#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template IStream &operator>>(IStream &is, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif #include #include #include #include #include #include #include #include template struct Point2d { static T_P EPS; static void set_eps(T_P e) { EPS = e; } T_P x, y; Point2d() : x(0), y(0) {} Point2d(T_P x, T_P y) : x(x), y(y) {} Point2d(const std::pair &p) : x(p.first), y(p.second) {} Point2d(const std::complex &p) : x(p.real()), y(p.imag()) {} std::complex to_complex() const noexcept { return {x, y}; } Point2d operator+(const Point2d &p) const noexcept { return Point2d(x + p.x, y + p.y); } Point2d operator-(const Point2d &p) const noexcept { return Point2d(x - p.x, y - p.y); } Point2d operator*(const Point2d &p) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x * p.x - y * p.y, x * p.y + y * p.x); } Point2d operator*(T_P d) const noexcept { return Point2d(x * d, y * d); } Point2d operator/(T_P d) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x / d, y / d); } Point2d inv() const { static_assert(std::is_floating_point::value == true); return conj() / norm2(); } Point2d operator/(const Point2d &p) const { return (*this) * p.inv(); } bool operator<(const Point2d &r) const noexcept { return x != r.x ? x < r.x : y < r.y; } bool operator==(const Point2d &r) const noexcept { return x == r.x and y == r.y; } bool operator!=(const Point2d &r) const noexcept { return !((*this) == r); } T_P dot(Point2d p) const noexcept { return x * p.x + y * p.y; } T_P det(Point2d p) const noexcept { return x * p.y - y * p.x; } T_P absdet(Point2d p) const noexcept { return std::abs(det(p)); } T_P norm() const noexcept { static_assert(std::is_floating_point::value == true); return std::sqrt(x * x + y * y); } T_P norm2() const noexcept { return x * x + y * y; } T_P arg() const noexcept { return std::atan2(y, x); } // rotate point/vector by rad Point2d rotate(T_P rad) const noexcept { static_assert(std::is_floating_point::value == true); return Point2d(x * std::cos(rad) - y * std::sin(rad), x * std::sin(rad) + y * std::cos(rad)); } Point2d normalized() const { static_assert(std::is_floating_point::value == true); return (*this) / this->norm(); } Point2d conj() const noexcept { return Point2d(x, -y); } template friend IStream &operator>>(IStream &is, Point2d &p) { T_P x, y; is >> x >> y; p = Point2d(x, y); return is; } template friend OStream &operator<<(OStream &os, const Point2d &p) { return os << '(' << p.x << ',' << p.y << ')'; } }; template <> double Point2d::EPS = 1e-9; template <> long double Point2d::EPS = 1e-12; template <> long long Point2d::EPS = 0; template int ccw(const Point2d &a, const Point2d &b, const Point2d &c) { // a->b->cの曲がり方 Point2d v1 = b - a; Point2d v2 = c - a; if (v1.det(v2) > Point2d::EPS) return 1; // 左折 if (v1.det(v2) < -Point2d::EPS) return -1; // 右折 if (v1.dot(v2) < -Point2d::EPS) return 2; // c-a-b if (v1.norm2() < v2.norm2()) return -2; // a-b-c return 0; // a-c-b } // Convex hull (凸包) // return: IDs of vertices used for convex hull, counterclockwise // include_boundary: If true, interior angle pi is allowed template std::vector convex_hull(const std::vector> &ps, bool include_boundary = false) { int n = ps.size(); if (n <= 1) return std::vector(n, 0); std::vector, int>> points(n); for (size_t i = 0; i < ps.size(); i++) points[i] = std::make_pair(ps[i], i); std::sort(points.begin(), points.end()); int k = 0; std::vector, int>> qs(2 * n); auto ccw_check = [&](int c) { return include_boundary ? (c == -1) : (c <= 