This documentation is automatically generated by online-judge-tools/verification-helper
#include "geo/count_points_in_triangles.hpp"
#include "geo/angle_sort.hpp"
#include "geo/base.hpp"
#include "random/base.hpp"
#include "ds/fenwicktree/fenwicktree_01.hpp"
// 点群 A, B を入力 (Point<ll>)
// query(i,j,k):三角形 AiAjAk 内部の Bl の個数(非負)を返す
// 前計算 O(NMlogM)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
// https://codeforces.com/contest/852/problem/H
struct Count_Points_In_Triangles {
using P = Point<ll>;
const int LIM = 1'000'000'000 + 10;
vc<P> A, B;
vc<int> new_idx; // O から見た偏角ソート順を管理
vc<int> point; // A[i] と一致する B[j] の数え上げ
vvc<int> seg; // 線分 A[i]A[j] 内にある B[k] の数え上げ
vvc<int> tri; // OA[i]A[j] 内部にある B[k] の数え上げ
Count_Points_In_Triangles(const vc<P>& A, const vc<P>& B) : A(A), B(B) {
for (auto&& p: A) assert(max(abs(p.x), abs(p.y)) < LIM);
for (auto&& p: B) assert(max(abs(p.x), abs(p.y)) < LIM);
build();
}
int count3(int i, int j, int k) {
i = new_idx[i], j = new_idx[j], k = new_idx[k];
if (i > j) swap(i, j);
if (j > k) swap(j, k);
if (i > j) swap(i, j);
assert(i <= j && j <= k);
ll d = (A[j] - A[i]).det(A[k] - A[i]);
if (d == 0) return 0;
if (d > 0) { return tri[i][j] + tri[j][k] - tri[i][k] - seg[i][k]; }
int x = tri[i][k] - tri[i][j] - tri[j][k];
return x - seg[i][j] - seg[j][k] - point[j];
}
// segment
int count2(int i, int j) {
i = new_idx[i], j = new_idx[j];
if (i > j) swap(i, j);
return seg[i][j];
}
private:
P take_origin() {
// OAiAj, OAiBj が同一直線上にならないようにする
// fail prob: at most N(N+M)/LIM
return P{-LIM, RNG(-LIM, LIM)};
}
void build() {
P O = take_origin();
for (auto&& p: A) p = p - O;
for (auto&& p: B) p = p - O;
int N = len(A), M = len(B);
vc<int> I = angle_sort(A);
A = rearrange(A, I);
new_idx.resize(N);
FOR(i, N) new_idx[I[i]] = i;
I = angle_sort(B);
B = rearrange(B, I);
point.assign(N, 0);
seg.assign(N, vc<int>(N));
tri.assign(N, vc<int>(N));
// point
FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];
int m = 0;
FOR(j, N) {
// OA[i]A[j], B[k]
while (m < M && A[j].det(B[m]) < 0) ++m;
vc<P> C(m);
FOR(k, m) C[k] = B[k] - A[j];
vc<int> I(m);
FOR(i, m) I[i] = i;
sort(all(I), [&](auto& a, auto& b) -> bool { return C[a].det(C[b]) > 0; });
C = rearrange(C, I);
vc<int> rk(m);
FOR(k, m) rk[I[k]] = k;
FenwickTree_01 bit(m);
int k = m;
FOR_R(i, j) {
while (k > 0 && A[i].det(B[k - 1]) > 0) { bit.add(rk[--k], 1); }
P p = A[i] - A[j];
int lb = binary_search([&](int n) -> bool { return (n == 0 ? true : C[n - 1].det(p) > 0); }, 0, m + 1);
int ub = binary_search([&](int n) -> bool { return (n == 0 ? true : C[n - 1].det(p) >= 0); }, 0, m + 1);
seg[i][j] += bit.sum(lb, ub), tri[i][j] += bit.sum(lb);
}
}
}
};
#line 1 "geo/count_points_in_triangles.hpp"
#line 2 "geo/angle_sort.hpp"
#line 2 "geo/base.hpp"
template <typename T>
struct Point {
T x, y;
Point() : x(0), y(0) {}
template <typename A, typename B>
Point(A x, B y) : x(x), y(y) {}
template <typename A, typename B>
Point(pair<A, B> p) : x(p.fi), y(p.se) {}
Point operator+=(const Point p) {
x += p.x, y += p.y;
return *this;
}
Point operator-=(const Point p) {
x -= p.x, y -= p.y;
return *this;
}
Point operator+(Point p) const { return {x + p.x, y + p.y}; }
