test/yosupo/find_linear_recurrence.test.cpp

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• Last update: 2024-10-01 17:03:34+09:00
• Problem: https://judge.yosupo.jp/problem/find_linear_recurrence

Depends on

• modintを扱うテンプレート (src/Internal/modint_traits.hpp)
• 剰余の高速化 (src/Internal/Remainder.hpp)
• Berlekamp-Massey (src/Math/berlekamp_massey.hpp)
• 逆元 ($\mathbb{Z}/m\mathbb{Z}$) (src/Math/mod_inv.hpp)
• ModInt (src/Math/ModInt.hpp)

Code

// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/find_linear_recurrence
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
#include <iostream>
#include <vector>
#include "src/Math/ModInt.hpp"
#include "src/Math/berlekamp_massey.hpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(0);
 using Mint= ModInt<998244353>;
 int N;
 cin >> N;
 vector<Mint> a(N);
 for (int i= 0; i < N; i++) cin >> a[i];
 vector<Mint> c= berlekamp_massey(a);
 int d= c.size();
 cout << d << '\n';
 for (int i= 0; i < d; i++) cout << c[i] << " \n"[i == d - 1];
 return 0;
}

#line 1 "test/yosupo/find_linear_recurrence.test.cpp"
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/find_linear_recurrence
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
#include <iostream>
#include <vector>
#line 2 "src/Math/mod_inv.hpp"
#include <utility>
#include <type_traits>
#include <cassert>
template <class Uint> constexpr inline Uint mod_inv(Uint a, Uint mod) {
 std::make_signed_t<Uint> x= 1, y= 0, z= 0;
 for (Uint q= 0, b= mod, c= 0; b;) z= x, x= y, y= z - y * (q= a / b), c= a, a= b, b= c - b * q;
 return assert(a == 1), x < 0 ? mod - (-x) % mod : x % mod;
}
#line 2 "src/Internal/Remainder.hpp"
namespace math_internal {
using namespace std;
using u8= unsigned char;
using u32= unsigned;
using i64= long long;
using u64= unsigned long long;
using u128= __uint128_t;
struct MP_Na {  // mod < 2^32
 u32 mod;
 constexpr MP_Na(): mod(0) {}
 constexpr MP_Na(u32 m): mod(m) {}
 constexpr inline u32 mul(u32 l, u32 r) const { return u64(l) * r % mod; }
 constexpr inline u32 set(u32 n) const { return n; }
 constexpr inline u32 get(u32 n) const { return n; }
 constexpr inline u32 norm(u32 n) const { return n; }
 constexpr inline u32 plus(u64 l, u32 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u32 diff(u64 l, u32 r) const { return l-= r, l >> 63 ? l + mod : l; }
};
template <class u_t, class du_t, u8 B> struct MP_Mo {  // mod < 2^32, mod < 2^62
 u_t mod;
 constexpr MP_Mo(): mod(0), iv(0), r2(0) {}
 constexpr MP_Mo(u_t m): mod(m), iv(inv(m)), r2(-du_t(mod) % mod) {}
 constexpr inline u_t mul(u_t l, u_t r) const { return reduce(du_t(l) * r); }
 constexpr inline u_t set(u_t n) const { return mul(n, r2); }
 constexpr inline u_t get(u_t n) const { return n= reduce(n), n >= mod ? n - mod : n; }
 constexpr inline u_t norm(u_t n) const { return n >= mod ? n - mod : n; }
 constexpr inline u_t plus(u_t l, u_t r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u_t diff(u_t l, u_t r) const { return l-= r, l >> (B - 1) ? l + (mod << 1) : l; }
private:
 u_t iv, r2;
 static constexpr u_t inv(u_t n, int e= 6, u_t x= 1) { return e ? inv(n, e - 1, x * (2 - x * n)) : x; }
 constexpr inline u_t reduce(const du_t &w) const { return u_t(w >> B) + mod - ((du_t(u_t(w) * iv) * mod) >> B); }
};
using MP_Mo32= MP_Mo<u32, u64, 32>;
using MP_Mo64= MP_Mo<u64, u128, 64>;
struct MP_Br {  // 2^20 < mod <= 2^41
