test/1_mytest/nimber.test.cpp

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• Last update: 2024-10-18 02:58:53+09:00
• Problem: https://judge.yosupo.jp/problem/aplusb

Depends on

• linalg/xor/basis.hpp
• my_template.hpp
• nt/nimber/base.hpp
• nt/nimber/nimber_impl.hpp
• nt/nimber/solve_quadratic.hpp
• random/base.hpp

Code

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

#include "random/base.hpp"
#include "nt/nimber/base.hpp"
#include "nt/nimber/solve_quadratic.hpp"

template <typename U>
void test() {
  using F = Nimber<U>;
  auto test = [&](F x) -> void {
    assert(x * x == x.square());
    assert(x.sqrt().square() == x);
    if (x != F(0)) assert(x * x.inverse() == F(1));
  };
  FOR(i, 1 << 20) { test(i); }
  FOR(10000) { test(F(RNG_64())); }

  auto test_q = [&](F a, F x) -> void {
    F b = x * x + a * x;
    vc<F> ANS = solve_quadratic(a, b);
    for (auto& z: ANS) { assert(z * z + a * z == b); }
    FOR(j, len(ANS)) FOR(i, j) { assert(ANS[i] != ANS[j]); }
    int exist = 0;
    FOR(i, len(ANS)) exist += (ANS[i] == x);
    assert(exist == 1);
  };
  // quadratic
  FOR(a, 100) {
    FOR(x, 100) { test_q(a, x); }
  }
  FOR(10000) { test_q(F(RNG_64()), F(RNG_64())); }
}

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

signed main() {
  test<u16>();
  test<u32>();
  test<u64>();
  solve();
}

#line 1 "test/1_mytest/nimber.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else

// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
// いまの CF だとこれ入れると動かない？
// #pragma GCC target("avx2,popcnt")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
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;
}

template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &... others) {
  vc<T> &res = first;
  (res.insert(res.end(), others.begin(), others.end()), ...);
}
#endif
#line 3 "test/1_mytest/nimber.test.cpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static u64 x_ = u64(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 "nt/nimber/nimber_impl.hpp"
namespace NIM_PRODUCT {
u16 E[65535 * 2 + 7];
u16 L[65536];
u64 S[4][65536];
u64 SR[4][65536];

u16 p16_15(u16 a, u16 b) { return (a && b ? E[u32(L[a]) + L[b] + 3] : 0); }
u16 p16_15_15(u16 a, u16 b) { return (a && b ? E[u32(L[a]) + L[b] + 6] : 0); }
u16 mul_15(u16 a) { return (a ? E[3 + L[a]] : 0); }
u16 mul_15_15(u16 a) { return (a ? E[6 + L[a]] : 0); }
u32 p32_mul_31(u32 a, u32 b) {
  u16 al = a & 65535, ah = a >> 16, bl = b & 65535, bh = b >> 16;
  u16 x = p16_15(al, bl);
  u16 y = p16_15_15(ah, bh);
  u16 z = p16_15(al ^ ah, bl ^ bh);
  return u32(y ^ z) << 16 | mul_15(z ^ x);
}
u32 mul_31(u32 a) {
  u16 al = a & 65535, ah = a >> 16;
  return u32(mul_15(al ^ ah)) << 16 | mul_15_15(ah);
}

u16 prod(u16 a, u16 b) { return (a && b ? E[u32(L[a]) + L[b]] : 0); }
u32 prod(u32 a, u32 b) {
  u16 al = a & 65535, ah = a >> 16, bl = b & 65535, bh = b >> 16;
  u16 c = prod(al, bl);
  return u32(prod(u16(al ^ ah), u16(bl ^ bh)) ^ c) << 16 | (p16_15(ah, bh) ^ c);
}
u64 prod(u64 a, u64 b) {
  u32 al = a & 0xffffffff, ah = a >> 32, bl = b & 0xffffffff, bh = b >> 32;
  u32 c = prod(al, bl);
  return u64(prod(al ^ ah, bl ^ bh) ^ c) << 32 ^ (p32_mul_31(ah, bh) ^ c);
}

u16 square(u16 a) { return S[0][a]; }
u32 square(u32 a) { return S[0][a & 65535] ^ S[1][a >> 16]; }
u64 square(u64 a) { return S[0][a & 65535] ^ S[1][a >> 16 & 65535] ^ S[2][a >> 32 & 65535] ^ S[3][a >> 48 & 65535]; }
u16 sqrt(u16 a) { return SR[0][a]; }
u32 sqrt(u32 a) { return SR[0][a & 65535] ^ SR[1][a >> 16]; }
u64 sqrt(u64 a) { return SR[0][a & 65535] ^ SR[1][a >> 16 & 65535] ^ SR[2][a >> 32 & 65535] ^ SR[3][a >> 48 & 65535]; }

