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QOJ
ID | 题目 | 提交者 | 结果 | 用时 | 内存 | 语言 | 文件大小 | 提交时间 | 测评时间 |
---|---|---|---|---|---|---|---|---|---|
#298884 | #7906. Almost Convex | ucup-team045# | WA | 14ms | 4144kb | C++20 | 16.5kb | 2024-01-06 15:36:16 | 2024-01-06 15:36:16 |
Judging History
answer
#include<bits/stdc++.h>
using namespace std;
using LL = long long;
using point_t = long long; //全局数据类型,可修改为 long long 等
const point_t eps = 1e-8;
const long double PI = acosl(-1);
//const long double PI = numbers::pi_v<long double>;
// 点与向量
template <typename T>
struct point{
T x, y;
bool operator==(const point &a) const { return (abs(x - a.x) <= eps && abs(y - a.y) <= eps); }
bool operator<(const point &a) const{
if (abs(x - a.x) <= eps)
return y < a.y - eps;
return x < a.x - eps;
}
bool operator>(const point &a) const { return !(*this < a || *this == a); }
point operator+(const point &a) const { return {x + a.x, y + a.y}; }
point operator-(const point &a) const { return {x - a.x, y - a.y}; }
point operator-() const { return {-x, -y}; }
point operator*(const T k) const { return {k * x, k * y}; }
point operator/(const T k) const { return {x / k, y / k}; }
T operator*(const point &a) const { return x * a.x + y * a.y; } // 点积
T operator^(const point &a) const { return x * a.y - y * a.x; } // 叉积,注意优先级
int toleft(const point &a) const
{
const auto t = (*this) ^ a;
return (t > eps) - (t < -eps);
} // to-left 测试
T len2() const { return (*this) * (*this); } // 向量长度的平方
T dis2(const point &a) const { return (a - (*this)).len2(); } // 两点距离的平方
// 涉及浮点数
long double len() const { return sqrtl(len2()); } // 向量长度
long double dis(const point &a) const { return sqrtl(dis2(a)); } // 两点距离
long double ang(const point &a) const { return acosl(max(-1.0l, min(1.0l, ((*this) * a) / (len() * a.len())))); } // 向量夹角
point rot(const long double rad) const { return {x * cosl(rad) - y * sinl(rad), x * sinl(rad) + y * cosl(rad)}; } // 逆时针旋转(给定角度)
point rot(const long double cosr, const long double sinr) const { return {x * cosr - y * sinr, x * sinr + y * cosr}; } // 逆时针旋转(给定角度的正弦与余弦)
};
using Point = point<point_t>;
// 极角排序
struct argcmp
{
bool operator()(const Point &a, const Point &b) const
{
const auto quad = [](const Point &a)
{
if (a.y < -eps)
return 1;
if (a.y > eps)
return 4;
if (a.x < -eps)
return 5;
if (a.x > eps)
return 3;
return 2;
};
const int qa = quad(a), qb = quad(b);
if (qa != qb)
return qa < qb;
const auto t = a ^ b;
// if (abs(t)<=eps) return a*a<b*b-eps; // 不同长度的向量需要分开
return t > eps;
}
};
// 直线
template <typename T>
struct line
{
point<T> p, v; // p 为直线上一点,v 为方向向量
bool operator==(const line &a) const { return v.toleft(a.v) == 0 && v.toleft(p - a.p) == 0; }
int toleft(const point<T> &a) const { return v.toleft(a - p); } // to-left 测试
bool operator<(const line &a) const // 半平面交算法定义的排序
{
if (abs(v ^ a.v) <= eps && v * a.v >= -eps)
return toleft(a.p) == -1;
return argcmp()(v, a.v);
}
// 涉及浮点数
point<T> inter(const line &a) const { return p + v * ((a.v ^ (p - a.p)) / (v ^ a.v)); } // 直线交点
long double dis(const point<T> &a) const { return abs(v ^ (a - p)) / v.len(); } // 点到直线距离
point<T> proj(const point<T> &a) const { return p + v * ((v * (a - p)) / (v * v)); } // 点在直线上的投影
point<T> symmetry(const point<T> &a) const { return proj(a) * 2 - a;} // 点关于直线的对称点
};
using Line = line<point_t>;
//线段
template <typename T>
struct segment
{
point<T> a, b;
