#963322 (C++23) No.2670 Sum of Products of Interval Lengths

提出ソース
結果

問題	No.2670 Sum of Products of Interval Lengths
ユーザー	Misuki
提出日時	2024-03-21 22:52:59
言語	C++23 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	172 ms / 2,000 ms
コード長	13,483 bytes
コンパイル時間	3,193 ms
コンパイル使用メモリ	214,268 KB
実行使用メモリ	22,804 KB
最終ジャッジ日時	2024-09-30 10:23:22
合計ジャッジ時間	5,511 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
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テストケース

テストケース表示
入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 5,248 KB
testcase_01	AC	171 ms 22,804 KB
testcase_02	AC	80 ms 12,316 KB
testcase_03	AC	20 ms 5,856 KB
testcase_04	AC	80 ms 12,428 KB
testcase_05	AC	167 ms 21,284 KB
testcase_06	AC	167 ms 21,352 KB
testcase_07	AC	81 ms 12,312 KB
testcase_08	AC	20 ms 5,856 KB
testcase_09	AC	80 ms 12,436 KB
testcase_10	AC	164 ms 21,160 KB
testcase_11	AC	169 ms 21,480 KB
testcase_12	AC	167 ms 22,804 KB
testcase_13	AC	170 ms 22,804 KB
testcase_14	AC	169 ms 22,800 KB
testcase_15	AC	169 ms 22,796 KB
testcase_16	AC	172 ms 22,800 KB
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ソースコード

raw source code
#pragma GCC optimize("O2")
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#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 <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <variant>

#if __cplusplus >= 202002L

#include <bit>
#include <compare>
#include <concepts>
#include <numbers>
#include <ranges>
#include <span>

#else

#define ssize(v) (int)(v).size()
#define popcount(x) __builtin_popcountll(x)
constexpr int bit_width(const unsigned int x) { return x == 0 ? 0 : ((sizeof(unsigned int) * CHAR_BIT) - __builtin_clz(x)); }
constexpr int bit_width(const unsigned long long x) { return x == 0 ? 0 : ((sizeof(unsigned long long) * CHAR_BIT) - __builtin_clzll(x)); }
constexpr int countr_zero(const unsigned int x) { return x == 0 ? sizeof(unsigned int) * CHAR_BIT : __builtin_ctz(x); }
constexpr int countr_zero(const unsigned long long x) { return x == 0 ? sizeof(unsigned long long) * CHAR_BIT : __builtin_ctzll(x); }
constexpr unsigned int bit_ceil(const unsigned int x) { return x == 0 ? 1 : (popcount(x) == 1 ? x : (1u << bit_width(x))); }
constexpr unsigned long long bit_ceil(const unsigned long long x) { return x == 0 ? 1 : (popcount(x) == 1 ? x : (1ull << bit_width(x))); }

#endif

//#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

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>;

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

//lagrange inversion formula:
//  let f(x) be composition inverse of g(x) (i.e. f(g(x)) = x) and [x^0]f(x) = [x^0]g(x) = 0, [x^1]f(x) != 0, [x^1]g(x) != 0, then
//  [x^n]g(x)^k = k/n [x^{n - k}] (x / f(x))^n
//  [x^n]g(x) = 1/n [x^{n - 1}] (x / f(x))^n (for k = 1)

template<class Mint>
struct FPS : vector<Mint> {

  static function<vector<Mint>(vector<Mint>, vector<Mint>)> conv;

  FPS(vector<Mint> v) : vector<Mint>(v) {}

  using vector<Mint>::vector;
  FPS& operator+=(FPS b) {
    if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0);
    for(int i = 0; i < ssize(b); i++)
      (*this)[i] += b[i];
    return *this;
  }

  FPS& operator-=(FPS b) {
    if (ssize(*this) < ssize(b)) this -> resize(ssize(b), 0);
    for(int i = 0; i < ssize(b); i++)
      (*this)[i] -= b[i];
    return *this;
  }

  FPS& operator*=(FPS b) {
    auto c = conv(*this, b);
    this -> resize(ssize(*this) + ssize(b) - 1);
    copy(c.begin(), c.end(), this -> begin());
    return *this;
  }

  FPS& operator*=(Mint b) {
    for(int i = 0; i < ssize(*this); i++)
      (*this)[i] *= b;
    return *this;
  }

  FPS& operator/=(Mint b) {
    b = Mint(1) / b;
    for(int i = 0; i < ssize(*this); i++)
      (*this)[i] *= b;
    return *this;
  }

  FPS shrink() {
    FPS F = *this;
    int size = ssize(F);
    while(size and F[size - 1] == 0) size -= 1;
    F.resize(size);
    return F;
  }

  FPS integral() {
    if (this -> empty()) return {0};
    vector<Mint> Inv(ssize(*this) + 1);
    Inv[1] = 1;
    for(int i = 2; i < ssize(Inv); i++)
      Inv[i] = (Mint::get_mod() - Mint::get_mod() / i) * Inv[Mint::get_mod() % i];
    FPS Q(ssize(*this) + 1, 0);
    for(int i = 0; i < ssize(*this); i++)
      Q[i + 1] = (*this)[i] * Inv[i + 1];
    return Q;
  }

  FPS derivative() {
    assert(!this -> empty());
    FPS Q(ssize(*this) - 1);
    for(int i = 1; i < ssize(*this); i++)
      Q[i - 1] = (*this)[i] * i;
    return Q;
  }

