#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else #pragma GCC optimize("Ofast") #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using pi = pair; using vi = vector; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vec(type, name, ...) vector name(__VA_ARGS__) #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define FOR4_R(i, a, b, c) for (ll i = (b)-1; i >= ll(a); i -= (c)) #define overload4(a, b, c, d, e, ...) e #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) \ overload4(__VA_ARGS__, FOR4_R, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) for (ll t = s; t >= 0; t = (t == 0 ? -1 : (t - 1) & s)) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll template T SUM(const vector &A) { T sum = 0; for (auto &&a: A) sum += a; return sum; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T pick(deque &que) { T a = que.front(); que.pop_front(); return a; } template T pick(pq &que) { T a = que.top(); que.pop(); return a; } template T pick(pqg &que) { assert(que.size()); T a = que.top(); que.pop(); return a; } template T pick(vc &que) { assert(que.size()); T a = que.back(); que.pop_back(); return a; } template T ceil(T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } template pair divmod(T x, U y) { T q = floor(x, y); return {q, x - q * y}; } template ll binary_search(F check, ll ok, ll ng) { assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x)); } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x)); } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = S[i] - first_char; } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } template vc bincount(const vc &A, int size) { vc C(size); for (auto &&x: A) { ++C[x]; } return C; } // stable template vector argsort(const vector &A) { vector ids(A.size()); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return A[i] < A[j] || (A[i] == A[j] && i < j); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { int n = len(I); vc B(n); FOR(i, n) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io.hpp" // based on yosupo's fastio #include namespace fastio { // クラスが read(), print() を持っているかを判定するメタ関数 struct has_write_impl { template static auto check(T &&x) -> decltype(x.write(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_write : public decltype(has_write_impl::check(std::declval())) { }; struct has_read_impl { template static auto check(T &&x) -> decltype(x.read(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_read : public decltype(has_read_impl::check(std::declval())) {}; struct Scanner { FILE *fp; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) { bool sep = false; for (size_t i = st; i < ed; i++) { if (isspace(line[i])) { sep = true; break; } } if (!sep) reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); } if (neg) ref = -ref; return true; } template ::value>::type * = nullptr> inline bool read_single(T &x) { x.read(); return true; } bool read_single(double &ref) { string s; if (!read_single(s)) return false; ref = std::stod(s); return true; } bool read_single(char &ref) { string s; if (!read_single(s) || s.size() != 1) return false; ref = s[0]; return true; } template bool read_single(vector &ref) { for (auto &d: ref) { if (!read_single(d)) return false; } return true; } template bool read_single(pair &p) { return (read_single(p.first) && read_single(p.second)); } template void read_single_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); read_single(x); read_single_tuple(t); } } template bool read_single(tuple &tpl) { read_single_tuple(tpl); return true; } void read() {} template void read(H &h, T &... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE *fp) : fp(fp) {} }; struct Printer { Printer(FILE *_fp) : fp(_fp) {} ~Printer() { flush(); } static constexpr size_t SIZE = 1 << 15; FILE *fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write(const char val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write('0'); return; } if (val < 0) { write('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char(0x30 | (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write(const string s) { for (char c: s) write(c); } void write(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write(s[i]); } void write(const double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } void write(const long