0); }; for (int i = 0; i < n; i++) { while (k > 1 and ccw_check(ccw(qs[k - 2].first, qs[k - 1].first, points[i].first))) k--; qs[k++] = points[i]; } for (int i = n - 2, t = k; i >= 0; i--) { while (k > t and ccw_check(ccw(qs[k - 2].first, qs[k - 1].first, points[i].first))) k--; qs[k++] = points[i]; } std::vector ret(k - 1); for (int i = 0; i < k - 1; i++) ret[i] = qs[i].second; return ret; } // Solve r1 + t1 * v1 == r2 + t2 * v2 template ::value>::type * = nullptr> Point2d lines_crosspoint(Point2d r1, Point2d v1, Point2d r2, Point2d v2) { static_assert(std::is_floating_point::value == true); assert(v2.det(v1) != 0); return r1 + v1 * (v2.det(r2 - r1) / v2.det(v1)); } // Whether two segments s1t1 & s2t2 intersect or not (endpoints not included) // Google Code Jam 2013 Round 3 - Rural Planning // Google Code Jam 2021 Round 3 - Fence Design template bool intersect_open_segments(Point2d s1, Point2d t1, Point2d s2, Point2d t2) { if (s1 == t1 or s2 == t2) return false; // Not segment but point int nbad = 0; for (int t = 0; t < 2; t++) { Point2d v1 = t1 - s1, v2 = t2 - s2; T den = v2.det(v1); if (den == 0) { if (s1.det(v1) == s2.det(v1)) { auto L1 = s1.dot(v1), R1 = t1.dot(v1); auto L2 = s2.dot(v1), R2 = t2.dot(v1); if (L1 > R1) std::swap(L1, R1); if (L2 > R2) std::swap(L2, R2); if (L1 > L2) std::swap(L1, L2), std::swap(R1, R2); return R1 > L2; } else { return false; } } else { auto num = v2.det(s2 - s1); if ((0 < num and num < den) or (den < num and num < 0)) nbad++; } std::swap(s1, s2); std::swap(t1, t2); } return nbad == 2; } // Whether point p is on segment (s, t) (endpoints not included) // Google Code Jam 2013 Round 3 - Rural Planning template bool is_point_on_open_segment(PointNd s, PointNd t, PointNd p) { if (s == t) return false; // not segment but point if (p == s or p == t) return false; auto v = t - s, w = p - s; if (v.absdet(w)) return false; auto vv = v.dot(v), vw = v.dot(w); return vw > 0 and vw < vv; } // Convex cut // Cut the convex polygon g by line p1->p2 and return the leftward one template std::vector> convex_cut(const std::vector> &g, Point2d p1, Point2d p2) { static_assert(std::is_floating_point::value == true); assert(p1 != p2); std::vector> ret; for (int i = 0; i < (int)g.size(); i++) { const Point2d &now = g[i], &nxt = g[(i + 1) % g.size()]; if (ccw(p1, p2, now) != -1) ret.push_back(now); if ((ccw(p1, p2, now) == -1) xor (ccw(p1, p2, nxt) == -1)) { ret.push_back(lines_crosspoint(now, nxt - now, p1, p2 - p1)); } } return ret; } // 2円の交点 (ABC157F, SRM 559 Div.1 900) template std::vector> IntersectTwoCircles(const Point2d &Ca, T_P Ra, const Point2d &Cb, T_P Rb) { static_assert(std::is_floating_point::value == true); T_P d = (Ca - Cb).norm(); if (Ra + Rb < d) return {}; T_P rc = (d * d + Ra * Ra - Rb * Rb) / (2 * d); T_P rs2 = Ra * Ra - rc * rc; if (rs2 < 0) return {}; T_P rs = std::sqrt(rs2); Point2d diff = (Cb - Ca) / d; return {Ca + diff * Point2d(rc, rs), Ca + diff * Point2d(rc, -rs)}; } // Solve |x0 + vt| = R (SRM 543 Div.1 1000, GCJ 2016 R3 C) template std::vector IntersectCircleLine(const PointNd &x0, const PointNd &v, Float R) { static_assert(std::is_floating_point::value == true); Float b = Float(x0.dot(v)) / v.norm2(); Float c = Float(x0.norm2() - Float(R) * R) / v.norm2(); if (b * b - c < 0) return {}; Float ret1 = -b + sqrtl(b * b - c) * (b > 0 ? -1 : 1); Float ret2 = c / ret1; return ret1 < ret2 ? std::vector{ret1, ret2} : std::vector{ret2, ret1}; } // Distance between point p <-> line ab template decltype(PointFloat::x) DistancePointLine(const