 u64 mod;
 constexpr MP_Br(): mod(0), x(0) {}
 constexpr MP_Br(u64 m): mod(m), x((u128(1) << 84) / m) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem(u128(l) * r); }
 static constexpr inline u64 set(u64 n) { return n; }
 constexpr inline u64 get(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 norm(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 plus(u64 l, u64 r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u64 diff(u64 l, u64 r) const { return l-= r, l >> 63 ? l + (mod << 1) : l; }
private:
 u64 x;
 constexpr inline u128 quo(const u128 &n) const { return (n * x) >> 84; }
 constexpr inline u64 rem(const u128 &n) const { return n - quo(n) * mod; }
};
template <class du_t, u8 B> struct MP_D2B1 {  // mod < 2^63, mod < 2^64
 u64 mod;
 constexpr MP_D2B1(): mod(0), s(0), d(0), v(0) {}
 constexpr MP_D2B1(u64 m): mod(m), s(__builtin_clzll(m)), d(m << s), v(u128(-1) / d) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem((u128(l) * r) << s) >> s; }
 constexpr inline u64 set(u64 n) const { return n; }
 constexpr inline u64 get(u64 n) const { return n; }
 constexpr inline u64 norm(u64 n) const { return n; }
 constexpr inline u64 plus(du_t l, u64 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u64 diff(du_t l, u64 r) const { return l-= r, l >> B ? l + mod : l; }
private:
 u8 s;
 u64 d, v;
 constexpr inline u64 rem(const u128 &u) const {
  u128 q= (u >> 64) * v + u;
  u64 r= u64(u) - (q >> 64) * d - d;
  if (r > u64(q)) r+= d;
  if (r >= d) r-= d;
  return r;
 }
};
using MP_D2B1_1= MP_D2B1<u64, 63>;
using MP_D2B1_2= MP_D2B1<u128, 127>;
template <class u_t, class MP> constexpr u_t pow(u_t x, u64 k, const MP &md) {
 for (u_t ret= md.set(1);; x= md.mul(x, x))
  if (k & 1 ? ret= md.mul(ret, x) : 0; !(k>>= 1)) return ret;
}
}
#line 3 "src/Internal/modint_traits.hpp"
namespace math_internal {
struct m_b {};
struct s_b: m_b {};
}
template <class mod_t> constexpr bool is_modint_v= std::is_base_of_v<math_internal::m_b, mod_t>;
template <class mod_t> constexpr bool is_staticmodint_v= std::is_base_of_v<math_internal::s_b, mod_t>;
#line 6 "src/Math/ModInt.hpp"
namespace math_internal {
template <class MP, u64 MOD> struct SB: s_b {
protected:
 static constexpr MP md= MP(MOD);
};
template <class U, class B> struct MInt: public B {
 using Uint= U;
 static constexpr inline auto mod() { return B::md.mod; }
 constexpr MInt(): x(0) {}
 template <class T, typename= enable_if_t<is_modint_v<T> && !is_same_v<T, MInt>>> constexpr MInt(T v): x(B::md.set(v.val() % B::md.mod)) {}
 constexpr MInt(__int128_t n): x(B::md.set((n < 0 ? ((n= (-n) % B::md.mod) ? B::md.mod - n : n) : n % B::md.mod))) {}
 constexpr MInt operator-() const { return MInt() - *this; }
#define FUNC(name, op) \
 constexpr MInt name const { \
  MInt ret; \
  return ret.x= op, ret; \
 }
 FUNC(operator+(const MInt & r), B::md.plus(x, r.x))
 FUNC(operator-(const MInt & r), B::md.diff(x, r.x))
 FUNC(operator*(const MInt & r), B::md.mul(x, r.x))
 FUNC(pow(u64 k), math_internal::pow(x, k, B::md))
#undef FUNC
 constexpr MInt operator/(const MInt &r) const { return *this * r.inv(); }
 constexpr MInt &operator+=(const MInt &r) { return *this= *this + r; }
 constexpr MInt &operator-=(const MInt &r) { return *this= *this - r; }
 constexpr MInt &operator*=(const MInt &r) { return *this= *this * r; }
 constexpr MInt &operator/=(const MInt &r) { return *this= *this / r; }
