#959519 (C++23) No.2503 Typical Path Counting Problem on a Grid

• 問題: No.2503 Typical Path Counting Problem on a Grid
• ユーザー: Misuki
• 提出日時: 2024-03-13 13:56:48
• 言語: C++23 (gcc 13.3.0 + boost 1.87.0)
• 結果: AC
• 実行時間: 369 ms / 2,000 ms
• コード長: 8,836 bytes
• コンパイル時間: 2,756 ms
• コンパイル使用メモリ: 205,616 KB
• 実行使用メモリ: 42,612 KB
• 最終ジャッジ日時: 2024-09-29 22:51:27
• 合計ジャッジ時間: 5,840 ms
• ジャッジサーバーID （参考情報）: judge3 / judge5

ファイルパターン	結果
other	AC * 10

ソースコード

#pragma GCC optimize("O2")
#include <algorithm>
#include <array>
#include <bit>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <compare>
#include <complex>
#include <concepts>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numbers>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <ranges>
#include <set>
#include <span>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <variant>

//#define int ll
#define INT128_MAX (__int128)(((unsigned __int128) 1 << ((sizeof(__int128) * __CHAR_BIT__) - 1)) - 1)
#define INT128_MIN (-INT128_MAX - 1)

#define clock chrono::steady_clock::now().time_since_epoch().count()

#ifdef DEBUG
#define dbg(x) cout << (#x) << " = " << x << '\n'
#else
#define dbg(x)
#endif

namespace R = std::ranges;
namespace V = std::views;

using namespace std;

using ll = long long;
using ull = unsigned long long;
using ldb = long double;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
//#define double ldb

template<class T>
ostream& operator<<(ostream& os, const pair<T, T> pr) {
  return os << pr.first << ' ' << pr.second;
}
template<class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N> &arr) {
  for(const T &X : arr)
    os << X << ' ';
  return os;
}
template<class T>
ostream& operator<<(ostream& os, const vector<T> &vec) {
  for(const T &X : vec)
    os << X << ' ';
  return os;
}
template<class T>
ostream& operator<<(ostream& os, const set<T> &s) {
  for(const T &x : s)
    os << x << ' ';
  return os;
}

//reference: https://github.com/NyaanNyaan/library/blob/master/modint/montgomery-modint.hpp#L10
//note: mod should be a prime less than 2^30.

template<uint32_t mod>
struct MontgomeryModInt {
  using mint = MontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 res = 1, base = mod;
    for(i32 i = 0; i < 31; i++)
      res *= base, base *= base;
    return -res;
  }

  static constexpr u32 get_mod() {
    return mod;
  }

  static constexpr u32 n2 = -u64(mod) % mod; //2^64 % mod
  static constexpr u32 r = get_r(); //-P^{-1} % 2^32

  u32 a;

  static u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * r) * mod) >> 32;
  }

  static u32 transform(const u64 &b) {
    return reduce(u64(b) * n2);
  }

  MontgomeryModInt() : a(0) {}
  MontgomeryModInt(const int64_t &b) 
    : a(transform(b % mod + mod)) {}

  mint pow(u64 k) const {
    mint res(1), base(*this);
    while(k) {
      if (k & 1) 
        res *= base;
      base *= base, k >>= 1;
    }
    return res;
  }

  mint inverse() const { return (*this).pow(mod - 2); }

  u32 get() const {
    u32 res = reduce(a);
    return res >= mod ? res - mod : res;
  }

  mint& operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  mint& operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  mint& operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  mint& operator/=(const mint &b) {
    a = reduce(u64(a) * b.inverse().a);
    return *this;
  }

  mint operator-() { return mint() - mint(*this); }
  bool operator==(mint b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(mint b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }

  friend mint operator+(mint a, mint b) { return a += b; }
  friend mint operator-(mint a, mint b) { return a -= b; }
  friend mint operator*(mint a, mint b) { return a *= b; }
  friend mint operator/(mint a, mint b) { return a /= b; }

  friend ostream& operator<<(ostream& os, const mint& b) {
    return os << b.get();
  }
  friend istream& operator>>(istream& is, mint& b) {
    int64_t val;
    is >> val;
    b = mint(val);
    return is;
  }
};

using mint = MontgomeryModInt<998244353>;