// inv: 2^16 の共役が 2^16+1 であることなどを使う. x^{-1}=y(xy)^{-1} という要領.

u16 inverse(u16 a) { return E[65535 - L[a]]; }
u32 inverse(u32 a) {
  if (a < 65536) return inverse(u16(a));
  u16 al = a & 65535, ah = a >> 16;
  u16 norm = prod(al, al ^ ah) ^ E[L[ah] * 2 + 3];
  int k = 65535 - L[norm];
  al = (al ^ ah ? E[L[al ^ ah] + k] : 0), ah = E[L[ah] + k];
  return al | u32(ah) << 16;
}
u64 inverse(u64 a) {
  if (a <= u32(-1)) return inverse(u32(a));
  u32 al = a & 0xffffffff, ah = a >> 32;
  u32 norm = prod(al, al ^ ah) ^ mul_31(square(ah));
  u32 i = inverse(norm);
  return prod(al ^ ah, i) | u64(prod(ah, i)) << 32;
}

void __attribute__((constructor)) init_nim_table() {
  // 2^16 未満のところについて原始根 10279 での指数対数表を作る

  // 2^k との積

  u16 tmp[] = {10279, 15417, 35722, 52687, 44124, 62628, 15661, 5686, 3862, 1323, 334, 647, 61560, 20636, 4267, 8445};
  u16 nxt[65536];
  FOR(i, 16) {
    FOR(s, 1 << i) { nxt[s | 1 << i] = nxt[s] ^ tmp[i]; }
  }
  E[0] = 1;
  FOR(i, 65534) E[i + 1] = nxt[E[i]];
  memcpy(E + 65535, E, 131070);
  memcpy(E + 131070, E, 14);
  FOR(i, 65535) L[E[i]] = i;
  FOR(t, 4) {
    FOR(i, 16) {
      int k = 16 * t + i;
      u64 X = prod(u64(1) << k, u64(1) << k);
      FOR(s, 1 << i) S[t][s | 1 << i] = S[t][s] ^ X;
    }
  }
  FOR(t, 4) {
    FOR(i, 16) {
      int k = 16 * t + i;
      u64 X = u64(1) << k;
      FOR(63) X = square(X);
      FOR(s, 1 << i) SR[t][s | 1 << i] = SR[t][s] ^ X;
    }
  }
}
} // namespace NIM_PRODUCT

#line 3 "nt/nimber/base.hpp"

template <typename UINT>
struct Nimber {
  using F = Nimber;
  UINT val;

  constexpr Nimber(UINT x = 0) : val(x) {}
  F &operator+=(const F &p) {
    val ^= p.val;
    return *this;
  }
  F &operator-=(const F &p) {
    val ^= p.val;
    return *this;
  }
  F &operator*=(const F &p) {
    val = NIM_PRODUCT::prod(val, p.val);
    return *this;
  }
  F &operator/=(const F &p) {
    *this *= p.inverse();
    return *this;
  }
  F operator-() const { return *this; }
  F operator+(const F &p) const { return F(*this) += p; }
  F operator-(const F &p) const { return F(*this) -= p; }
  F operator*(const F &p) const { return F(*this) *= p; }
  F operator/(const F &p) const { return F(*this) /= p; }
  bool operator==(const F &p) const { return val == p.val; }
  bool operator!=(const F &p) const { return val != p.val; }
  F inverse() const { return NIM_PRODUCT::inverse(val); }
  F pow(u64 n) const {
    assert(n >= 0);
    UINT ret = 1, mul = val;
    while (n > 0) {
      if (n & 1) ret = NIM_PRODUCT::prod(ret, mul);
      mul = NIM_PRODUCT::square(mul);
      n >>= 1;
    }
    return F(ret);
  }
  F square() { return F(NIM_PRODUCT::square(val)); }
  F sqrt() { return F(NIM_PRODUCT::sqrt(val)); }
};

#ifdef FASTIO
template <typename T>
void rd(Nimber<T> &x) {
  fastio::rd(x.val);
}
template <typename T>
void wt(Nimber<T> &x) {
  fastio::wt(x.val);
}
#endif

using Nimber16 = Nimber<u16>;
using Nimber32 = Nimber<u32>;
using Nimber64 = Nimber<u64>;
#line 1 "linalg/xor/basis.hpp"

