結果

問題 No.2705 L to R Graph (Another ver.)
ユーザー 👑 hos.lyric
提出日時 2024-03-29 22:27:38
言語 C++14
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 2,880 ms / 3,000 ms
コード長 9,382 bytes
コンパイル時間 1,683 ms
コンパイル使用メモリ 127,076 KB
実行使用メモリ 7,424 KB
最終ジャッジ日時 2024-09-30 16:21:06
合計ジャッジ時間 103,616 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge4
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ファイルパターン 結果
sample AC * 3
other AC * 50
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ソースコード

diff #
プレゼンテーションモードにする

#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using Int = long long;
template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i 
    >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
#define COLOR(s) ("\x1b[" s "m")
////////////////////////////////////////////////////////////////////////////////
// Barrett
struct ModInt {
  static unsigned M;
  static unsigned long long NEG_INV_M;
  static void setM(unsigned m) { M = m; NEG_INV_M = -1ULL / M; }
  unsigned x;
  ModInt() : x(0U) {}
  ModInt(unsigned x_) : x(x_ % M) {}
  ModInt(unsigned long long x_) : x(x_ % M) {}
  ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
  ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
  ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
  ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
  ModInt &operator*=(const ModInt &a) {
    const unsigned long long y = static_cast<unsigned long long>(x) * a.x;
    const unsigned long long q = static_cast<unsigned long long>((static_cast<unsigned __int128>(NEG_INV_M) * y) >> 64);
    const unsigned long long r = y - M * q;
    x = r - M * (r >= M);
    return *this;
  }
  ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
  ModInt pow(long long e) const {
    if (e < 0) return inv().pow(-e);
    ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
  }
  ModInt inv() const {
    unsigned a = M, b = x; int y = 0, z = 1;
    for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
    assert(a == 1U); return ModInt(y);
  }
  ModInt operator+() const { return *this; }
  ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
  ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
  ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
  ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
  ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
  template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
  template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
  template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
  template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
  explicit operator bool() const { return x; }
  bool operator==(const ModInt &a) const { return (x == a.x); }
  bool operator!=(const ModInt &a) const { return (x != a.x); }
  friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
unsigned ModInt::M;
unsigned long long ModInt::NEG_INV_M;
// !!!Use ModInt::setM!!!
////////////////////////////////////////////////////////////////////////////////
using Mint = ModInt;
constexpr int LIM_INV = 200'010;
Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV];
void prepare() {
  inv[1] = 1;
  for (int i = 2; i < LIM_INV; ++i) {
    inv[i] = -((Mint::M / i) * inv[Mint::M % i]);
  }
  fac[0] = invFac[0] = 1;
  for (int i = 1; i < LIM_INV; ++i) {
    fac[i] = fac[i - 1] * i;
    invFac[i] = invFac[i - 1] * inv[i];
  }
}
Mint binom(Int n, Int k) {
  if (n < 0) {
    if (k >= 0) {
      return ((k & 1) ? -1 : +1) * binom(-n + k - 1, k);
    } else if (n - k >= 0) {
      return (((n - k) & 1) ? -1 : +1) * binom(-k - 1, n - k);
    } else {
      return 0;
    }
  } else {
    if (0 <= k && k <= n) {
      assert(n < LIM_INV);
      return fac[n] * invFac[k] * invFac[n - k];
    } else {
      return 0;
    }
  }
}
// N^N * N(N+1)/2 * N(N-1)
// F, [L, R], S != T
Mint slow(int N) {
  Mint ans = 0;
  for (int L = 1; L <= N; ++L) for (int R = L; R <= N; ++R) {
    vector<int> can(N + 1, 0);
    for (int x = 1; L * x <= N; ++x) {
      for (int d = L * x; d <= R * x && d <= N; ++d) can[d] = 1;
    }
    Mint here = 0;
    // direct
    for (int d = 1; d <= N - 1; ++d) if (can[d]) {
      here += fac[N] * invFac[N - (d + 1)] * Mint(N).pow(N - d);
    }
    // cycle
    for (int a = 0; a <= N - 1; ++a) for (int b = 0; a + b <= N - 1; ++b) for (int c = 1; a + b + c <= N; ++c) {
      const int d = a + b, m = b + c;
      if (d == 0) continue;
      if (can[d]) continue;
      if (L < R || d % __gcd(L, m) == 0) {
        here += fac[N] * invFac[N - (a + b + c)] * Mint(N).pow(N - (a + b + c));
      }
    }
    ans += here;
  }
  return ans;
}
Mint slow2(int N) {
  vector<Mint> path(N, 0);
  for (int L = 1; L <= N; ++L) for (int R = L; R <= N; ++R) {
    vector<int> can(N, 0);
    for (int x = 0; L * x <= N; ++x) {
      for (int d = L * x; d <= min(R * x, N - 1); ++d) {
        can[d] = 1;
      }
    }
    for (int d = 0; d <= N - 1; ++d) if (can[d]) {
      path[d] += 1;
    }
  }
cerr<<"N = "<<N<<": path = "<<path<<endl;
  for (int d = 1; d <= N - 1; ++d) path[d] += path[d - 1];
  Mint ans = 0;
  // fix rho
  for (int h = 0; h <= N - 1; ++h) for (int m = 1; h + m <= N; ++m) {
    const Mint way = fac[N] * invFac[N - (h + m)] * Mint(N).pow(N - (h + m));
    // before cycle
    if (h) ans += way * path[h - 1];
    // cycle
    for (int L = 1; L <= N; ++L) ans += way * Mint(m / __gcd(L, m));
    ans += way * Mint((Int)N * (N - 1) / 2) * Mint(m);
  }
  // S = T
  ans -= Mint(N).pow(N) * Mint((Int)N * (N + 1) / 2) * N;
  return ans;
}
Mint fast(int N) {
  vector<Mint> path(N, 0);
  // R x < d < L (x+1)
  for (int d = 0; d <= N - 1; ++d) {
    path[d] = Mint((Int)N * (N + 1) / 2);
    if (d == 0) continue;
    /*
    for (int x = 0; x <= N; ++x) {
      const int L = d / (x + 1) + 1;
      const int R = x ? ((d + x - 1) / x - 1) : N;
      if (L <= R) {
        path[d] -= (Int)(R - L + 1) * (R - L + 2) / 2;
      }
    }
    */
    vector<int> xs;
    xs.push_back(0);
    for (int a = 0, b; a < d; a = b) {
      const int k = d / (a + 1);
      b = d / k;
      xs.push_back(b - 1);
    }
    const int mid = xs.size();
    for (int a = 0, b; a < d - 1; a = b) {
      const int k = (d - 1) / (a + 1) + 1;
      b = (d - 1) / (k - 1);
      xs.push_back(b);
    }
    xs.push_back(N);
    inplace_merge(xs.begin(), xs.begin() + mid, xs.end());
// cerr<<"d = "<<d<<": xs = "<<xs<<endl;
    for (int i = 0; i < (int)xs.size(); ++i) if (i == 0 || xs[i - 1] < xs[i]) {
      // (xs[i - 1], xs[i]]
      const int x = xs[i];
      const int L = d / (x + 1) + 1;
      const int R = x ? ((d + x - 1) / x - 1) : N;
      if (L <= R) {
        path[d] -= (i ? Mint(xs[i] - xs[i - 1]) : 1) * Mint((Int)(R - L + 1) * (R - L + 2) / 2);
      }
    }
  }
// cerr<<"N = "<<N<<": path = "<<path<<endl;
  for (int d = 1; d <= N - 1; ++d) path[d] += path[d - 1];
  
