// >>> TEMPLATES #include using namespace std; using ll = long long; using ld = long double; using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; #define int ll #define rep(i, n) for (int i = 0; i < (int)(n); i++) #define rep1(i, n) for (int i = 1; i <= (int)(n); i++) #define repR(i, n) for (int i = (int)(n)-1; i >= 0; i--) #define rep1R(i, n) for (int i = (int)(n); i >= 1; i--) #define loop(i, a, B) for (int i = a; i B; i++) #define loopR(i, a, B) for (int i = a; i B; i--) #define all(x) begin(x), end(x) #define allR(x) rbegin(x), rend(x) #define rng(x, l, r) begin(x) + (l), begin(x) + (r) #define pb push_back #define eb emplace_back #define fst first #define snd second template constexpr auto mp(A &&a, B &&b) { return make_pair(forward(a), forward(b)); } template constexpr auto mt(T&&... x) { return make_tuple(forward(x)...); } template auto constexpr inf_ = numeric_limits::max()/2-1; auto constexpr INF32 = inf_; auto constexpr INF64 = inf_; auto constexpr INF = inf_; #ifdef LOCAL #include "debug.hpp" #else #define dump(...) (void)(0) #define say(x) (void)(0) #define debug if (0) #endif template struct pque : priority_queue, Comp> { vector &data() { return this->c; } void clear() { this->c.clear(); } }; template using pque_max = pque>; template using pque_min = pque>; template ::value, int> = 0> ostream& operator<<(ostream& os, T const& a) { bool f = true; for (auto const& x : a) os << (f ? "" : " ") << x, f = false; return os; } template ::value, int> = 0> ostream& operator<<(ostream& os, const T (&a)[N]) { bool f = true; for (auto const& x : a) os << (f ? "" : " ") << x, f = false; return os; } template ())), class = typename enable_if::value>::type> istream& operator>>(istream& is, T &a) { for (auto& x : a) is >> x; return is; } template ostream& operator<<(ostream& os, pair const& p) { return os << p.first << " " << p.second; } template istream& operator>>(istream& is, pair& p) { return is >> p.first >> p.second; } struct IOSetup { IOSetup() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); } } iosetup; template struct FixPoint : private F { constexpr FixPoint(F&& f) : F(forward(f)) {} template constexpr auto operator()(T&&... x) const { return F::operator()(*this, forward(x)...); } }; struct MakeFixPoint { template constexpr auto operator|(F&& f) const { return FixPoint(forward(f)); } }; #define MFP MakeFixPoint()| #define def(name, ...) auto name = MFP [&](auto &&name, __VA_ARGS__) template struct vec_impl { using type = vector::type>; template static type make_v(size_t n, U&&... x) { return type(n, vec_impl::make_v(forward(x)...)); } }; template struct vec_impl { using type = T; static type make_v(T const& x = {}) { return x; } }; template using vec = typename vec_impl::type; template auto make_v(Args&&... args) { return vec_impl::make_v(forward(args)...); } template void quit(T const& x) { cout << x << endl; exit(0); } template constexpr bool chmin(T& x, U const& y) { if (x > (T)y) { x = (T)y; return true; } return false; } template constexpr bool chmax(T& x, U const& y) { if (x < (T)y) { x = (T)y; return true; } return false; } template constexpr auto sumof(It b, It e) { return accumulate(b, e, typename iterator_traits::value_type{}); } template int sz(T const& x) { return x.size(); } template int lbd(C const& v, T const& x) { return lower_bound(begin(v), end(v), x)-begin(v); } template int ubd(C const& v, T const& x) { return upper_bound(begin(v), end(v), x)-begin(v); } constexpr int64_t mod(int64_t x, int64_t m) { assert(m > 0); return (x %= m) < 0 ? x+m : x; } constexpr int64_t div_floor(int64_t x, int64_t y) { assert(y != 0); return x/y - ((x^y) < 0 and x%y); } constexpr int64_t div_ceil(int64_t x, int64_t y) { assert(y != 0); return x/y + ((x^y) > 0 and x%y); } constexpr int dx[] = { 1, 0, -1, 0, 1, -1, -1, 1 }; constexpr int dy[] = { 0, 1, 0, -1, 1, 1, -1, -1 }; constexpr int popcnt(ll x) { return __builtin_popcountll(x); } mt19937_64 seed_{random_device{}()}; template Int rand(Int a, Int b) { return uniform_int_distribution(a, b)(seed_); } i64 