Point operator-(Point p) const { return {x - p.x, y - p.y}; }
bool operator==(Point p) const { return x == p.x && y == p.y; }
bool operator!=(Point p) const { return x != p.x || y != p.y; }
Point operator-() const { return {-x, -y}; }
Point operator*(T t) const { return {x * t, y * t}; }
Point operator/(T t) const { return {x / t, y / t}; }
bool operator<(Point p) const {
if (x != p.x) return x < p.x;
return y < p.y;
}
T dot(const Point& other) const { return x * other.x + y * other.y; }
T det(const Point& other) const { return x * other.y - y * other.x; }
double norm() { return sqrtl(x * x + y * y); }
double angle() { return atan2(y, x); }
Point rotate(double theta) {
static_assert(!is_integral<T>::value);
double c = cos(theta), s = sin(theta);
return Point{c * x - s * y, s * x + c * y};
}
Point rot90(bool ccw) { return (ccw ? Point{-y, x} : Point{y, -x}); }
};
#ifdef FASTIO
template <typename T>
void rd(Point<T>& p) {
fastio::rd(p.x), fastio::rd(p.y);
}
template <typename T>
void wt(Point<T>& p) {
fastio::wt(p.x);
fastio::wt(' ');
fastio::wt(p.y);
}
#endif
// A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1
template <typename T>
int ccw(Point<T> A, Point<T> B, Point<T> C) {
T x = (B - A).det(C - A);
if (x > 0) return 1;
if (x < 0) return -1;
return 0;
}
template <typename REAL, typename T, typename U>
REAL dist(Point<T> A, Point<U> B) {
REAL dx = REAL(A.x) - REAL(B.x);
REAL dy = REAL(A.y) - REAL(B.y);
return sqrt(dx * dx + dy * dy);
}
// ax+by+c
template <typename T>
struct Line {
T a, b, c;
Line(T a, T b, T c) : a(a), b(b), c(c) {}
Line(Point<T> A, Point<T> B) { a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x; }
Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {}
template <typename U>
U eval(Point<U> P) {
return a * P.x + b * P.y + c;
}
template <typename U>
T eval(U x, U y) {
return a * x + b * y + c;
}
// 同じ直線が同じ a,b,c で表現されるようにする
void normalize() {
static_assert(is_same_v<T, int> || is_same_v<T, long long>);
T g = gcd(gcd(abs(a), abs(b)), abs(c));
a /= g, b /= g, c /= g;
if (b < 0) { a = -a, b = -b, c = -c; }
if (b == 0 && a < 0) { a = -a, b = -b, c = -c; }
}
bool is_parallel(Line other) { return a * other.b - b * other.a == 0; }
bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; }
};
template <typename T>
struct Segment {
Point<T> A, B;
Segment(Point<T> A, Point<T> B) : A(A), B(B) {}
Segment(T x1, T y1, T x2, T y2) : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {}
bool contain(Point<T> C) {
T det = (C - A).det(B - A);
if (det != 0) return 0;
return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0;
}
Line<T> to_Line() { return Line(A, B); }
};
template <typename REAL>
struct Circle {
Point<REAL> O;
REAL r;
Circle(Point<REAL> O, REAL r) : O(O), r(r) {}
Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {}
template <typename T>
bool contain(Point<T> p) {
REAL dx = p.x - O.x, dy = p.y - O.y;
return dx * dx + dy * dy <= r * r;
}
};
#line 4 "geo/angle_sort.hpp"
// lower: -1, origin: 0, upper: 1, (-pi,pi]
template <typename T> int lower_or_upper(const Point<T> &p) {
if (p.y != 0)
return (p.y > 0 ? 1 : -1);
if (p.x > 0)
return -1;
if (p.x < 0)
return 1;
return 0;
}
// L<R:-1, L==R:0, L>R:1, (-pi,pi]