// basis[i]: i 番目に追加成功したもの. 別のラベルがあるなら外で管理する.
// array<UINT, MAX_DIM> rbasis: 上三角化された基底. [i][i]==1.
// way<UINT,UINT> rbasis[i] を basis[j] で作る方法
template <int MAX_DIM>
struct Basis {
  static_assert(MAX_DIM <= 128);
  using UINT = conditional_t<(MAX_DIM <= 32), u32, conditional_t<(MAX_DIM <= 64), u64, u128>>;
  int rank;
  array<UINT, MAX_DIM> basis;
  array<UINT, MAX_DIM> rbasis;
  array<UINT, MAX_DIM> way;
  Basis() : rank(0), basis{}, rbasis{}, way{} {}

  // return : (sum==x にできるか, その方法)
  pair<bool, UINT> solve(UINT x) {
    UINT c = 0;
    FOR(i, MAX_DIM) {
      if ((x >> i & 1) && (rbasis[i] != 0)) { c ^= way[i], x ^= rbasis[i]; }
    }
    if (x == 0) return {true, c};
    return {false, 0};
  }

  // return : (sum==x にできるか, その方法). false の場合には追加する
  pair<bool, UINT> solve_or_add(UINT x) {
    UINT y = x, c = 0;
    FOR(i, MAX_DIM) {
      if ((x >> i & 1) && (rbasis[i] != 0)) { c ^= way[i], x ^= rbasis[i]; }
    }
    if (x == 0) return {true, c};
    int k = lowbit(x);
    basis[rank] = y, rbasis[k] = x, way[k] = c | UINT(1) << rank, ++rank;
    return {false, 0};
  }
};
#line 3 "nt/nimber/solve_quadratic.hpp"

namespace NIMBER_QUADRATIC {
// x^2+x==a を解く. Trace(a)==0 が必要.
// Nimber では Trace は topbit.
// topbit==0 である空間から偶数全体への全単射がある.
// これを前計算したい. 線形写像なので連立方程式を解いて埋め込むだけでよい.

u64 Q[4][65536];

void __attribute__((constructor)) precalc() {
  Basis<63> B;
  FOR(i, 63) {
    Nimber64 x(u64(1) << (i + 1));
    x = x.square() + x;
    assert(!B.solve_or_add(x.val).fi);
  }
  FOR(k, 63) {
    int t = k / 16, i = k % 16;
    u64 X = B.way[k] * 2;
    FOR(s, 1 << i) Q[t][s | 1 << i] = Q[t][s] ^ X;
  }
}

u16 f(u16 a) { return Q[0][a]; }
u32 f(u32 a) { return Q[0][a & 65535] ^ Q[1][a >> 16]; }
u64 f(u64 a) { return Q[0][a & 65535] ^ Q[1][a >> 16 & 65535] ^ Q[2][a >> 32 & 65535] ^ Q[3][a >> 48 & 65535]; }

template <typename U>
vc<U> solve_quadratic_1(U a) {
  constexpr int k = numeric_limits<U>::digits - 1;
  if (a >> k & 1) return {};
  return {f(a), U(f(a) | 1)};
}
} // namespace NIMBER_QUADRATIC

template <typename F>
vc<F> solve_quadratic(F a, F b) {
  if (a == F(0)) return {b.sqrt()};
  b /= a.square();
  vc<F> ANS;
  for (auto& x: NIMBER_QUADRATIC::solve_quadratic_1(b.val)) { ANS.eb(a * F(x)); }
  return ANS;
}
#line 7 "test/1_mytest/nimber.test.cpp"

template <typename U>
void test() {
  using F = Nimber<U>;
  auto test = [&](F x) -> void {
    assert(x * x == x.square());
    assert(x.sqrt().square() == x);
    if (x != F(0)) assert(x * x.inverse() == F(1));
  };
  FOR(i, 1 << 20) { test(i); }
  FOR(10000) { test(F(RNG_64())); }

  auto test_q = [&](F a, F x) -> void {
    F b = x * x + a * x;
    vc<F> ANS = solve_quadratic(a, b);
    for (auto& z: ANS) { assert(z * z + a * z == b); }
    FOR(j, len(ANS)) FOR(i, j) { assert(ANS[i] != ANS[j]); }
    int exist = 0;
    FOR(i, len(ANS)) exist += (ANS[i] == x);
    assert(exist == 1);
  };
  // quadratic
  FOR(a, 100) {
    FOR(x, 100) { test_q(a, x); }
  }
  FOR(10000) { test_q(F(RNG_64()), F(RNG_64())); }
}

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

signed main() {
  test<u16>();
  test<u32>();
  test<u64>();
  solve();
}

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