//source: KACTL(for det() and inv())

template<class Mint>
struct matrix : vector<vector<Mint>> {
  matrix(int n, int m) : vector<vector<Mint>>(n, vector<Mint>(m, 0)) {}
  matrix(int n) : vector<vector<Mint>>(n, vector<Mint>(n, 0)) {}

  int n() const { return ssize(*this); }
  int m() const { return ssize((*this)[0]); }

  static matrix I(int n) {
    auto res = matrix(n, n);
    for(int i = 0; i < n; i++)
      res[i][i] = 1;
    return res;
  }

  matrix& operator+=(const matrix &b) {
    assert(n() == b.n());
    assert(m() == b.m());
    for(int i = 0; i < n(); i++)
      for(int j = 0; j < m(); j++)
        (*this)[i][j] += b[i][j];
    return *this;
  }

  matrix& operator-=(const matrix &b) {
    assert(n() == b.n());
    assert(m() == b.m());
    for(int i = 0; i < n(); i++)
      for(int j = 0; j < m(); j++)
        (*this)[i][j] -= b[i][j];
    return *this;
  }

  matrix& operator*=(const matrix &b) {
    assert(m() == b.n());
    auto res = matrix(n(), b.m());
    for(int i = 0; i < n(); i++)
      for(int j = 0; j < b.m(); j++)
        for(int k = 0; k < m(); k++)
          res[i][j] += (*this)[i][k] * b[k][j];
    this -> swap(res);
    return *this;
  }

  matrix pow(ll k) const {
    assert(n() == m());
    auto res = I(n()), base = *this;
    while(k) {
      if (k & 1) res *= base;
      base *= base, k >>= 1;
    }
    return res;
  }

  Mint det() const {
    Mint res = 1;
    auto a = *this;
    for(int i = 0; i < n(); i++) {
      for(int j = i + 1; j < m(); j++) {
        while(a[j][i] != 0) {
          Mint t = a[i][i] / a[j][i];
          if (t != 0)
            for(int k = i; k < n(); k++)
              a[i][k] -= a[j][k] * t;
          swap(a[i], a[j]);
          res = -res;
        }
      }
      res *= a[i][i];
      if (res == 0) return 0;
    }
    return res;
  }

  matrix inv() const {
    assert(n() == m());
    matrix a = *this, tmp = I(n());
    vector<int> col(n());
    for(int i = 0; i < n(); i++) col[i] = i;

    for(int i = 0; i < n(); i++) {
      int r = i, c = i;
      for(int j = i; j < n(); j++) {
        for(int k = i; k < n(); k++) {
          if (a[j][k] != 0) {
            r = j, c = k;
            goto found;
          }
        }
      }
      return matrix(0);
      found:
      a[i].swap(a[r]), tmp[i].swap(tmp[r]);
      for(int j = 0; j < n(); j++)
        swap(a[j][i], a[j][c]), swap(tmp[j][i], tmp[j][c]);
      swap(col[i], col[c]);
      Mint v = 1 / a[i][i];
      for(int j = i + 1; j < n(); j++) {
        Mint f = a[j][i] * v;
        a[j][i] = 0;
        for(int k = i + 1; k < n(); k++)
          a[j][k] -= f * a[i][k];
        for(int k = 0; k < n(); k++)
          tmp[j][k] -= f * tmp[i][k];
      }
      for(int j = i + 1; j < n(); j++) 
        a[i][j] *= v;
      for(int j = 0; j < n(); j++) 
        tmp[i][j] *= v;
      a[i][i] = 1;
    }

    for(int i = n() - 1; i > 0; i--) {
      for(int j = 0; j < i; j++) {
        Mint v = a[j][i];
        for(int k = 0; k < n(); k++)
          tmp[j][k] -= v * tmp[i][k];
      }
    }

    for(int i = 0; i < n(); i++)
      for(int j = 0; j < n(); j++)
        a[col[i]][col[j]] = tmp[i][j];
    return a;
  }

  matrix operator-() { return matrix(n(), m()) - (*this); }
  
  friend matrix operator+(matrix a, matrix b) { return a += b; }
  friend matrix operator-(matrix a, matrix b) { return a -= b; }
  friend matrix operator*(matrix a, matrix b) { return a *= b; }
  
  friend ostream& operator<<(ostream& os, const matrix& b) {
    for(int i = 0; i < b.n(); i++) {
      os << '\n';
      for(int j = 0; j < b.m(); j++)
        os << b[i][j] << ' ';
    }
    return os;
  }
  friend istream& operator>>(istream& is, matrix& b) {
    for(int i = 0; i < b.n(); i++)
      for(int j = 0; j < b.m(); j++)
        is >> b[i][j];
    return is;
  }
};

const int MAX = 10000001;
mint f[MAX];

signed main() {
  ios::sync_with_stdio(false), cin.tie(NULL);
  
  f[0] = 1, f[1] = 2;
  for(int i = 2; i < MAX; i++)
    f[i] = f[i - 1] * (2 * i) + f[i - 2] * (i - 1);

  int t; cin >> t;
  while(t--) {
    ll n, m; cin >> n >> m;

    if (min(n, m) == 0) {
      cout << 1 << '\n';
      continue;
    }

    if (n > m) swap(n, m);

    matrix<mint> M(2, 2), x(2, 1);
    M[0][1] = 1, M[1][0] = n, M[1][1] = 2 * n + 1;
    x[0][0] = f[n - 1], x[1][0] = f[n];
    auto b = M.pow(m - n) * x;

    cout << f[n] * b[1][0] + f[n - 1] * b[0][0] * n << '\n';
  }

  return 0;
}