template <typename T> int angle_comp_3(const Point<T> &L, const Point<T> &R) {
int a = lower_or_upper(L), b = lower_or_upper(R);
if (a != b)
return (a < b ? -1 : +1);
T det = L.det(R);
if (det > 0)
return -1;
if (det < 0)
return 1;
return 0;
}
// 偏角ソートに対する argsort, (-pi,pi]
template <typename T> vector<int> angle_sort(vector<Point<T>> &P) {
vc<int> I(len(P));
FOR(i, len(P)) I[i] = i;
sort(all(I), [&](auto &L, auto &R) -> bool {
return angle_comp_3(P[L], P[R]) == -1;
});
return I;
}
// 偏角ソートに対する argsort, (-pi,pi]
template <typename T> vector<int> angle_sort(vector<pair<T, T>> &P) {
vc<Point<T>> tmp(len(P));
FOR(i, len(P)) tmp[i] = Point<T>(P[i]);
return angle_sort<T>(tmp);
}
#line 2 "random/base.hpp"
u64 RNG_64() {
static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
x_ ^= x_ << 7;
return x_ ^= x_ >> 9;
}
u64 RNG(u64 lim) { return RNG_64() % lim; }
ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "ds/fenwicktree/fenwicktree_01.hpp"
#line 2 "alg/monoid/add.hpp"
template <typename E>
struct Monoid_Add {
using X = E;
using value_type = X;
static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
static constexpr X inverse(const X &x) noexcept { return -x; }
static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
static constexpr X unit() { return X(0); }
static constexpr bool commute = true;
};
#line 3 "ds/fenwicktree/fenwicktree.hpp"
template <typename Monoid>
struct FenwickTree {
using G = Monoid;
using MX = Monoid;
using E = typename G::value_type;
int n;
vector<E> dat;
E total;
FenwickTree() {}
FenwickTree(int n) { build(n); }
template <typename F>
FenwickTree(int n, F f) {
build(n, f);
}
FenwickTree(const vc<E>& v) { build(v); }
void build(int m) {
n = m;
dat.assign(m, G::unit());
total = G::unit();
}
void build(const vc<E>& v) {
build(len(v), [&](int i) -> E { return v[i]; });
}
template <typename F>
void build(int m, F f) {
n = m;
dat.clear();
dat.reserve(n);
total = G::unit();
FOR(i, n) { dat.eb(f(i)); }
for (int i = 1; i <= n; ++i) {
int j = i + (i & -i);
if (j <= n) dat[j - 1] = G::op(dat[i - 1], dat[j - 1]);
}
total = prefix_sum(m);
}
E prod_all() { return total; }
E sum_all() { return total; }
E sum(int k) { return prefix_sum(k); }
E prod(int k) { return prefix_prod(k); }
E prefix_sum(int k) { return prefix_prod(k); }
E prefix_prod(int k) {
chmin(k, n);
E ret = G::unit();
for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
return ret;
}
E sum(int L, int R) { return prod(L, R); }
E prod(int L, int R) {
chmax(L, 0), chmin(R, n);
if (L == 0) return prefix_prod(R);
assert(0 <= L && L <= R && R <= n);
E pos = G::unit(), neg = G::unit();
while (L < R) { pos = G::op(pos, dat[R - 1]), R -= R & -R; }
while (R < L) { neg = G::op(neg, dat[L - 1]), L -= L & -L; }
return G::op(pos, G::inverse(neg));
}
vc<E> get_all() {
vc<E> res(n);
FOR(i, n) res[i] = prod(i, i + 1);
return res;
}
void add(int k, E x) { multiply(k, x); }
void multiply(int k, E x) {
static_assert(G::commute);
total = G::op(total, x);
for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
}
void set(int k, E x) { add(k, G::op(G::inverse(prod(k, k + 1)), x)); }
template <class F>
int max_right(const F check, int L = 0) {
