test/mytest/min_of_linear_segments.test.cpp

View this file on GitHub
Last update: 2023-11-09 01:44:55+09:00
Problem: https://judge.yosupo.jp/problem/aplusb

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "my_template.hpp"
#include "random/base.hpp"
#include "mod/min_of_linear_segments.hpp"

pair<vc<int>, vc<int>> naive(int a, int b, int mod) {
  assert(0 <= a && a < mod);
  assert(0 <= b && b < mod);
  vc<int> A;
  int last_y = b;
  FOR(x, 1, mod + 1) {
    int y = (ll(a) * x + b) % mod;
    if (chmin(last_y, y)) A.eb(x);
  }
  vc<int> X = {0};
  vc<int> DX;
  int dx = -1;
  for (auto&& x: A) {
    if (X.back() + dx == x) {
      X.back() = x;
    } else {
      dx = x - X.back();
      DX.eb(dx);
      X.eb(x);
    }
  }
  return {X, DX};
}

void test() {
  FOR(mod, 1, 1000) {
    FOR(10) {
      int a = RNG(0, mod);
      int b = RNG(0, mod);
      auto [X1, DX1] = naive(a, b, mod);
      auto [X2, DX2] = min_of_linear_segments(a, b, mod);
      assert(X1 == X2);
      assert(DX1 == DX2);
    }
  }
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

signed main() {
  test();
  solve();

  return 0;
}

#line 1 "test/mytest/min_of_linear_segments.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__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 overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, 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

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); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(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 <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#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()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(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 "mod/min_of_linear_segments.hpp"

/*
ax + b (x>=0) が最小となるところの情報を返す。
prefix min を更新する x 全体が、等差数列の和集合。次を返す。
・等差数列の境界となる x_0, x_1, ..., x_n
・各境界の間での交差 dx_0, ..., dx_{n-1}
*/
pair<vc<int>, vc<int>> min_of_linear_segments(int a, int b, int mod) {
  assert(0 <= a && a < mod);
  assert(0 <= b && b < mod);
  vc<int> X = {0};
  vc<int> DX;
  int g = gcd(a, mod);
  a /= g, b /= g, mod /= g;
  // p/q <= (mod-a)/mod <= r/s
  int p = 0, q = 1, r = 1, s = 1;
  int det_l = mod - a, det_r = a;
  int x = 0, y = b;

  while (y) {
    // upd r/s
    int k = det_r / det_l;
    det_r %= det_l;
    if (det_r == 0) {
      --k;
      det_r = det_l;
    }
    r += k * p;
    s += k * q;
    while (1) {
      int k = max(0, ceil(det_l - y, det_r));
      if (det_l - k * det_r <= 0) break;
      det_l -= k * det_r;
      p += k * r;
      q += k * s;
      // p/q <= a/mod
      // (aq - pmod) = det_l を y から引く
      k = y / det_l;
      y -= k * det_l;
      x += q * k;
      X.eb(x);
      DX.eb(q);
    }
    k = det_l / det_r;
    det_l -= k * det_r;
    p += k * r;
    q += k * s;
    assert(min({p, q, r, s}) >= 0);
  }
  return {X, DX};
}
#line 5 "test/mytest/min_of_linear_segments.test.cpp"

pair<vc<int>, vc<int>> naive(int a, int b, int mod) {
  assert(0 <= a && a < mod);
  assert(0 <= b && b < mod);
  vc<int> A;
  int last_y = b;
  FOR(x, 1, mod + 1) {
    int y = (ll(a) * x + b) % mod;
    if (chmin(last_y, y)) A.eb(x);
  }
  vc<int> X = {0};
  vc<int> DX;
  int dx = -1;
  for (auto&& x: A) {
    if (X.back() + dx == x) {
      X.back() = x;
    } else {
      dx = x - X.back();
      DX.eb(dx);
      X.eb(x);
    }
  }
  return {X, DX};
}

void test() {
  FOR(mod, 1, 1000) {
    FOR(10) {
      int a = RNG(0, mod);
      int b = RNG(0, mod);
      auto [X1, DX1] = naive(a, b, mod);
      auto [X2, DX2] = min_of_linear_segments(a, b, mod);
      assert(X1 == X2);
      assert(DX1 == DX2);
    }
  }
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

signed main() {
  test();
  solve();

  return 0;
}