bool operator<(const segment &s) const { return make_pair(a, b) < make_pair(s.a, s.b); }
// 判定性函数建议在整数域使用
// 判断点是否在线段上
// -1 点在线段端点 | 0 点不在线段上 | 1 点严格在线段上
int is_on(const point<T> &p) const
{
if (p == a || p == b)
return -1;
return (p - a).toleft(p - b) == 0 && (p - a) * (p - b) < -eps;
}
// 判断线段直线是否相交
// -1 直线经过线段端点 | 0 线段和直线不相交 | 1 线段和直线严格相交
int is_inter(const line<T> &l) const
{
if (l.toleft(a) == 0 || l.toleft(b) == 0)
return -1;
return l.toleft(a) != l.toleft(b);
}
// 判断两线段是否相交
// -1 在某一线段端点处相交 | 0 两线段不相交 | 1 两线段严格相交
int is_inter(const segment<T> &s) const
{
if (is_on(s.a) || is_on(s.b) || s.is_on(a) || s.is_on(b))
return -1;
const line<T> l{a, b - a}, ls{s.a, s.b - s.a};
return l.toleft(s.a) * l.toleft(s.b) == -1 && ls.toleft(a) * ls.toleft(b) == -1;
}
// 点到线段距离
long double dis(const point<T> &p) const
{
if ((p - a) * (b - a) < -eps || (p - b) * (a - b) < -eps)
return min(p.dis(a), p.dis(b));
const line<T> l{a, b - a};
return l.dis(p);
}
// 两线段间距离
long double dis(const segment<T> &s) const
{
if (is_inter(s))
return 0;
return min({dis(s.a), dis(s.b), s.dis(a), s.dis(b)});
}
// 只求整点交点可以不使用浮点数,避免精度问题,使用前需要先判断是否有交点
pair<bool, point<T> > int_inter(const segment &s){
// 线段转为直线的一般式
auto seg2line = [&](const segment &s){
T A = s.a.y - s.b.y;
T B = s.b.x - s.a.x;
T C = -A * s.a.x - B * s.a.y;
return array{A, B, C};
};
auto [A1, B1, C1] = seg2line(*this);
auto [A2, B2, C2] = seg2line(s);
T dx = C1 * B2 - C2 * B1;
T dy = A1 * C2 - A2 * C1;
T d = B1 * A2 - B2 * A1;
if (d == 0) return {false, {}};
if (dy % d || dx % d) return {false, {}};
return {true, {dx / d, dy / d}};
}
};
using Segment = segment<point_t>;
// 多边形
template <typename T>
struct polygon
{
vector<point<T>> p; // 以逆时针顺序存储
size_t nxt(const size_t i) const { return i == p.size() - 1 ? 0 : i + 1; }
size_t pre(const size_t i) const { return i == 0 ? p.size() - 1 : i - 1; }
// 回转数
// 返回值第一项表示点是否在多边形边上
// 对于狭义多边形,回转数为 0 表示点在多边形外,否则点在多边形内
pair<bool, int> winding(const point<T> &a) const
{
int cnt = 0;
for (size_t i = 0; i < p.size(); i++)
{
const point<T> u = p[i], v = p[nxt(i)];
if (abs((a - u) ^ (a - v)) <= eps && (a - u) * (a - v) <= eps)
return {true, 0};
if (abs(u.y - v.y) <= eps)
continue;
const Line uv = {u, v - u};
if (u.y < v.y - eps && uv.toleft(a) <= 0)
continue;
if (u.y > v.y + eps && uv.toleft(a) >= 0)
continue;
if (u.y < a.y - eps && v.y >= a.y - eps)
cnt++;
if (u.y >= a.y - eps && v.y < a.y - eps)
cnt--;
}
return {false, cnt};
}
// 射线法 2表示在多边形内,1表示在多边形上,0表示在多边形外
int is_in(const point<T> &a){
int x = 0;
for(size_t i = 0; i < p.size(); i++){
segment<T> s = {p[i], p[nxt(i)]};
if (s.is_on(a)) return 1;
point<T> p1 = p[i] - a, p2 = p[nxt(i)] - a;
if(p1.y > p2.y) swap(p1, p2);
if(p1.y < eps && p2.y > eps && (p1 ^ p2) > eps) x = !x;
}
return x ? 2 : 0;
}
// 多边形面积的两倍
// 可用于判断点的存储顺序是顺时针或逆时针
T area() const
{
T sum = 0;
for (size_t i = 0; i < p.size(); i++)
sum += p[i] ^ p[nxt(i)];
return sum;
}
// 多边形的周长
long double circ() const
{
long double sum = 0;
for (size_t i = 0; i < p.size(); i++)
sum += p[i].dis(p[nxt(i)]);
return sum;
}
};
using Polygon = polygon<point_t>;
//凸多边形
template <typename T>
struct convex : polygon<T>
{
// 闵可夫斯基和
convex operator+(const convex &c) const
{
const auto &p = this->p;
vector<Segment> e1(p.size()), e2(c.p.size()), edge(p.size() + c.p.size());
vector<point<T>> res;
res.reserve(p.size() + c.p.size());
const auto cmp = [](const Segment &u, const Segment &v)
{ return argcmp()(u.b - u.a, v.b - v.a); };
