Gym 102460L Largest Quadrilateral
题意
给$n$个点从中选出四个点,使得面积最大
Solution
- 首先,肯定是求凸包,要求的点一定在凸包上。
- 不难联想到凸包对每个边求最大三角形面积的问题,也就是旋转卡壳。
- 可以将问题转化为,对凸包的每一个对角线$A_iA_j$,求最大面积的两个三角形,$\triangle{A_iA_jP}$, $\triangle{A_iA_jQ}$, 然后就可以枚举对角线,旋转卡壳算最大面积
细节
- 输出的格式
- 凸包上应该留下共线的点
#include <cstdio>
#include <stack>
#include <set>
#include <cmath>
#include <map>
#include <time.h>
#include <vector>
#include <iostream>
#include <string>
#include <cstring>
#include <algorithm>
//#include <memory.h>
#include <cstdlib>
#include <queue>
#include <iomanip>
#include <cassert>
// #include <unordered_map>
#define P pair<int, int>
#define LL long long
#define LD long double
#define PLL pair<LL, LL>
#define mset(a, b) memset(a, b, sizeof(a))
#define rep(i, a, b) for (int i = a; i < b; i++)
#define PI acos(-1.0)
#define random(x) rand() % x
#define debug(x) cout << #x << " " << x << "\n"
using namespace std;
const int inf = 0x3f3f3f3f;
const LL __64inf = 0x3f3f3f3f3f3f3f3f;
#ifdef DEBUG
const int MAX = 2e3 + 50;
#else
const int MAX = 1e6 + 50;
#endif
const int mod = 1e9+9;
void file_read(){
#ifdef DEBUG
freopen("in", "r", stdin);
// freopen("out", "w", stdout);
#endif
}
template<typename type>
struct Vec
{
type x, y;
Vec() {}
Vec(type x, type y) : x(x), y(y) {}
friend istream & operator >> (istream &in, Vec &A) {
in >> A.x >> A.y;
return in;
}
friend Vec operator - (const Vec &A, const Vec &B) {
return Vec(A.x-B.x, A.y-B.y);
}
friend Vec operator + (const Vec &A, const Vec &B) {
return Vec(A.x + B.x, A.y + B.y);
}
friend type det(const Vec &A, const Vec &B) {
return A.x * B.y - A.y * B.x;
}
friend type dot(const Vec &A, const Vec &B) {
return A.x * B.x + A.y * B.y;
}
friend bool operator < (const Vec &A, const Vec &B) {
if(A.x != B.x) return A.x < B.x;
return A.y < B.y;
}
friend type area(const Vec &A, const Vec &B) {
return abs(det(A, B));
}
friend type operator == (const Vec &A, const Vec &B) {
return A.x == B.x and A.y == B.y;
}
};
template<typename type>
vector<Vec<type>> convex_hull(vector<Vec<type>> &pt) {
sort(pt.begin(), pt.end());
int n = pt.size();
vector<Vec<type>> res(2*n);
int k = 0 ;
for(int i = 0; i < n; i++) {
while(k > 1 and det(res[k-1]-res[k-2], pt[i]-res[k-1]) < 0) // <=会wa
k--;
res[k++] = pt[i];
}
for(int i = n-2, t = k; i >= 0; i--) {
while(k > t and det(res[k-1]-res[k-2], pt[i]-res[k-1]) < 0) // <=会wa
k--;
res[k++] = pt[i];
}
res.resize(k-1);
return res;
}
struct ModI
{
int i, n;
ModI(int n ) : i(0), n(n) { assert(n > 0) ;}
ModI(int i, int n) : i(i%n), n(n) { assert(n > 0); }
ModI operator ++ ( int ) {
ModI row = ModI(i, n);
i = (i + 1) % n;
return row;
}
ModI operator + (int x) {
ModI res = ModI(i, n);
res.i = (res.i + x) % n;
return res;
}
int operator = (int x) {
return i = x;
}
bool operator < (int x) const {
return i < x;
}
bool operator == (const ModI &other) const {
return i == other.i;
}
operator int () {
return i;
}
};
template<typename type>
type area(const Vec<type> &A, const Vec<type> &B, const Vec<type> &C) {
return area(A-B, A-C);
}
template<typename type>
type rotateCalipers(vector<Vec<type>> pt) {
int n = pt.size();
type res = 0;
for(int i = 0; i < pt.size(); i++) {
ModI p1 = ModI(i+1, n);
ModI p2 = ModI(i+3, n);
for(ModI j = ModI(i+2, n); j+1 != i; j++) {
while(p1+1 != j and area(pt[p1], pt[i], pt[j]) < area(pt[p1+1], pt[i], pt[j]))
p1 ++;
if(j == p2) p2++;
while(p2+1 != i and area(pt[p2], pt[i], pt[j]) < area(pt[p2+1], pt[i], pt[j]))
p2 ++;
auto cur = area(pt[p1], pt[i], pt[j]) + area(pt[p2], pt[i], pt[j]);
res = max(res, cur);
}
}
return res;
}
void out(LL ans) {
if(ans & 1) {
printf("%lld.5\n", ans >> 1);
}
else {
printf("%lld\n", ans >> 1);
}
}
int main() {
file_read();
int T;
scanf("%d", &T);
while (T--)
{
int n;
scanf("%d", &n);
vector<Vec<LL>> pt(n);
for(int i = 0; i < n; i++) cin >> pt[i];
auto ch = convex_hull(pt);
if(ch.size() < 3) {
printf("0\n");
continue;
}
if(ch.size() == 3) {
LL ans = 0;
LL A = area(ch[0], ch[1], ch[2]);
for(auto p : pt) {
if(p == ch[0] or p == ch[1] or p == ch[2]) continue;
auto a = area(p, ch[1], ch[2]);
a = min(a, area(p, ch[0], ch[2]));
a = min(a, area(p, ch[0], ch[1]));
ans = max(ans, A-a);
}
out(ans);
continue;
}
LL res = rotateCalipers(ch);
out(res);
}
return 0;
}