r) return;
        int mid = HalF;
        ll dist = dis(q, tree[rt]);
        if(q.id ^ tree[rt].id)
        {
            if(dist > ans)
            {
                ans = dist;
                point = tree[rt];
            }
        }
        ll dl = _Get(lsn), dr = _Get(rsn);
        if(dl > dr)
        {
            if(dl > ans) query(rt << 1, l, mid - 1);
            if(dr > ans) query(Rson);
        }
        else
        {
            if(dr > ans) query(Rson);
            if(dl > ans) query(rt << 1, l, mid - 1);
        }
    }
} far_Tree;
int main()
{
    scanf("%d", &N);
    for(int i=1; i<=N; i++)
    {
        a[i].In();
        a[i].id = i;
        b[i] = a[i];
    }
    close_Tree.build(1, 1, N);
    far_Tree.build(1, 1, N);
    I_want = INF;
    for(int i=1; i<=N; i++)
    {
        q = b[i];
        close_Tree.ans = INF;
        close_Tree.query(1, 1, N);
        far_Tree.ans = 0;
        far_Tree.query(1, 1, N);
        I_want = min(I_want, far_Tree.ans - close_Tree.ans);
    }
    printf("%lld\n", I_want);
    return 0;
}