for (size_t i = 0; i < p.size(); i++)
e1[i] = {p[i], p[this->nxt(i)]};
for (size_t i = 0; i < c.p.size(); i++)
e2[i] = {c.p[i], c.p[c.nxt(i)]};
rotate(e1.begin(), min_element(e1.begin(), e1.end(), cmp), e1.end());
rotate(e2.begin(), min_element(e2.begin(), e2.end(), cmp), e2.end());
merge(e1.begin(), e1.end(), e2.begin(), e2.end(), edge.begin(), cmp);
const auto check = [](const vector<point<T>> &res, const point<T> &u)
{
const auto back1 = res.back(), back2 = *prev(res.end(), 2);
return (back1 - back2).toleft(u - back1) == 0 && (back1 - back2) * (u - back1) >= -eps;
};
auto u = e1[0].a + e2[0].a;
for (const auto &v : edge)
{
while (res.size() > 1 && check(res, u))
res.pop_back();
res.push_back(u);
u = u + v.b - v.a;
}
if (res.size() > 1 && check(res, res[0]))
res.pop_back();
return {res};
}
// 旋转卡壳
// func 为更新答案的函数,可以根据题目调整位置
template <typename F>
void rotcaliper(const F &func) const
{
const auto &p = this->p;
const auto area = [](const point<T> &u, const point<T> &v, const point<T> &w)
{ return (w - u) ^ (w - v); };
for (size_t i = 0, j = 1; i < p.size(); i++)
{
const auto nxti = this->nxt(i);
func(p[i], p[nxti], p[j]);
while (area(p[this->nxt(j)], p[i], p[nxti]) >= area(p[j], p[i], p[nxti]))
{
j = this->nxt(j);
func(p[i], p[nxti], p[j]);
}
}
}
// 凸多边形的直径的平方
T diameter2() const
{
const auto &p = this->p;
if (p.size() == 1)
return 0;
if (p.size() == 2)
return p[0].dis2(p[1]);
T ans = 0;
auto func = [&](const point<T> &u, const point<T> &v, const point<T> &w)
{ ans = max({ans, w.dis2(u), w.dis2(v)}); };
rotcaliper(func);
return ans;
}
// 判断点是否在凸多边形内
// 复杂度 O(logn)
// -1 点在多边形边上 | 0 点在多边形外 | 1 点在多边形内
int is_in(const point<T> &a) const
{
const auto &p = this->p;
if (p.size() == 1)
return a == p[0] ? -1 : 0;
if (p.size() == 2)
return segment<T>{p[0], p[1]}.is_on(a) ? -1 : 0;
if (a == p[0])
return -1;
if ((p[1] - p[0]).toleft(a - p[0]) == -1 || (p.back() - p[0]).toleft(a - p[0]) == 1)
return 0;
const auto cmp = [&](const Point &u, const Point &v)
{ return (u - p[0]).toleft(v - p[0]) == 1; };
const size_t i = lower_bound(p.begin() + 1, p.end(), a, cmp) - p.begin();
if (i == 1)
return segment<T>{p[0], p[i]}.is_on(a) ? -1 : 0;
if (i == p.size() - 1 && segment<T>{p[0], p[i]}.is_on(a))
return -1;
if (segment<T>{p[i - 1], p[i]}.is_on(a))
return -1;
return (p[i] - p[i - 1]).toleft(a - p[i - 1]) > 0;
}
// 凸多边形关于某一方向的极点
// 复杂度 O(logn)
// 参考资料:https://codeforces.com/blog/entry/48868
template <typename F>
size_t extreme(const F &dir) const
{
const auto &p = this->p;
const auto check = [&](const size_t i)
{ return dir(p[i]).toleft(p[this->nxt(i)] - p[i]) >= 0; };
const auto dir0 = dir(p[0]);
const auto check0 = check(0);
if (!check0 && check(p.size() - 1))
return 0;
const auto cmp = [&](const Point &v)
{
const size_t vi = &v - p.data();
if (vi == 0)
return 1;
const auto checkv = check(vi);
const auto t = dir0.toleft(v - p[0]);
if (vi == 1 && checkv == check0 && t == 0)
return 1;
return checkv ^ (checkv == check0 && t <= 0);
};
return partition_point(p.begin(), p.end(), cmp) - p.begin();
}
// 过凸多边形外一点求凸多边形的切线,返回切点下标
// 复杂度 O(logn)
// 必须保证点在多边形外
pair<size_t, size_t> tangent(const point<T> &a) const
{
const size_t i = extreme([&](const point<T> &u)
{ return u - a; });
const size_t j = extreme([&](const point<T> &u)
{ return a - u; });
return {i, j};
}
// 求平行于给定直线的凸多边形的切线,返回切点下标
// 复杂度 O(logn)
pair<size_t, size_t> tangent(const line<T> &a) const
{
const size_t i = extreme([&](...)