 constexpr bool operator==(const MInt &r) const { return B::md.norm(x) == B::md.norm(r.x); }
 constexpr bool operator!=(const MInt &r) const { return !(*this == r); }
 constexpr bool operator<(const MInt &r) const { return B::md.norm(x) < B::md.norm(r.x); }
 constexpr inline MInt inv() const { return mod_inv<U>(val(), B::md.mod); }
 constexpr inline Uint val() const { return B::md.get(x); }
 friend ostream &operator<<(ostream &os, const MInt &r) { return os << r.val(); }
 friend istream &operator>>(istream &is, MInt &r) {
  i64 v;
  return is >> v, r= MInt(v), is;
 }
private:
 Uint x;
};
template <u64 MOD> using MP_B= conditional_t < (MOD < (1 << 30)) & MOD, MP_Mo32, conditional_t < MOD < (1ull << 32), MP_Na, conditional_t<(MOD < (1ull << 62)) & MOD, MP_Mo64, conditional_t<MOD<(1ull << 41), MP_Br, conditional_t<MOD<(1ull << 63), MP_D2B1_1, MP_D2B1_2>>>>>;
template <u64 MOD> using ModInt= MInt < conditional_t<MOD<(1 << 30), u32, u64>, SB<MP_B<MOD>, MOD>>;
}
using math_internal::ModInt;
#line 3 "src/Math/berlekamp_massey.hpp"
// a[n] = c[0] * a[n-1] + c[1] * a[n-2] + ... + c[d-1] * a[n-d]
// return c
template <class K> std::vector<K> berlekamp_massey(const std::vector<K> &a) {
 size_t n= a.size(), d= 0, m= 0, i, j;
 if (n == 0) return {};
 std::vector<K> c(n), b(n), tmp;
 K x= 1, y, coef;
 for (c[0]= b[0]= 1, i= 0; i < n; ++i) {
  for (++m, y= a[i], j= 1; j <= d; ++j) y+= c[j] * a[i - j];
  if (y == K()) continue;
  for (tmp= c, coef= y / x, j= m; j < n; ++j) c[j]-= coef * b[j - m];
  if (2 * d <= i) d= i + 1 - d, b= tmp, x= y, m= 0;
 }
 c.resize(d + 1), c.erase(c.begin());
 for (auto &x: c) x= -x;
 return c;
}
#line 8 "test/yosupo/find_linear_recurrence.test.cpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(0);
 using Mint= ModInt<998244353>;
 int N;
 cin >> N;
 vector<Mint> a(N);
 for (int i= 0; i < N; i++) cin >> a[i];
 vector<Mint> c= berlekamp_massey(a);
 int d= c.size();
 cout << d << '\n';
 for (int i= 0; i < d; i++) cout << c[i] << " \n"[i == d - 1];
 return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++-13	example_00	AC	5 ms	4 MB
g++-13	example_01	AC	4 ms	4 MB
g++-13	example_02	AC	4 ms	4 MB
g++-13	example_03	AC	4 ms	4 MB
g++-13	issue_1253_00	AC	4 ms	4 MB
g++-13	long_00	AC	5 ms	4 MB
g++-13	long_01	AC	104 ms	4 MB
g++-13	long_02	AC	107 ms	4 MB
g++-13	random_00	AC	251 ms	4 MB
g++-13	random_01	AC	251 ms	4 MB
g++-13	random_02	AC	81 ms	4 MB
g++-13	random_03	AC	141 ms	4 MB
g++-13	suffix_zero_00	AC	251 ms	4 MB
g++-13	suffix_zero_01	AC	251 ms	4 MB
g++-13	suffix_zero_02	AC	81 ms	4 MB
g++-13	suffix_zero_03	AC	141 ms	4 MB
g++-13	zero_00	AC	6 ms	4 MB
g++-13	zero_01	AC	5 ms	4 MB
clang++-18	example_00	AC	5 ms	4 MB
clang++-18	example_01	AC	4 ms	4 MB
clang++-18	example_02	AC	4 ms	4 MB
clang++-18	example_03	AC	4 ms	4 MB
clang++-18	issue_1253_00	AC	4 ms	4 MB
clang++-18	long_00	AC	5 ms	4 MB
clang++-18	long_01	AC	95 ms	4 MB
clang++-18	long_02	AC	97 ms	4 MB
clang++-18	random_00	AC	228 ms	4 MB
clang++-18	random_01	AC	228 ms	4 MB
clang++-18	random_02	AC	73 ms	4 MB
clang++-18	random_03	AC	127 ms	4 MB
clang++-18	suffix_zero_00	AC	228 ms	4 MB
clang++-18	suffix_zero_01	AC	229 ms	4 MB
clang++-18	suffix_zero_02	AC	73 ms	4 MB
clang++-18	suffix_zero_03	AC	128 ms	4 MB
clang++-18	zero_00	AC	5 ms	4 MB
clang++-18	zero_01	AC	5 ms	4 MB

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