irand(i64 a, i64 b) { return rand(a, b); } // [a, b] u64 urand(u64 a, u64 b) { return rand(a, b); } // template void shuffle(It l, It r) { shuffle(l, r, seed_); } template vector &operator--(vector &v) { for (T &x : v) --x; return v; } template vector &operator++(vector &v) { for (T &x : v) ++x; return v; } // <<< // >>> LCA (RMQ) #ifndef EDGE_INFO #define EDGE_INFO constexpr int dest(int v) { return v; } template ().to)> constexpr int dest(E const& e) { return e.to; } constexpr int cost(int) { return 1; } template ().cost)> constexpr auto cost(E const& e) { return e.cost; } template using cost_t = decltype(cost(declval())); #endif namespace RMQ { // >>> DST template struct DST : Handler { using Value = typename Handler::Value; using Handler::unit; // () -> Value using Handler::merge; // (Value, Value) -> Value static constexpr int bsr(int32_t x) { return x ? 31-__builtin_clz(x) : -1; } vector> v; DST() {} template DST(T&&... x) { build(forward(x)...); } void build(int n) { build(n, unit()); } void build(int n, Value const& x) { build(n, [&](int) { return x; }); } void build(vector const& v) { build(v.size(), [&](int i) { return v[i]; }); } template ()(0))> void build(int n, F gen) { assert(n >= 0); v.resize(1); v[0].resize(n, unit()); for (int i = 0; i < n; i++) v[0][i] = gen(i); if (n <= 1) return; const int lg = __lg(2*n-1); v.resize(lg, vector(n, unit())); for (int h = 1; h < lg; ++h) { const int w = 1<= b-w; --j) { v[h][j] = x = merge(v[0][j], x); } x = unit(); for (int j = b; j < min(n, b+w); ++j) { v[h][j] = x = merge(x, v[0][j]); } } } } int size() const { return v.empty() ? 0 : v[0].size(); } Value get(int l, int r) const { assert(0 <= l); assert(l <= r); assert(r <= size()); if (l == r) return unit(); --r; if (l == r) return v[0][l]; const int h = bsr(l^r); return merge(v[h][l], v[h][r]); } Value operator[](int idx) const { assert(0 <= idx); assert(idx < size()); return v[0][idx]; } }; // <<< struct Min { using Value = pair; static constexpr auto inf = numeric_limits::max(); constexpr static Value unit() { return { inf, inf }; } constexpr static Value merge(Value const& x, Value const& y) { return min(x, y); } }; using RMQ = DST; } template struct LCA { vector> const* g; int32_t n, root; vector id; vector> d; RMQ rmq; LCA() {} LCA(vector> const* g, int root) : g(g), n(g->size()), root(root), id(n), d(n) { vector> a; a.reserve(n-1); auto dfs = [&](auto dfs, int x, int p, int d0, cost_t d1) -> void { id[x] = a.size(); d[x] = d1; for (auto const& e : (*g)[x]) { if (dest(e) == p) continue; a.emplace_back(d0, x); dfs(dfs, dest(e), x, d0 + 1, d1 + cost(e)); } }; dfs(dfs, root, -1, 0, 0); rmq = RMQ(a); } cost_t dep(int a) const { return d[a]; } int operator()(int a, int b) const { return lca(a, b); } int lca(int a, int b) const { if (a == b) return a; if (id[a] > id[b]) swap(a, b); return rmq.get(id[a], id[b]).second; } cost_t dist(int a, int b) const { return d[a] + d[b] - 2*d[lca(a, b)]; } }; template auto get_lca(vector> const& g, int root = 0) { return LCA(&g, root); } // <<< int32_t main() { int n, k; cin >> n >> k; if (k == n-1) quit("No"); if (k == 1) quit("No"); if (k == 2 and n % 2 == 1) quit("No"); cout << "Yes" << '\n'; vector> es; if (k == 2) { rep (i, n-1) { int w = (i % 2 == 0 ? n-2 : -(n-1)); es.eb(i, i+1, w); } } else if (k == 3) { if (n == 5) { es.eb(0, 1, +3); es.eb(1, 2, -2); es.eb(2, 3, -2); es.eb(3, 4, +3); } else { es.eb(0, 1, +1); es.eb(1, 2, -3); es.eb(2, 3, +1); loop (x, 4, < n) { es.eb(2, x, +1); } } } else { rep (x, k) es.eb(x, x+1, -1); loop (x, k+1, < n) { es.eb(2, x, n); } } for (auto [x, y, w] : es) { cout << x+1 << " " << y+1 << " " << w << '\n'; } debug { struct edge { int to, cost; }; vector> g(n), wg(n); int sum = 0; for (auto [x, y, w] : es) { sum += w; g[x].pb({y, 1}); g[y].pb({x, 1}); wg[x].pb({y, w}); wg[y].pb({x, w}); } assert(sum > 0); auto g_lca = get_lca(g); auto wg_lca = get_lca(wg); bool found = false; rep (y, n) rep (x, y) if (g_lca.dist(x, y) == k) { found = true; assert(wg_lca.dist(x, y) < 0); } assert(found); } }