double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } template ::value>::type * = nullptr> inline void write(T x) { x.write(); } template void write(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } template void write(const pair val) { write(val.first); write(' '); write(val.second); } template void write_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { write(' '); } const auto x = std::get(t); write(x); write_tuple(t); } } template bool write(tuple tpl) { write_tuple(tpl); return true; } template void write(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } void write(i128 val) { string s; bool negative = 0; if (val < 0) { negative = 1; val = -val; } while (val) { s += '0' + int(val % 10); val /= 10; } if (negative) s += "-"; reverse(all(s)); if (len(s) == 0) s = "0"; write(s); } }; Scanner scanner = Scanner(stdin); Printer printer = Printer(stdout); void flush() { printer.flush(); } void print() { printer.write('\n'); } template void print(Head &&head, Tail &&... tail) { printer.write(head); if (sizeof...(Tail)) printer.write(' '); print(forward(tail)...); } void read() {} template void read(Head &head, Tail &... tail) { scanner.read(head); read(tail...); } } // namespace fastio using fastio::print; using fastio::flush; using fastio::read; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 3 "main.cpp" #line 2 "/home/maspy/compro/library/geo/base.hpp" template struct Point { T x, y; Point() = default; template Point(A x, B y) : x(x), y(y) {} template Point(pair p) : x(p.fi), y(p.se) {} 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; } Point operator-() const { return {-x, -y}; } bool operator<(Point p) const { if (x != p.x) return x < p.x; return y < p.y; } T dot(Point other) { return x * other.x + y * other.y; } T det(Point other) { return x * other.y - y * other.x; } void read() { fastio::read(x), fastio::read(y); } void write() { fastio::printer.write(pair({x, y})); } }; template REAL dist(Point A, Point B) { A = A - B; T p = A.dot(A); return sqrt(REAL(p)); } template struct Line { T a, b, c; Line(T a, T b, T c) : a(a), b(b), c(c) {} Line(Point A, Point 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(x1, y1), Point(x2, y2)) {} template U eval(Point P) { return a * P.x + b * P.y + c; } template T eval(U x, U y) { return a * x + b * y + 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 struct Segment { Point A, B; Segment(Point A, Point B) : A(A), B(B) {} Segment(T x1, T y1, T x2, T y2) : Segment(Point(x1, y1), Point(x2, y2)) {} Line to_Line() { return Line(A, B); } }; template struct Circle { Point O; T r; Circle(Point O, T r) : O(O), r(r) {} Circle(T x, T y, T r) : O(Point(x, y)), r(r) {} }; template struct Polygon { vc> points; T a; template Polygon(vc> pairs) { for (auto&& [a, b]: pairs) points.eb(Point(a, b)); build(); } Polygon(vc> points) : points(points) { build(); } int size() { return len(points); } template REAL area() { return a * 0.5; } template ::value, int> = 0> T area_2() { return a; } bool is_convex() { FOR(j, len(points)) { int i = (j == 0 ? len(points) - 1 : j - 1); int k = (j == len(points) - 1 ? 0 : j + 1); if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false; } return true; } private: void build() { a = 0; FOR(i, len(points)) { int j = (i + 1 == len(points) ? 0 : i + 1); a += points[i].det(points[j]); } if (a < 0) { a = -a; reverse(all(points)); } } }; #line 2 "/home/maspy/compro/library/geo/cross_point.hpp" // 平行でないことを仮定 template Point cross_point(const Line L1, const Line L2) { T det = L1.a * L2.b - L1.b * L2.a; assert(det != 0); REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c; REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a; return Point(x / det, y / det); } // 0: 交点なし // 1: 一意な交点 // 2:2 つ以上の交点(整数型を利用して厳密にやる) template ::value, int> = 0> int count_cross(Segment S1, Segment S2, bool include_ends) { Line L1 = S1.to_Line(); Line L2 = S2.to_Line(); if (L1.is_parallel(L2)) { if (L1.eval(S2.A) != 0) return 0; // 4 点とも同一直線上にある T a1 = S1.A.x, b1 = S1.B.x; T a2 = S2.A.x, b2 = S2.B.x; if (a1 == b1) { a1 = S1.A.y, b1 = S1.B.y; a2 = S2.A.y, b2 = S2.B.y; } if (a1 > b1) swap(a1, b1); if (a2 > b2) swap(a2, b2); T a = max(a1, a2); T b = min(b1, b2); if (a < b) return 2; if (a > b) return 0; return (include_ends ? 