LightOJ 1418 Trees on My Island (Pick定理)
Posted wulitaotao
tags:
篇首语:本文由小常识网(cha138.com)小编为大家整理,主要介绍了LightOJ 1418 Trees on My Island (Pick定理)相关的知识,希望对你有一定的参考价值。
题目链接:LightOJ 1418
Problem Description
I have bought an island where I want to plant trees in rows and columns. So, the trees will form a rectangular grid and each of them can be thought of having integer coordinates by taking a suitable grid point as the origin.
But, the problem is that the island itself is not rectangular. So, I have identified a simple polygonal area inside the island with vertices on the grid points and have decided to plant trees on grid points lying strictly inside the polygon.
Figure: A sample of my island
For example, in the above figure, the green circles form the polygon, and the blue circles show the position of the trees.
Now, I seek your help for calculating the number of trees that can be planted on my island.
Input
Input starts with an integer (T (≤ 100)), denoting the number of test cases.
Each case starts with a line containing an integer (N (3 ≤ N ≤ 10000)) denoting the number of vertices of the polygon.
Each of the next (N) lines contains two integers (x_i y_i (-10^6 ≤ x_i, y_i ≤ 10^6)) denoting the co-ordinate of a vertex. The vertices will be given in clockwise or anti-clockwise order. And they will form a simple polygon.
Output
For each case, print the case number and the total number of trees that can be planted inside the polygon.
Sample Input
1
9
1 2
2 1
4 1
4 3
6 2
6 4
4 5
1 5
2 3
Sample Output
Case 1: 8
Note
Dataset is huge, use faster I/O methods.
Solution
题意:
给定一个多边形,顶点都在格点上,求多边形内部的格点个数。
思路
Pick 定理 裸题。
#include <cstdio>
#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>
using namespace std;
typedef long long ll;
typedef double db;
const db eps = 1e-10;
const db pi = acos(-1.0);
const ll inf = 0x3f3f3f3f3f3f3f3f;
const ll maxn = 1e5 + 10;
inline int dcmp(db x) {
if(fabs(x) < eps) return 0;
return x > 0? 1: -1;
}
class Point {
public:
ll x, y;
Point(ll x = 0, ll y = 0) : x(x), y(y) {}
void input() {
scanf("%lld%lld", &x, &y);
}
Point operator-(const Point a) {
return Point(x - a.x, y - a.y);
}
ll cross(const Point a) {
return x * a.y - y * a.x;
}
};
Point p[maxn];
ll gcd(ll a, ll b) {
return b == 0? a: gcd(b, a % b);
}
int main() {
int T;
scanf("%d", &T);
for(int _ = 1; _ <= T; ++_) {
int n;
scanf("%d", &n);
ll on = 0;
ll s = 0;
for(int i = 0; i < n; ++i) {
p[i].input();
}
p[n] = p[0];
for(int i = 0; i < n; ++i) {
s += (p[i + 1] - p[0]).cross(p[i] - p[0]);
on += gcd(abs(p[i].x - p[i + 1].x), abs(p[i].y - p[i + 1].y));
}
s = abs(s);
ll in = s / 2 - on / 2 + 1;
printf("Case %d: ", _);
printf("%lld
", in);
}
return 0;
}
以上是关于LightOJ 1418 Trees on My Island (Pick定理)的主要内容,如果未能解决你的问题,请参考以下文章
lightoj 1119 - Pimp My Ride(状压dp)