結果

問題 No.2613 Sum of Combination
コンテスト
ユーザー Misuki
提出日時 2024-01-26 04:49:33
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 207 ms / 4,500 ms
コード長 9,306 bytes
コンパイル時間 2,254 ms
コンパイル使用メモリ 199,428 KB
実行使用メモリ 15,192 KB
最終ジャッジ日時 2024-09-28 07:27:10
合計ジャッジ時間 8,270 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge4
このコードへのチャレンジ
（要ログイン）
ファイルパターン 結果
sample AC * 3
other AC * 49
権限があれば一括ダウンロードができます

ソースコード

diff # 

#define PROBLEM "https://yukicoder.me/problems/no/2613"

//#include "../default/t.cpp";
//#include "../modint/MontgomeryModInt.cpp"
//#include "../poly/NTTmint.cpp"
//#include "../numtheory/fastFactorize.cpp"
//#include "../numtheory/primitiveRoot.cpp"
//#include "../poly/mulConvolution.cpp"

#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 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()

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>;

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>;

//reference: https://judge.yosupo.jp/submission/69896
//remark: MOD = 2^K * C + 1, R is a primitive root modulo MOD
//remark: a.size() <= 2^K must be satisfied
//some common modulo: 998244353  = 2^23 * 119 + 1, R = 3
//                    469762049  = 2^26 * 7   + 1, R = 3
//                    1224736769 = 2^24 * 73  + 1, R = 3

template<int32_t k = 23, int32_t c = 119, int32_t r = 3, class Mint = MontgomeryModInt<998244353>>
struct NTT {

  using u32 = uint32_t;
  static constexpr u32 mod = (1 << k) * c + 1;
  static constexpr u32 get_mod() { return mod; }

  static void ntt(vector<Mint> &a, bool inverse) {
    static array<Mint, 30> w, w_inv;
    if (w[0] == 0) {
      Mint root = 2;
      while(root.pow((mod - 1) / 2) == 1) root += 1;
      for(int i = 0; i < 30; i++)
        w[i] = -(root.pow((mod - 1) >> (i + 2))), w_inv[i] = 1 / w[i];
    }
    int n = ssize(a);
    if (not inverse) {
      for(int m = n; m >>= 1; ) {
        Mint ww = 1;
        for(int s = 0, l = 0; s < n; s += 2 * m) {
          for(int i = s, j = s + m; i < s + m; i++, j++) {
            Mint x = a[i], y = a[j] * ww;
            a[i] = x + y, a[j] = x - y;
          }
          ww *= w[__builtin_ctz(++l)];
        }
      }
    } else {
      for(int m = 1; m < n; m *= 2) {
        Mint ww = 1;
        for(int s = 0, l = 0; s < n; s += 2 * m) {
          for(int i = s, j = s + m; i < s + m; i++, j++) {
            Mint x = a[i], y = a[j];
            a[i] = x + y, a[j] = (x - y) * ww;
          }
          ww *= w_inv[__builtin_ctz(++l)];
        }
      }
      Mint inv = 1 / Mint(n);
      for(Mint &x : a) x *= inv;
    }
  }

  static vector<Mint> conv(vector<Mint> a, vector<Mint> b) {
    int sz = ssize(a) + ssize(b) - 1;
    int n = bit_ceil((u32)sz);

    a.resize(n, 0);
    ntt(a, false);
    b.resize(n, 0);
    ntt(b, false);

    for(int i = 0; i < n; i++)
      a[i] *= b[i];

    ntt(a, true);

    a.resize(sz);

    return a;
  }
};

//#include "poly/NTTmint.cpp"
//#include "modint/MontgomeryModInt.cpp"

struct mulConvolution {
  const int P, root;
  vector<int> powR, logR;

  int primitiveRoot(int p) {
    vector<int> pf;
    {
      int tmp = p - 1;
      for(int i = 2; i * i <= (p - 1); i++) {
        if (tmp % i != 0) continue;
        pf.emplace_back(i);
        while(tmp % i == 0) tmp /= i;
      }
      if (tmp != 1)
        pf.emplace_back(tmp);
    }

    auto modPow = [p](ll a, int x) -> int {
      if (x == 0) return 1;
      if (a == 0) return 0;
      ll b = 1;
      while(x) {
        if (x & 1) b = b * a % p;
        a = a * a % p, x >>= 1;
      }
      return b;
    };

    for(int r = 1; ; r++) {
      bool isRoot = true;
      for(int d : pf) {
        if (modPow(r, (p - 1) / d) == 1) {
          isRoot = false;
          break;
        }
      }
      if (isRoot)
        return r;
    }
  }

  mulConvolution(int _P) : P(_P), root(primitiveRoot(_P)), powR(P - 1), logR(P, -1) {
    for(int i = 0, tmp = 1; i < P - 1; i++, tmp = (ll)tmp * root % P)
      powR[i] = tmp, logR[tmp] = i;
  }

  template<class Mint>
  vector<Mint> transform(vector<Mint> &f) {
    assert(ssize(f) == P);
    vector<Mint> g(P - 1);
    for(int i = 1; i < P; i++)
      g[logR[i]] = f[i];
    return g;
  }

  template<class Mint>
  vector<Mint> invTransform(vector<Mint> &f) {
    assert(ssize(f) == P - 1);
    vector<Mint> g(P);
    for(int i = 0; i < P - 1; i++)
      g[powR[i]] = f[i];
    return g;
  }

  template<class Mint>
  vector<Mint> mulConv(vector<Mint> a, vector<Mint> b, vector<Mint>(*conv)(vector<Mint>, vector<Mint>)) {
    Mint zero = accumulate(a.begin(), a.end(), mint(0)) * b[0] + accumulate(b.begin() + 1, b.end(), mint(0)) * a[0];
    a = transform(a), b = transform(b);
    a = conv(a, b);
    for(int i = P - 1; i < 2 * P - 3; i++)
      a[i - (P - 1)] += a[i];
    a.resize(P - 1);
    a = invTransform(a);
    a[0] = zero;
    return a;
  }
};

int p;
int fac[200000], facInv[200000];
int C(int a, int b) {
  if (b > a or b < 0) return 0;
  else return (ll)fac[a] * facInv[b] % p * facInv[a - b] % p;
}

NTT ntt;

signed main() {
  ios::sync_with_stdio(false), cin.tie(NULL);

  ll n; cin >> n >> p;

  fac[0] = 1;
  for(int i = 1; i < p; i++)
    fac[i] = (ll)fac[i - 1] * i % p;
  facInv[p - 1] = 1;
  for(int i = 0; i < p - 2; i++)
    facInv[p - 1] = (ll)facInv[p - 1] * fac[p - 1] % p;
  for(int i = p - 2; i >= 0; i--)
    facInv[i] = (ll)facInv[i + 1] * (i + 1) % p;

  mulConvolution mu(p);

  vector<mint> f(p);
  f[1] = 1;
  while(n) {
    int nd = n % p; n /= p;
    vector<mint> g(p);
    for(int i = 0; i < p; i++)
      g[C(nd, i)] += 1;
    f = mu.mulConv(f, g, ntt.conv);
  }

  mint ans = 0;
  for(int i = 1; i < p; i++)
    ans += f[i] * i;

  cout << ans << '\n';

  return 0;
}
0