{ return a.v; });
const size_t j = extreme([&](...)
{ return -a.v; });
return {i, j};
}
};
using Convex = convex<point_t>;
Convex convexhull(vector<Point> p)
{
vector<Point> st;
if (p.empty())
return Convex{st};
sort(p.begin(), p.end());
const auto check = [](const vector<Point> &st, const Point &u)
{
const auto back1 = st.back(), back2 = *prev(st.end(), 2);
return (back1 - back2).toleft(u - back1) <= 0;
};
for (const Point &u : p)
{
while (st.size() > 1 && check(st, u))
st.pop_back();
st.push_back(u);
}
size_t k = st.size();
p.pop_back();
reverse(p.begin(), p.end());
for (const Point &u : p)
{
while (st.size() > k && check(st, u))
st.pop_back();
st.push_back(u);
}
st.pop_back();
return Convex{st};
}
int main(){
#ifdef LOCAL
freopen("data.in", "r", stdin);
freopen("data.out", "w", stdout);
#endif
cin.tie(0);
cout.tie(0);
ios::sync_with_stdio(0);
int n;
cin >> n;
vector<Point> p(n);
for(int i = 0; i < n; i++){
cin >> p[i].x >> p[i].y;
}
auto convex = convexhull(p);
set<Point> s(convex.p.begin(), convex.p.end());
vector<Point> v;
for(auto &pt : p){
if (!s.contains(pt)){
v.push_back(pt);
}
}
vector<int> id1(v.size()), id2(v.size());
iota(id1.begin(), id1.end(), 0);
iota(id2.begin(), id2.end(), 0);
int ans = 1;
for(int i = 0; i < convex.p.size(); i++){
auto p1 = convex.p[i], p2 = convex.p[(i + 1) % convex.p.size()];
sort(id1.begin(), id1.end(), [&](int x, int y){
Point a = v[x] - p1;
Point b = v[y] - p1;
return argcmp()(a, b);
});
sort(id2.begin(), id2.end(), [&](int x, int y){
Point a = v[x] - p2;
Point b = v[y] - p2;
return !argcmp()(a, b);
});
vector<bitset<2000> > bs(v.size());
bitset<2000> mask;
for(auto x : id1){
mask.set(x);
bs[x] |= mask;
}
mask.reset();
for(auto x : id2){
mask.set(x);
bs[x] |= mask;
}
for(int j = 0; j < v.size(); j++){
if (bs[j].count() == v.size()){
ans += 1;
}
}
}
cout << ans << '\n';
}
详细
Test #1:
score: 100
Accepted
time: 0ms
memory: 3608kb
input:
7 1 4 4 0 2 3 3 1 3 5 0 0 2 4
output:
9
result:
ok 1 number(s): "9"
Test #2:
score: 0
Accepted
time: 0ms
memory: 3744kb
input:
5 4 0 0 0 2 1 3 3 3 1
output:
5
result:
ok 1 number(s): "5"
Test #3:
score: 0
Accepted
time: 0ms
memory: 3528kb
input:
3 0 0 3 0 0 3
output:
1
result:
ok 1 number(s): "1"
Test #4:
score: 0
Accepted
time: 0ms
memory: 3560kb
input:
6 0 0 3 0 3 2 0 2 1 1 2 1
output:
7
result:
ok 1 number(s): "7"
Test #5:
score: 0
Accepted
time: 0ms
memory: 3532kb
input:
4 0 0 0 3 3 0 3 3
output:
1
result:
ok 1 number(s): "1"
Test #6:
score: -100
Wrong Answer
time: 14ms
memory: 4144kb
input:
2000 86166 617851 383354 -277127 844986 386868 -577988 453392 -341125 -386775 -543914 -210860 -429613 606701 -343534 893727 841399 339305 446761 -327040 -218558 -907983 787284 361823 950395 287044 -351577 -843823 -198755 138512 -306560 -483261 -487474 -857400 885637 -240518 -297576 603522 -748283 33...
output:
10306
result:
wrong answer 1st numbers differ - expected: '718', found: '10306'