PointFloat &p, const PointFloat &a, const PointFloat &b) { assert(a != b); return (b - a).absdet(p - a) / (b - a).norm(); } // Distance between point p <-> line segment ab template decltype(PointFloat::x) DistancePointSegment(const PointFloat &p, const PointFloat &a, const PointFloat &b) { if (a == b) { return (p - a).norm(); } else if ((p - a).dot(b - a) <= 0) { return (p - a).norm(); } else if ((p - b).dot(a - b) <= 0) { return (p - b).norm(); } else { return DistancePointLine(p, a, b); } } // Area of polygon (might be negative) template T_P signed_area_of_polygon(const std::vector> &poly) { static_assert(std::is_floating_point::value == true); T_P area = 0; for (size_t i = 0; i < poly.size(); i++) area += poly[i].det(poly[(i + 1) % poly.size()]); return area * 0.5; } using Pt = Point2d; void Yes() { puts("Yes"); exit(0); } // Point on grid, sortable by its argument struct Point { constexpr static double eps = 1e-2; long long X, Y; double theta; Point() = default; Point(long long x, long long y) : X(x), Y(y), theta(std::atan2(y, x)) {} bool operator<(const Point &r) const { double b = theta - r.theta; return std::abs(b) > eps ? (b < 0) : (X * r.Y > r.X * Y); } bool operator==(const Point &r) const { return std::abs(theta - r.theta) < eps and X * r.Y == r.X * Y; } bool operator!=(const Point &r) const { return !(*this == r); } void rotate_pi() { theta += M_PI; X *= -1; Y *= -1; } }; // Key class must be comparable // Example: [(2, 3), (1, 5), (2, 2), (2, 3), (1, 4)] -> [(1, [5, 4]), (2, [3, 2, 3])] // Verified: https://atcoder.jp/contests/abc259/tasks/abc259_h template std::vector>> groupby(std::vector> xs) { using KV = std::pair; std::stable_sort( xs.begin(), xs.end(), [&](const KV &l, const KV &r) { return l.first < r.first; }); std::vector>> ret; for (const auto &x : xs) { if (ret.empty() or ret.back().first != x.first) { ret.emplace_back(x.first, std::vector()); } ret.back().second.push_back(x.second); } return ret; } int main() { int N, K; cin >> N >> K; vector> vs(2); REP(i, N) { int x, y, c; cin >> x >> y >> c; vs.at(c - 1).emplace_back(x, y); } dbg(make_tuple(vs.front().size(), vs.back().size())); if (vs.front().size() > vs.back().size()) swap(vs.front(), vs.back()); for (const Pt &p : vs.front()) { vector>> as; for (const Pt &q : vs.front()) { if (p == q) continue; auto dx = q - p; as.emplace_back(Point(dx.x, dx.y), make_pair(0, dx.norm2())); } for (const Pt &q : vs.back()) { auto dx = q - p; as.emplace_back(Point(dx.x, dx.y), make_pair(1, dx.norm2())); as.emplace_back(Point(-dx.x, -dx.y), make_pair(2, dx.norm2())); } for (const auto &[key, vals] : groupby(as)) { lint min1 = ((1LL << 62) - 1) * 2 , max0 = -1; int fall = 0; for (auto [flg, norm2] : vals) { fall |= flg; if (flg == 1) { chmin(min1, norm2); } else if (flg == 0) { chmax(max0, norm2); } } if (min1 < max0 or fall == 3) Yes(); } } for (auto &v : vs) { vector w; for (int i : convex_hull(v)) { w.push_back(v.at(i)); } v = w; } dbg(make_tuple(vs.front().size(), vs.back().size())); if (K > 3 and vs.front().size() and vs.back().size()) { REP(_, 2) { for (const auto &p : vs.back()) { auto flast = vs.front().back(); bool good = true; for (const auto &f : vs.front()) { int c = ccw(p, flast, f); if (c != 1) good = false; flast = f; } if (good) Yes(); break; } Pt flast = vs.front().back(); for (const auto &f : vs.front()) { Pt blast = vs.back().back(); for (const auto &b : vs.back()) { if (intersect_open_segments(flast, f, blast, b)) Yes(); blast = b; } flast = f; } swap(vs.front(), vs.back()); } } puts("No"); }