assert(check(G::unit()));
E s = G::unit();
int i = L;
// 2^k 進むとダメ
int k = [&]() {
while (1) {
if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
if (i == 0) { return topbit(n) + 1; }
int k = lowbit(i) - 1;
if (i + (1 << k) > n) return k;
E t = G::op(s, dat[i + (1 << k) - 1]);
if (!check(t)) { return k; }
s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
}
}();
while (k) {
--k;
if (i + (1 << k) - 1 < len(dat)) {
E t = G::op(s, dat[i + (1 << k) - 1]);
if (check(t)) { i += (1 << k), s = t; }
}
}
return i;
}
// check(i, x)
template <class F>
int max_right_with_index(const F check, int L = 0) {
assert(check(L, G::unit()));
E s = G::unit();
int i = L;
// 2^k 進むとダメ
int k = [&]() {
while (1) {
if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
if (i == 0) { return topbit(n) + 1; }
int k = lowbit(i) - 1;
if (i + (1 << k) > n) return k;
E t = G::op(s, dat[i + (1 << k) - 1]);
if (!check(i + (1 << k), t)) { return k; }
s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
}
}();
while (k) {
--k;
if (i + (1 << k) - 1 < len(dat)) {
E t = G::op(s, dat[i + (1 << k) - 1]);
if (check(i + (1 << k), t)) { i += (1 << k), s = t; }
}
}
return i;
}
template <class F>
int min_left(const F check, int R) {
assert(check(G::unit()));
E s = G::unit();
int i = R;
// false になるところまで戻る
int k = 0;
while (i > 0 && check(s)) {
s = G::op(s, dat[i - 1]);
k = lowbit(i);
i -= i & -i;
}
if (check(s)) {
assert(i == 0);
return 0;
}
// 2^k 進むと ok になる
// false を維持して進む
while (k) {
--k;
E t = G::op(s, G::inverse(dat[i + (1 << k) - 1]));
if (!check(t)) { i += (1 << k), s = t; }
}
return i + 1;
}
int kth(E k, int L = 0) {
return max_right([&k](E x) -> bool { return x <= k; }, L);
}
};
#line 4 "ds/fenwicktree/fenwicktree_01.hpp"
struct FenwickTree_01 {
int N, n;
vc<u64> dat;
FenwickTree<Monoid_Add<int>> bit;
FenwickTree_01() {}
FenwickTree_01(int n) { build(n); }
template <typename F>
FenwickTree_01(int n, F f) {
build(n, f);
}
void build(int m) {
N = m;
n = ceil<int>(N + 1, 64);
dat.assign(n, u64(0));
bit.build(n);
}
template <typename F>
void build(int m, F f) {
N = m;
n = ceil<int>(N + 1, 64);
dat.assign(n, u64(0));
FOR(i, N) { dat[i / 64] |= u64(f(i)) << (i % 64); }
bit.build(n, [&](int i) -> int { return popcnt(dat[i]); });
}
int sum_all() { return bit.sum_all(); }
int sum(int k) { return prefix_sum(k); }
int prefix_sum(int k) {
int ans = bit.sum(k / 64);
ans += popcnt(dat[k / 64] & ((u64(1) << (k % 64)) - 1));
return ans;
}
int sum(int L, int R) {
if (L == 0) return prefix_sum(R);
int ans = 0;
ans -= popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
ans += popcnt(dat[R / 64] & ((u64(1) << (R % 64)) - 1));
ans += bit.sum(L / 64, R / 64);
return ans;
}
void add(int k, int x) {
if (x == 1) add(k);
elif (x == -1) remove(k);
else assert(0);
}
void add(int k) {
dat[k / 64] |= u64(1) << (k % 64);
bit.add(k / 64, 1);
}
void remove(int k) {
dat[k / 64] &= ~(u64(1) << (k % 64));
bit.add(k / 64, -1);
}
int kth(int k, int L = 0) {
if (k >= sum_all()) return N;
k += popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
L /= 64;
int mid = 0;
auto check = [&](auto e) -> bool {
if (e <= k) chmax(mid, e);
return e <= k;
};
int idx = bit.max_right(check, L);
