【HDU5721 BestCoder 2nd AnniversaryD】【平面最近点对 分治写法+KD-tree写法】Palace 平面最近点对

Palace

Time Limit: 8000/4000 MS (Java/Others)    Memory Limit: 262144/262144 K (Java/Others)

Total Submission(s): 260    Accepted Submission(s): 72

Problem Description

The last trial Venus imposes on Psyche is a quest to the underworld. She is to take a box and obtain in it a dose of the beauty of Prosperina, queen of the underworld.

There are n
palaces in the underworld, which can be located on a 2-Dimension plane with
(x,y)
coordinates (where x,y
are integers). Psyche would like to find the distance of the closest pair of two palaces. It is the password to enter the main palace.

However, the underworld is mysterious and changes all the time. At different times, exactly one of then
palaces disappears.

Psyche wonders what the distance of the closest pair of two palaces is after some palace has disappeared.

Print the sum of the distance after every single palace has disappeared.

To avoid floating point error, define the distance
d
between palace (x1,y1)
and (x2,y2)
as d=(x1?x2)2+(y1?y2)2.

Input

The first line of the input contains an integer
T(1≤T≤5),
which denotes the number of testcases.

For each testcase, the first line contains an integers
n(3≤n≤105),
which denotes the number of temples in this testcase.

The following n
lines contains n
pairs of integers, the i-th
pair (x,y)(?105≤x,y≤105)
denotes the position of the i-th
palace.

Output

For each testcase, print an integer which denotes the sum of the distance after every single palace has disappeared.

Sample Input

1
3
0 0
1 1
2 2

Sample Output

12

Hint

If palace $ (0,0) $ disappears，$ d = (1-2) ^ 2 + (1 - 2) ^ 2 = 2 $;

If palace $ (1,1) $ disappears，$ d = (0-2) ^ 2 + (0 - 2) ^ 2 = 8 $;

If palace $ (2,2) $ disappears，$ d = (0-1) ^ 2 + (0-1) ^ 2 = 2 $;

Thus the answer is $ 2 + 8 + 2 = 12 $。

Source

BestCoder 2nd Anniversary

Recommend

wange2014

分治写法：

#include<stdio.h>
#include<iostream>
#include<string.h>
#include<string>
#include<ctype.h>
#include<math.h>
#include<set>
#include<map>
#include<vector>
#include<queue>
#include<bitset>
#include<algorithm>
#include<time.h>
using namespace std;
void fre() { freopen("c://test//input.in", "r", stdin); freopen("c://test//output.out", "w", stdout); }
#define MS(x,y) memset(x,y,sizeof(x))
#define MC(x,y) memcpy(x,y,sizeof(x))
#define MP(x,y) make_pair(x,y)
#define ls o<<1
#define rs o<<1|1
typedef long long LL;
typedef unsigned long long UL;
typedef unsigned int UI;
template <class T1, class T2>inline void gmax(T1 &a, T2 b) { if (b>a)a = b; }
template <class T1, class T2>inline void gmin(T1 &a, T2 b) { if (b<a)a = b; }
const int N = 1e5 + 10, M = 0, Z = 1e9 + 7, ms63 = 0x3f3f3f3f;
int casenum, casei;
struct Point
{
	LL x, y; int o;
}p[N], tmpp[N];
int n;
LL K(LL x)
{
	return x*x;
}
struct Ans
{
	int x, y; LL dis;
	Ans(int x = -1, int y = -1, LL dis = 1e18) :x(x), y(y), dis(dis) {};
	void update(Ans b)
	{
		if (b.dis < dis)
		{
			x = b.x;
			y = b.y;
			dis = b.dis;
		}
	}
};
Ans getDistance(Point &a, Point &b)
{
	LL dis = K(a.x - b.x) + K(a.y - b.y);
	return Ans(a.o, b.o, dis);
}
bool cmpxy(const Point& a, const Point& b)
{
	if (a.x != b.x)return a.x < b.x;
	return a.y < b.y;
}
bool cmpy(const Point& a, const Point& b)
{
	return a.y < b.y;
}
Ans res;
void Closest_Pair(int l, int r)
{
	if (l + 1 == r)
	{
		res.update(getDistance(p[l], p[r]));
	}
	else if (l + 2 == r)
	{
		res.update(getDistance(p[l], p[l + 1]));
		res.update(getDistance(p[l + 1], p[r]));
		res.update(getDistance(p[l], p[r]));
	}
	else
	{
		int mid = (l + r) >> 1;
		//先分治求左右子区间内部的最近公共点对
		Closest_Pair(l, mid);
		Closest_Pair(mid + 1, r);
		//再求左右子区间之间的最近公共点对
		int g = 0;
		for (int i = l; i <= r; ++i)
		{
			if (K(p[i].x - p[mid].x) < res.dis)tmpp[g++] = p[i];
		}
		sort(tmpp, tmpp + g, cmpy);
		for (int i = 0; i < g; ++i)
		{
			for (int j = i + 1; j < g && K(tmpp[j].y - tmpp[i].y) < res.dis; ++j)
			{
				res.update(getDistance(tmpp[j], tmpp[i]));
			}
		}
	}
}
int main()
{
	scanf("%d", &casenum);
	for (casei = 1; casei <= casenum; ++casei)
	{
		scanf("%d", &n);
		for (int i = 0; i < n; i++)scanf("%lld%lld", &p[i].x, &p[i].y);
		sort(p, p + n, cmpxy);
		for (int i = 0; i < n; i++)p[i].o = i;
		res = Ans(); Closest_Pair(0, n - 1); LL ans = res.dis * (n - 2);
		int x = res.x; int y = res.y;
		int tmpx = p[x].x; int tmpy = p[x].y;
		p[x].x = p[x].y = 1e9;
		res = Ans(); Closest_Pair(0, n - 1); ans += res.dis;
		p[x].x = tmpx; p[x].y = tmpy;
		p[y].x = p[y].y = 1e9;
		res = Ans(); Closest_Pair(0, n - 1); ans += res.dis;
		printf("%lld\n", ans);
	}
	return 0;
}
/*
【trick&&吐槽】
这题一眼标算，可惜不会模板TwT
最后学习了别人的模板改动过了初测然后FST，血崩！