1 : 0); } // 平行でない場合 T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B); T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B); if (a1 > b1) swap(a1, b1); if (a2 > b2) swap(a2, b2); bool ok1 = 0, ok2 = 0; if (include_ends) { ok1 = (a1 <= 0) && (0 <= b1); ok2 = (a2 <= 0) && (0 <= b2); } else { ok1 = (a1 < 0) && (0 < b1); ok2 = (a2 < 0) && (0 < b2); } return (ok1 && ok2 ? 1 : 0); } // 0 または 1 // real だと誤差によっては間違った答を返す可能性あり。 template int count_cross(Segment S1, Segment S2) { Line L1 = S1.to_Line(); Line L2 = S2.to_Line(); T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B); T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B); if (a1 > b1) swap(a1, b1); if (a2 > b2) swap(a2, b2); bool ok1 = 0, ok2 = 0; ok1 = (a1 <= 0) && (0 <= b1); ok2 = (a2 <= 0) && (0 <= b2); return (ok1 && ok2 ? 1 : 0); } // 唯一の交点を持つことを仮定 template Point cross_point(Segment S1, Segment S2) { Line L1 = S1.to_Line(); Line L2 = S2.to_Line(); return cross_point(L1, L2); } #line 6 "main.cpp" template vector ConvexHull(vector>& XY, string mode = "full", bool inclusive = false, bool sorted = false) { assert(mode == "full" || mode == "lower" || mode == "upper"); ll N = XY.size(); if (N == 1) return {0}; if (N == 2) return {0, 1}; vc I = argsort(XY); auto check = [&](ll i, ll j, ll k) -> bool { auto xi = XY[i].fi, yi = XY[i].se; auto xj = XY[j].fi, yj = XY[j].se; auto xk = XY[k].fi, yk = XY[k].se; auto dx1 = xj - xi, dy1 = yj - yi; auto dx2 = xk - xj, dy2 = yk - yj; ll det = dx1 * dy2 - dy1 * dx2; return (inclusive ? det >= 0 : det > 0); }; auto calc = [&]() { vector P; for (auto&& k: I) { while (P.size() > 1) { auto i = P[P.size() - 2]; auto j = P[P.size() - 1]; if (check(i, j, k)) break; P.pop_back(); } P.eb(k); } return P; }; vc P; if (mode == "full" || mode == "lower") { vc Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "full" || mode == "upper") { if (!P.empty()) P.pop_back(); reverse(all(I)); vc Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "upper") reverse(all(P)); if (len(P) >= 2 && P[0] == P.back()) P.pop_back(); return P; } template vc ConvexHull(vector>& XY, string mode = "full", bool inclusive = false, bool sorted = false) { assert(mode == "full" || mode == "lower" || mode == "upper"); ll N = XY.size(); if (N == 1) return {0}; if (N == 2) return {0, 1}; vc I = argsort(XY); auto check = [&](ll i, ll j, ll k) -> bool { auto xi = XY[i].x, yi = XY[i].y; auto xj = XY[j].x, yj = XY[j].y; auto xk = XY[k].x, yk = XY[k].y; auto dx1 = xj - xi, dy1 = yj - yi; auto dx2 = xk - xj, dy2 = yk - yj; ll det = dx1 * dy2 - dy1 * dx2; return (inclusive ? det >= 0 : det > 0); }; auto calc = [&]() { vc P; for (auto&& k: I) { while (P.size() > 1) { auto i = P[P.size() - 2]; auto j = P[P.size() - 1]; if (check(i, j, k)) break; P.pop_back(); } P.eb(k); } return P; }; vc P; if (mode == "full" || mode == "lower") { vc Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "full" || mode == "upper") { if (!P.empty()) P.pop_back(); reverse(all(I)); vc Q = calc(); P.insert(P.end(), all(Q)); } if (mode == "upper") reverse(all(P)); if (len(P) >= 2 && P[0] == P.back()) P.pop_back(); return P; } using P = Point; void solve() { LL(N, K); vc

A, B; FOR(N) { LL(a, b, c); if (c == 1) A.eb(a, b); else B.eb(a, b); } if (A.empty() || B.empty()) return No(); if (K == 3) { FOR(2) { swap(A, B); for (auto&& a: A) { set ss; for (auto&& p: B) { if (p == a) continue; auto [dx, dy] = mp(p.x - a.x, p.y - a.y); ll g = gcd(dx, dy); dx /= g, dy /= g; ss.insert(mp(dx, dy)); } for (auto&& p: B) { if (p == a) continue; auto [dx, dy] = mp(-p.x + a.x, -p.y + a.y); ll g = gcd(dx, dy); dx /= g, dy /= g; pi k = {dx, dy}; if (ss.count(k)) return Yes(); } } } No(); return; } auto IA = ConvexHull(A, "full", false); auto IB = ConvexHull(B, "full", false); A = rearrange(A, IA); B = rearrange(B, IB); FOR(2) { swap(A, B); for (auto&& p: A) { // p が B に含まれるならば、ある三角形に含まれるので、よい 4 点が見つかる bool ok = 1; FOR(i, len(B)) { int j = (i + 1 < len(B) ? i + 1 : 0); auto b1 = B[i], b2 = B[j]; if ((b2 - b1).det(p - b1) < 0) ok = 0; } if (ok) return Yes(); } FOR(i, len(A)) { P a1 = A[i], a2 = (i + 1 == len(A) ? A[0] : A[i + 1]); if (a1 == a2) continue; FOR(j, len(B)) { P b1 = B[j], b2 = (j + 1 == len(B) ? B[0] : B[j + 1]); if (b1 == b2) continue; Segment SA(a1, a2); Segment SB(b1, b2); if (count_cross(SA, SB)) return Yes(); } } } No(); } signed main() { int T = 1; // INT(T); FOR(T) solve(); return 0; }