  Mint eval(Mint x) {
    Mint base = 1, res = 0;
    for(int i = 0; i < ssize(*this); i++, base *= x)
      res += (*this)[i] * base;
    return res;
  }

  FPS inv(int k) { // 1 / FPS (mod x^k)
    assert(!this -> empty() and (*this)[0] != 0);
    FPS Q(1, 1 / (*this)[0]);
    for(int i = 1; (1 << (i - 1)) < k; i++) {
      FPS P = (*this);
      P.resize(1 << i, 0);
      Q = Q * (FPS(1, 2) - P * Q);
      Q.resize(1 << i, 0);
    }
    Q.resize(k);
    return Q;
  }

  array<FPS, 2> div(FPS G) {
    FPS F = this -> shrink();
    G = G.shrink();
    assert(!G.empty());
    if (ssize(G) > ssize(F))
      return {{{}, F}};
    int n = ssize(F) - ssize(G) + 1;
    auto FR = F, GR = G;
    ranges::reverse(FR);
    ranges::reverse(GR);
    FPS Q = FR * GR.inv(n);
    Q.resize(n);
    ranges::reverse(Q);
    return {Q, (F - G * Q).shrink()};
  }

  FPS log(int k) {
    assert(!this -> empty() and (*this)[0] == 1);
    FPS Q = *this;
    Q = (Q.derivative() * Q.inv(k));
    Q.resize(k - 1);
    return Q.integral();
  }

  FPS exp(int k) {
    assert(!this -> empty() and (*this)[0] == 0);
    FPS Q(1, 1);
    for(int i = 1; (1 << (i - 1)) < k; i++) {
      FPS P = (*this);
      P.resize(1 << i, 0);
      Q = Q * (FPS(1, 1) + P - Q.log(1 << i));
      Q.resize(1 << i, 0);
    }
    Q.resize(k);
    return Q;
  }

  FPS pow(ll idx, int k) {
    if (idx == 0) {
      FPS res(k, 0);
      res[0] = 1;
      return res;
    }
    for(int i = 0; i < ssize(*this) and i * idx < k; i++) {
      if ((*this)[i] != 0) {
        Mint Inv = 1 / (*this)[i];
        FPS Q(ssize(*this) - i);
        for(int j = i; j < ssize(*this); j++)
          Q[j - i] = (*this)[j] * Inv;
        Q = (Q.log(k) * idx).exp(k);
        FPS Q2(k, 0);
        Mint Pow = (*this)[i].pow(idx);
        for(int j = 0; j + i * idx < k; j++)
          Q2[j + i * idx] = Q[j] * Pow;
        return Q2;
      }
    } 
    return FPS(k, 0);
  }

  vector<Mint> multieval(vector<Mint> xs) {
    int n = ssize(xs);
    vector<FPS> data(2 * n);
    for(int i = 0; i < n; i++)
      data[n + i] = {-xs[i], 1};
    for(int i = n - 1; i > 0; i--)
      data[i] = data[i << 1] * data[i << 1 | 1];
    data[1] = (this -> div(data[1]))[1];
    for(int i = 1; i < n; i++) {
      data[i << 1] = data[i].div(data[i << 1])[1];
      data[i << 1 | 1] = data[i].div(data[i << 1 | 1])[1];
    }
    vector<Mint> res(n);
    for(int i = 0; i < n; i++)
      res[i] = data[n + i].empty() ? 0 : data[n + i][0];
    return res;
  }

  static vector<Mint> interpolate(vector<Mint> xs, vector<Mint> ys) {
    assert(ssize(xs) == ssize(ys));
    int n = ssize(xs);
    vector<FPS> data(2 * n), res(2 * n);
    for(int i = 0; i < n; i++)
      data[n + i] = {-xs[i], 1};
    for(int i = n - 1; i > 0; i--)
      data[i] = data[i << 1] * data[i << 1 | 1];
    res[1] = data[1].derivative().div(data[1])[1];
    for(int i = 1; i < n; i++) {
      res[i << 1] = res[i].div(data[i << 1])[1];
      res[i << 1 | 1] = res[i].div(data[i << 1 | 1])[1];
    }
    for(int i = 0; i < n; i++)
      res[n + i][0] = ys[i] / res[n + i][0];
    for(int i = n - 1; i > 0; i--)
      res[i] = res[i << 1] * data[i << 1 | 1] + res[i << 1 | 1] * data[i << 1];
    return res[1];
  }

  static vector<Mint> allProd(vector<FPS> &fs) {
    if (fs.empty()) return {1};
    auto dfs = [&](int l, int r, auto self) -> FPS {
      if (l + 1 == r)
        return fs[l];
      else
        return self(l, (l + r) / 2, self) * self((l + r) / 2, r, self);
    };
    return dfs(0, ssize(fs), dfs);
  }

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

NTT ntt;
using fps = FPS<mint>;
template<>
function<vector<mint>(vector<mint>, vector<mint>)> fps::conv = ntt.conv;

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

  int n; cin >> n;
  ll m; cin >> m;

  fps f(n + 1);
  f[1] = 1;
  for(int i = 2; i <= n; i++)
    f[i] = -f[i - 2] + f[i - 1];

  for(int i = 0; i <= n; i++)
    f[i] *= max(m + 1 - i, 0ll);
  f *= -1;
  f[0] = 1;

  cout << f.inv(n + 1).back() << '\n';

  return 0;
}
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結果

テストケース

ソースコード