【题意】
二维平面上有n个点
让你求，在每个点消失一次的情况下，剩余点的最近公共点对距离，并求和。

【类型】
最近公共点对
分治法 or KD-Tree

【分析】
显然，我们只要先求出不删点条件下的最近公共点对距离。
这个距离对答案的贡献系数是(n-2)
然后再分别删掉这2个端点，并再次求最近公共点对求和，答案就出来了。

【时间复杂度&&优化】
O(T * 3nlognlogn)

【数据】
1
5
3 1 1 1 1
3 5 7 5 5

*/

KD-tree写法

#include<iostream>
#include<stdio.h>
#include<string.h>
#include<ctype.h>
#include<string>
#include<math.h>
#include<map>
#include<set>
#include<vector>
#include<queue>
#include<algorithm>
using namespace std;
template <class T1, class T2>inline void gmax(T1 &a, T2 b) { if (b>a)a = b; }
template <class T1, class T2>inline void gmin(T1 &a, T2 b) { if (b<a)a = b; }
#define MS(x, y) memset(x, y, sizeof(x))
#define MC(x, y) memcpy(x, y, sizeof(x))
#define lson l,mid-1,(key+1)%Dim
#define rson mid+1,r,(key+1)%Dim
typedef long long LL;
const int Dim = 2;
const int N = 100010;
int Spe;
struct Point
{
	LL p[2]; int o;
}p[N], P;
int n;
struct Ans
{
	int x, y; LL dis;
	Ans(int x = -1, int y = -1, LL dis = 1e18) :x(x), y(y), dis(dis) {};
	void update(Ans b)
	{
		if (b.dis < dis)
		{
			x = b.x;
			y = b.y;
			dis = b.dis;
		}
	}
}ans;
LL K(LL x) { return x*x; }
Ans getDistance(Point &a, Point &b)
{
	if (b.o == Spe || a.o == b.o)return Ans();
	LL dis = K(a.p[0] - b.p[0]) + K(a.p[1] - b.p[1]);
	return Ans(a.o, b.o, dis);
}
int cmpkey;
bool cmp(Point x, Point y)
{
	return x.p[cmpkey] < y.p[cmpkey];
}
void build(int l, int r, int key)
{
	if (l >= r)return;
	int mid = (l + r) >> 1;
	cmpkey = key;
	nth_element(p + l, p + mid, p + r + 1, cmp);
	build(lson);
	build(rson);
}
void find(int l, int r, int key)
{
	if (l > r)return;
	int mid = (l + r) >> 1;
	ans.update(getDistance(P, p[mid]));
	if (P.p[key] < p[mid].p[key])
	{
		find(lson);
		if (K(P.p[key] - p[mid].p[key]) < ans.dis)find(rson);
	}
	if (P.p[key] > p[mid].p[key])
	{
		find(rson);
		if (K(P.p[key] - p[mid].p[key]) < ans.dis)find(lson);
	}
}
void solve()
{
	int n;
	scanf("%d", &n);
	for (int i = 1; i <= n; ++i)
	{
		for (int j = 0; j < Dim; ++j)scanf("%lld", &p[i].p[j]);
	}
	build(1, n, 0);
	for (int i = 1; i <= n; ++i)p[i].o = i;
	ans = Ans(); Spe = 0; LL sum = 0;
	for (int i = 1; i <= n; ++i)
	{
		P = p[i]; find(1, n, 0);
	}
	sum += ans.dis*(n - 2);
	int x = ans.x; int y = ans.y;
	ans = Ans(); Spe = x;
	for (int i = 1; i <= n; ++i)if (i != x)
	{
		P = p[i]; find(1, n, 0);
	}
	sum += ans.dis;
	ans = Ans(); Spe = y;
	for (int i = 1; i <= n; ++i)if (i != y)
	{
		P = p[i]; find(1, n, 0);
	}
	sum += ans.dis;
	printf("%lld\n", sum);

}
void fre() { freopen("c://test//input.in", "r", stdin); freopen("c://test//output.out", "w", stdout); }
int casenum, casei;
int main()
{
	//fre();
	scanf("%d", &casenum);
	for (casei = 1; casei <= casenum; ++casei)solve();
	return 0;
}
/*
【trick&&吐槽】
这题一眼标算，可惜不会模板TwT
最后学习了别人的模板改动过了初测，然后FST，血崩！

这题还有另外一种KD-tree写法

【题意】
二维平面上有n个点
让你求，在每个点消失一次的情况下，剩余点的最近公共点对距离，并求和。

【类型】
最近公共点对
分治法 or KD-Tree

【分析】
显然，我们只要先求出不删点条件下的最近公共点对距离。
这个距离对答案的贡献系数是(n-2)
然后再分别删掉这2个端点，并再次求最近公共点对求和，答案就出来了。

【时间复杂度&&优化】
O(T * 3nsqrt(n))

【数据】
1
5
100000 100000
200000 200000
300000 300000
400000 400000
500000 500000

*/
时间： 2024-08-28 14:27:36

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