if (idx == n) return N;
k -= mid;
u64 x = dat[idx];
int p = popcnt(x);
if (p <= k) return N;
k = binary_search([&](int n) -> bool { return (p - popcnt(x >> n)) <= k; }, 0, 64, 0);
return 64 * idx + k;
}
int next(int k) {
int idx = k / 64;
k %= 64;
u64 x = dat[idx] & ~((u64(1) << k) - 1);
if (x) return 64 * idx + lowbit(x);
idx = bit.kth(0, idx + 1);
if (idx == n || !dat[idx]) return N;
return 64 * idx + lowbit(dat[idx]);
}
int prev(int k) {
if (k == N) --k;
int idx = k / 64;
k %= 64;
u64 x = dat[idx];
if (k < 63) x &= (u64(1) << (k + 1)) - 1;
if (x) return 64 * idx + topbit(x);
idx = bit.min_left([&](auto e) -> bool { return e <= 0; }, idx) - 1;
if (idx == -1) return -1;
return 64 * idx + topbit(dat[idx]);
}
};
#line 6 "geo/count_points_in_triangles.hpp"
// 点群 A, B を入力 (Point<ll>)
// query(i,j,k):三角形 AiAjAk 内部の Bl の個数(非負)を返す
// 前計算 O(NMlogM)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
// https://codeforces.com/contest/852/problem/H
struct Count_Points_In_Triangles {
using P = Point<ll>;
const int LIM = 1'000'000'000 + 10;
vc<P> A, B;
vc<int> new_idx; // O から見た偏角ソート順を管理
vc<int> point; // A[i] と一致する B[j] の数え上げ
vvc<int> seg; // 線分 A[i]A[j] 内にある B[k] の数え上げ
vvc<int> tri; // OA[i]A[j] 内部にある B[k] の数え上げ
Count_Points_In_Triangles(const vc<P>& A, const vc<P>& B) : A(A), B(B) {
for (auto&& p: A) assert(max(abs(p.x), abs(p.y)) < LIM);
for (auto&& p: B) assert(max(abs(p.x), abs(p.y)) < LIM);
build();
}
int count3(int i, int j, int k) {
i = new_idx[i], j = new_idx[j], k = new_idx[k];
if (i > j) swap(i, j);
if (j > k) swap(j, k);
if (i > j) swap(i, j);
assert(i <= j && j <= k);
ll d = (A[j] - A[i]).det(A[k] - A[i]);
if (d == 0) return 0;
if (d > 0) { return tri[i][j] + tri[j][k] - tri[i][k] - seg[i][k]; }
int x = tri[i][k] - tri[i][j] - tri[j][k];
return x - seg[i][j] - seg[j][k] - point[j];
}
// segment
int count2(int i, int j) {
i = new_idx[i], j = new_idx[j];
if (i > j) swap(i, j);
return seg[i][j];
}
private:
P take_origin() {
// OAiAj, OAiBj が同一直線上にならないようにする
// fail prob: at most N(N+M)/LIM
return P{-LIM, RNG(-LIM, LIM)};
}
void build() {
P O = take_origin();
for (auto&& p: A) p = p - O;
for (auto&& p: B) p = p - O;
int N = len(A), M = len(B);
vc<int> I = angle_sort(A);
A = rearrange(A, I);
new_idx.resize(N);
FOR(i, N) new_idx[I[i]] = i;
I = angle_sort(B);
B = rearrange(B, I);
point.assign(N, 0);
seg.assign(N, vc<int>(N));
tri.assign(N, vc<int>(N));
// point
FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];
int m = 0;
FOR(j, N) {
// OA[i]A[j], B[k]
while (m < M && A[j].det(B[m]) < 0) ++m;
vc<P> C(m);
FOR(k, m) C[k] = B[k] - A[j];
vc<int> I(m);
FOR(i, m) I[i] = i;
sort(all(I), [&](auto& a, auto& b) -> bool { return C[a].det(C[b]) > 0; });
C = rearrange(C, I);
vc<int> rk(m);
FOR(k, m) rk[I[k]] = k;
FenwickTree_01 bit(m);
int k = m;
FOR_R(i, j) {
while (k > 0 && A[i].det(B[k - 1]) > 0) { bit.add(rk[--k], 1); }
P p = A[i] - A[j];
int lb = binary_search([&](int n) -> bool { return (n == 0 ? true : C[n - 1].det(p) > 0); }, 0, m + 1);
int ub = binary_search([&](int n) -> bool { return (n == 0 ? true : C[n - 1].det(p) >= 0); }, 0, m + 1);
seg[i][j] += bit.sum(lb, ub), tri[i][j] += bit.sum(lb);
}
}
}
};