verify/math/LC_rational_approximation.test.cpp

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• Last update: 2026-08-11 23:01:09+09:00
• Problem: https://judge.yosupo.jp/problem/rational_approximation

Depends on

• Stern-Brocot Tree (math/stern-brocot-tree.hpp)
• template/debug.hpp
• template/inout.hpp
• template/macro.hpp
• template/template.hpp
• template/util.hpp

Code

#define PROBLEM "https://judge.yosupo.jp/problem/rational_approximation"

#include "template/template.hpp"
#include "math/stern-brocot-tree.hpp"
using sbt = SternBrocotTreeNode<ll>;

int main() {
  int t;
  in(t);
  while (t--) {
    ll n, x, y;
    in(n, x, y);
    auto [lower, upper] = sbt::binary_search(n, [&](ll p, ll q) {
      return q == 0 || x * q < y * p;
    });
    if (lower.first * y == lower.second * x) upper = lower;
    out(lower, upper);
  }
}

#line 1 "verify/math/LC_rational_approximation.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/rational_approximation"

#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;

#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"

#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T>
using priority_queue_asc = priority_queue<T, vector<T>, greater<T>>;

template <class T, class S = T>
S SUM(const vector<T>& a) {
  return accumulate(ALL(a), S(0));
}
template <class T1, class T2>
inline bool chmin(T1& a, T2 b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}
template <class T1, class T2>
inline bool chmax(T1& a, T2 b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}
template <class T1, class T2>
inline bool chmin_opt(optional<T1>& a, T2 b) {
  if (!a || a > b) {
    a = b;
    return true;
  }
  return false;
}
template <class T1, class T2>
inline bool chmax_opt(optional<T1>& a, T2 b) {
  if (!a || a < b) {
    a = b;
    return true;
  }
  return false;
}

template <class T>
int popcnt(T x) {
  return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
  return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
  return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"

#line 2 "template/inout.hpp"
struct Fast {
  Fast() {
    cin.tie(nullptr);
    ios_base::sync_with_stdio(false);
    cout << fixed << setprecision(15);
  }
} fast;

ostream& operator<<(ostream& os, __uint128_t x) {
  char buf[40];
  size_t k = 0;
  while (x > 0) buf[k++] = (char)(x % 10 + '0'), x /= 10;
  if (k == 0) buf[k++] = '0';
  while (k) os << buf[--k];
  return os;
}
ostream& operator<<(ostream& os, __int128_t x) {
  return x < 0 ? (os << '-' << (__uint128_t)(-x)) : (os << (__uint128_t)x);
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a);
template <class T1, class T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
  return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
  return os << p.first << " " << p.second;
}
template <class T>
istream& operator>>(istream& is, vector<T>& a) {
  for (auto& v : a) is >> v;
  return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& a) {
  for (auto it = a.begin(); it != a.end();) {
    os << *it;
    if (++it != a.end()) os << " ";
  }
  return os;
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a) {
  for (auto it = a.begin(); it != a.end();) {
    os << *it;
    if (++it != a.end()) os << " ";
  }
  return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& st) {
  os << "{";
  for (auto it = st.begin(); it != st.end();) {
    os << *it;
    if (++it != st.end()) os << ",";
  }
  os << "}";
  return os;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const map<T1, T2>& mp) {
  os << "{";
  for (auto it = mp.begin(); it != mp.end();) {
    os << it->first << ":" << it->second;
    if (++it != mp.end()) os << ",";
  }
  os << "}";
  return os;
}

void in() {}
template <typename T, class... U>
void in(T& t, U&... u) {
  cin >> t;
  in(u...);
}
template <class... T>
void in_zip(int n, T&... t) {
  assert(n >= 0 && ((size(t) >= static_cast<size_t>(n)) && ...));
  for (int i = 0; i < n; i++) in(t[i]...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T& t, const U&... u) {
  cout << t;
  if (sizeof...(u)) cout << sep;
  out(u...);
}
template <class T, class U>
void out_opt(const optional<T>& opt, const U& fallback, ostream& os = cout) {
  if (opt.has_value())
    os << opt.value();
  else
    os << fallback;
  os << "\n";
}
template <class T, class U>
void out_opt(const vector<optional<T>>& vec, const U& fallback, ostream& os = cout) {
  for (auto it = vec.begin(); it != vec.end();) {
    if ((*it).has_value())
      os << (*it).value();
    else
      os << fallback;
    if (++it != vec.end()) os << " ";
  }
  os << "\n";
}

namespace IO {
template <class T, class... U>
T read(U&&... u) {
  T t = T(forward<U>(u)...);
  in(t);
  return t;
}
namespace Graph {
vector<vector<int>> unweighted(int n, int m, bool directed = false, int offset = 1) {
  vector<vector<int>> g(n);
  for (int i = 0; i < m; i++) {
    int u, v;
    cin >> u >> v;
    u -= offset, v -= offset;
    g[u].push_back(v);
    if (!directed) g[v].push_back(u);
  }
  return g;
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, int m, bool directed = false, int offset = 1) {
  vector<vector<pair<int, T>>> g(n);
  for (int i = 0; i < m; i++) {
    int u, v;
    T w;
    cin >> u >> v >> w;
    u -= offset, v -= offset;
    g[u].push_back({v, w});
    if (!directed) g[v].push_back({u, w});
  }
  return g;
}
}  // namespace Graph
namespace Tree {
vector<vector<int>> unweighted(int n, bool directed = false, int offset = 1) {
  return Graph::unweighted(n, n - 1, directed, offset);
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, bool directed = false, int offset = 1) {
  return Graph::weighted<T>(n, n - 1, directed, offset);
}
vector<vector<int>> rooted(int n, bool to_root = true, bool to_leaf = true, int offset = 1) {
  vector<vector<int>> g(n);
  for (int i = 1; i < n; i++) {
    int p;
    cin >> p;
    p -= offset;
    if (to_root) g[i].push_back(p);
    if (to_leaf) g[p].push_back(i);
  }
  return g;
}
}  // namespace Tree
}  // namespace IO
#line 10 "template/template.hpp"