  vector<Mint> ws(N + 1);
  ws[N] = 1;
  for (int n = N; --n >= 0; ) ws[n] = ws[n + 1] * N;
  for (int n = 0; n <= N; ++n) ws[n] *= fac[N] * invFac[N - n];
  ws[0] = 0;
  for (int n = 1; n <= N; ++n) ws[n] += ws[n - 1];
  auto get = [&](int nL, int nR) -> Mint {
    return (nL <= nR) ? (ws[nR] - ws[nL - 1]) : 0;
  };
  
  Mint ans = 0;
  // before cycle
  for (int h = 1; h <= N - 1; ++h) {
    ans += get(h + 1, N) * path[h - 1];
  }
  // cycle
  vector<Mint> fs(N + 1, 0), gs(N + 1, 1);
  for (int m = 1; m <= N; ++m) {
    const Mint w = get(m, N);
    /*
    Int sum = 0;
    for (int L = 1; L <= N; ++L) sum += m / __gcd(L, m);
    ans += w * sum;
    */
    fs[m] += w * Mint(m);
    ans += w * Mint((Int)N * (N - 1) / 2) * Mint(m);
  }
  // gcd conv
  for (int i = 1; i <= N; ++i) for (int j = 2 * i; j <= N; j += i) fs[i] += fs[j];
  for (int i = 1; i <= N; ++i) for (int j = 2 * i; j <= N; j += i) gs[i] += gs[j];
  for (int i = 1; i <= N; ++i) fs[i] *= gs[i];
  for (int i = N; i >= 1; --i) for (int j = 2 * i; j <= N; j += i) fs[i] -= fs[j];
  for (int i = 1; i <= N; ++i) ans += fs[i] * inv[i];
  
  // S = T
  ans -= Mint(N).pow(N) * Mint((Int)N * (N + 1) / 2) * N;
  return ans;
}
int main() {
  int N, P;
  for (; ~scanf("%d%d", &N, &P); ) {
    Mint::setM(P);
    prepare();
    
    const Mint ans = fast(N);
    printf("%u\n", ans.x);
#ifdef LOCAL
/*
const Mint slw=slow(N);
const Mint slw2=slow2(N);
cerr<<"slw  = "<<slw<<endl;
cerr<<"slw2 = "<<slw2<<endl;
//*/
#endif
  }
  return 0;
}
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