Codeforces 340B - Maximal Area Quadrilateral (计算几何)
Codeforces Round #198 (Div. 2)
题目链接:Maximal Area Quadrilateral
Iahub has drawn a set of \(n\) points in the cartesian plane which he calls "special points". A quadrilateral is a simple polygon without self-intersections with four sides (also called edges) and four vertices (also called corners). Please note that a quadrilateral doesn't have to be convex. A special quadrilateral is one which has all four vertices in the set of special points. Given the set of special points, please calculate the maximal area of a special quadrilateral.
Input
The first line contains integer \(n (4 \le n \le 300)\). Each of the next \(n\) lines contains two integers: \(x_i, y_i ( - 1000 \le xi, yi \le 1000)\) — the cartesian coordinates of ith special point. It is guaranteed that no three points are on the same line. It is guaranteed that no two points coincide.
Output
Output a single real number — the maximal area of a special quadrilateral. The answer will be considered correct if its absolute or relative error does't exceed \(10 ^{- 9}\).
Examples
input
5
0 0
0 4
4 0
4 4
2 3
output
16.000000
Note
In the test example we can choose first \(4\) points to be the vertices of the quadrilateral. They form a square by side \(4\), so the area is \(4\cdot 4 = 16\).
Solution
题意
给定 \(n\) 个点的坐标,选择其中 \(4\) 个点构成四边形,求最大四边形面积。
题解
四边形的面积等于两个三角形面积的和。枚举四边形的对角线,以及左右两边的点,选择两个面积最大的三角形,更新 \(ans\)。
三重循环枚举,时间复杂度 \(O(n^3)\)。
Code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef double db;
const db eps = 1e-10;
const db pi = acos(-1.0);
const int inf = 0x3f3f3f3f;
const int maxn = 300 + 5;
inline int dcmp(db x) {
if(fabs(x) < eps) return 0;
return x > 0? 1: -1;
}
class Point {
public:
double x, y;
Point(double x = 0, double y = 0) : x(x), y(y) {}
void input() {
scanf("%lf%lf", &x, &y);
}
bool operator<(const Point &a) const {
return (!dcmp(x - a.x))? dcmp(y - a.y) < 0: x < a.x;
}
bool operator==(const Point &a) const {
return dcmp(x - a.x) == 0 && dcmp(y - a.y) == 0;
}
db dis2(const Point a) {
return pow(x - a.x, 2) + pow(y - a.y, 2);
}
db dis(const Point a) {
return sqrt(dis2(a));
}
db dis2() {
return x * x + y * y;
}
db dis() {
return sqrt(dis2());
}
Point operator+(const Point a) {
return Point(x + a.x, y + a.y);
}
Point operator-(const Point a) {
return Point(x - a.x, y - a.y);
}
Point operator*(double p) {
return Point(x * p, y * p);
}
Point operator/(double p) {
return Point(x / p, y / p);
}
db dot(const Point a) {
return x * a.x + y * a.y;
}
db cross(const Point a) {
return x * a.y - y * a.x;
}
};
typedef Point Vector;
vector<Point> p;
map<pair<int, int>, int> mp;
int main() {
double ans = 0;
int n;
scanf("%d", &n);
for(int i = 0; i < n; ++i) {
Point tmp;
tmp.input();
p.push_back(tmp);
}
sort(p.begin(), p.end());
for(int i = 0; i < p.size(); ++i) {
for(int j = i + 1; j < p.size(); ++j) {
if(!mp[{i, j}]) {
mp[{i, j}] = 1;
double s1 = 0, s2 = 0;
for(int k = 0; k < p.size(); ++k) {
if(k == i || k == j) continue;
if(dcmp((p[j] - p[i]).cross(p[k] - p[i])) > 0) {
s1 = max(s1, fabs((p[j] - p[i]).cross(p[k] - p[i])) * 0.5);
} else {
s2 = max(s2, fabs((p[j] - p[i]).cross(p[k] - p[i])) * 0.5);
}
}
if(dcmp(s1) == 0 || dcmp(s2) == 0) continue;
ans = max(ans, s1 + s2);
}
}
}
printf("%.10lf\n", ans);
return 0;
}
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