#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int, T) {
  cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1& a, const T2& b, const T3&... c) {
  for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
  cerr << ":" << b << " ";
  _show(i + 1, a, c...);
}
#line 2 "math/stern-brocot-tree.hpp"

template <class T>
struct SternBrocotTreeNode {
  using Node = SternBrocotTreeNode;
  T la, lb, a, b, ra, rb;
  vector<T> seq;
  SternBrocotTreeNode() : la(0), lb(1), a(1), b(1), ra(1), rb(0) {}
  SternBrocotTreeNode(T x, T y) : SternBrocotTreeNode() {
    assert(x > 0 && y > 0);
    T g = gcd(x, y);
    x /= g, y /= g;
    bool is_right = true;
    while (x > 0 && y > 0) {
      T d = x / y;
      x -= d * y;
      if (is_right)
        go_right(d - (x == 0 ? 1 : 0));
      else
        go_left(d - (x == 0 ? 1 : 0));
      swap(x, y);
      is_right = !is_right;
    }
  }
  SternBrocotTreeNode(pair<T, T> p) : SternBrocotTreeNode(p.first, p.second) {}
  SternBrocotTreeNode(const vector<T> seq_) {
    for (auto& v : seq_) {
      assert(v != 0);
      if (v > 0)
        go_right(v);
      else
        go_left(v);
    }
    assert(seq == seq_);
  }
  pair<T, T> get() const { return {a, b}; }
  pair<T, T> lower_bound() const { return {la, lb}; }
  pair<T, T> upper_bound() const { return {ra, rb}; }

  void go_left(const T d = 1) {
    if (d <= 0) return;
    if (seq.empty() || seq.back() > 0) seq.push_back(0);
    seq.back() -= d;
    ra += la * d, rb += lb * d;
    a = la + ra, b = lb + rb;
  }
  void go_right(const T d = 1) {
    if (d <= 0) return;
    if (seq.empty() || seq.back() < 0) seq.push_back(0);
    seq.back() += d;
    la += ra * d, lb += rb * d;
    a = la + ra, b = lb + rb;
  }
  T depth() const {
    T d = 0;
    for (auto& v : seq) d += abs(v);
    return d;
  }
  static Node lca(const Node& x, const Node& y) {
    Node res;
    int sz = min(x.seq.size(), y.seq.size());
    for (int i = 0; i < sz; i++) {
      T d1 = x.seq[i], d2 = y.seq[i];
      if ((d1 > 0) != (d2 > 0)) break;
      if (d1 > 0)
        res.go_right(min(d1, d2));
      else
        res.go_left(min(-d1, -d2));
      if (d1 != d2) break;
    }
    return res;
  }
  bool go_parent(T d = 1) {
    if (d <= 0) return true;
    while (d > 0) {
      if (seq.empty()) return false;
      T d1 = min(d, abs(seq.back()));
      if (seq.back() > 0) {
        la -= ra * d1, lb -= rb * d1;
        seq.back() -= d1;
      } else {
        ra -= la * d1, rb -= lb * d1;
        seq.back() += d1;
      }
      a = la + ra, b = lb + rb;
      if (seq.back() <= 0) seq.pop_back();
      d -= d1;
    }
    return true;
  }
  template <class F>
  static pair<pair<T, T>, pair<T, T>> binary_search(T n, F f) {
    assert(0 <= n);
    Node m;
    if (n == 0) return {m.lower_bound(), m.upper_bound()};
    auto over = [&](bool return_value) {
      auto [p, q] = m.get();
      return max(m.a, m.b) > n || f(p, q) == return_value;
    };
    if (f(0, 1)) return {m.lower_bound(), m.lower_bound()};
    for (int go_left = over(true); true; go_left ^= 1) {
      if (go_left) {
        T a = 1;
        for (; true; a *= 2) {
          m.go_left(a);
          if (over(false)) {
            m.go_parent(a);
            break;
          }
        }
        for (a /= 2; a != 0; a /= 2) {
          m.go_left(a);
          if (over(false)) m.go_parent(a);
        }
        m.go_left(1);
        if (max(m.get().first, m.get().second) > n)
          return {m.lower_bound(), m.upper_bound()};
      } else {
        T a = 1;
        for (; true; a *= 2) {
          m.go_right(a);
          if (over(true)) {
            m.go_parent(a);
            break;
          }
        }
        for (a /= 2; a != 0; a /= 2) {
          m.go_right(a);
          if (over(true)) m.go_parent(a);
        }
        m.go_right(1);
        if (max(m.get().first, m.get().second) > n)
          return {m.lower_bound(), m.upper_bound()};
      }
    }
  }
};

/**
 * @brief Stern-Brocot Tree
 * @docs docs/math/stern-brocot-tree.md
 */
#line 5 "verify/math/LC_rational_approximation.test.cpp"
using sbt = SternBrocotTreeNode<ll>;

int main() {
  int t;
  in(t);
  while (t--) {
    ll n, x, y;
    in(n, x, y);
    auto [lower, upper] = sbt::binary_search(n, [&](ll p, ll q) {
      return q == 0 || x * q < y * p;
    });
    if (lower.first * y == lower.second * x) upper = lower;
    out(lower, upper);
  }
}

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