結果
| 問題 |
No.1907 DETERMINATION
|
| ユーザー |
 drken1215
|
| 提出日時 | 2025-08-19 23:13:30 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 782 ms / 4,000 ms |
| コード長 | 60,367 bytes |
| コンパイル時間 | 4,612 ms |
| コンパイル使用メモリ | 333,284 KB |
| 実行使用メモリ | 7,720 KB |
| 最終ジャッジ日時 | 2025-08-19 23:14:06 |
| 合計ジャッジ時間 | 34,116 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 63 |
ソースコード
//
// det(M1x + M0) の計算, for given N x N matrix M0, M1, in O(N^3)
//
// verified:
// yukicoder No.1907 DETERMINATION
// https://yukicoder.me/problems/no/1907
//
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
//------------------------------//
// Utility
//------------------------------//
template<class S, class T> inline bool chmax(S &a, T b) { return (a < b ? a = b, 1 : 0); }
template<class S, class T> inline bool chmin(S &a, T b) { return (a > b ? a = b, 1 : 0); }
using pint = pair<int, int>;
using pll = pair<long long, long long>;
using tint = array<int, 3>;
using tll = array<long long, 3>;
using fint = array<int, 4>;
using fll = array<long long, 4>;
using qint = array<int, 5>;
using qll = array<long long, 5>;
using vint = vector<int>;
using vll = vector<long long>;
using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T>
using min_priority_queue = priority_queue<T, vector<T>, greater<T>>;
#define REP(i, a) for (long long i = 0; i < (long long)(a); i++)
#define REP2(i, a, b) for (long long i = a; i < (long long)(b); i++)
#define RREP(i, a) for (long long i = (a)-1; i >= (long long)(0); --i)
#define RREP2(i, a, b) for (long long i = (b)-1; i >= (long long)(a); --i)
#define EB emplace_back
#define PB push_back
#define MP make_pair
#define MT make_tuple
#define FI first
#define SE second
#define ALL(x) x.begin(), x.end()
#define COUT(x) cout << #x << " = " << (x) << " (L" << __LINE__ << ")" << endl
// debug stream
template<class T1, class T2> ostream& operator << (ostream &s, pair<T1,T2> P)
{ return s << '<' << P.first << ", " << P.second << '>'; }
template<class T> ostream& operator << (ostream &s, array<T, 3> P)
{ return s << '<' << P[0] << ", " << P[1] << ", " << P[2] << '>'; }
template<class T> ostream& operator << (ostream &s, array<T, 4> P)
{ return s << '<' << P[0] << ", " << P[1] << ", " << P[2] << ", " << P[3] << '>'; }
template<class T> ostream& operator << (ostream &s, vector<T> P)
{ for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; }
template<class T> ostream& operator << (ostream &s, deque<T> P)
{ for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; }
template<class T> ostream& operator << (ostream &s, vector<vector<T> > P)
{ for (int i = 0; i < P.size(); ++i) { s << endl << P[i]; } return s << endl; }
template<class T> ostream& operator << (ostream &s, set<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T> ostream& operator << (ostream &s, multiset<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T> ostream& operator << (ostream &s, unordered_set<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T1, class T2> ostream& operator << (ostream &s, map<T1,T2> P)
{ for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; }
template<class T1, class T2> ostream& operator << (ostream &s, unordered_map<T1,T2> P)
{ for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; }
// 4-neighbor
const vector<int> dx = {1, 0, -1, 0};
const vector<int> dy = {0, 1, 0, -1};
// 8-neighbor
const vector<int> dx8 = {1, 0, -1, 0, 1, -1, 1, -1};
const vector<int> dy8 = {0, 1, 0, -1, 1, 1, -1, -1};
// min non-negative i such that n <= 2^i
int ceil_pow2(int n) {
int i = 0;
while ((1U << i) < (unsigned int)(n)) i++;
return i;
}
// num of i such that (x & (1 << i)) != 0
int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(unsigned int x) { return __builtin_popcount(x); }
int popcnt(long long x) { return __builtin_popcountll(x); }
int popcnt(unsigned long long x) { return __builtin_popcountll(x); }
// min non-negative i such that (x & (1 << i)) != 0
int bsf(int x) { return __builtin_ctz(x); }
int bsf(unsigned int x) { return __builtin_ctz(x); }
int bsf(long long x) { return __builtin_ctzll(x); }
int bsf(unsigned long long x) { return __builtin_ctzll(x); }
// max non-negative i such that (x & (1 << i)) != 0
int bsr(int x) { return 8 * (int)sizeof(int) - 1 - __builtin_clz(x); }
int bsr(unsigned int x) { return 8 * (int)sizeof(unsigned int) - 1 - __builtin_clz(x); }
int bsr(long long x) { return 8 * (int)sizeof(long long) - 1 - __builtin_clzll(x); }
int bsr(unsigned long long x) { return 8 * (int)sizeof(unsigned long long) - 1 - __builtin_clzll(x); }
// floor, ceil
template<class T> T floor(T a, T b) {
if (a % b == 0 || a >= 0) return a / b;
else return -((-a) / b) - 1;
}
template<class T> T ceil(T x, T y) {
return floor(x + y - 1, y);
}
// kth root
// N < 2^64, K <= 64
uint64_t kth_root(uint64_t N, uint64_t K) {
assert(K >= 1);
if (N <= 1 || K == 1) return N;
if (K >= 64) return 1;
if (N == uint64_t(-1)) --N;
auto mul = [&](uint64_t x, uint64_t y) -> uint64_t {
if (x < UINT_MAX && y < UINT_MAX) return x * y;
if (x == uint64_t(-1) || y == uint64_t(-1)) return uint64_t(-1);
return (x <= uint64_t(-1) / y ? x * y : uint64_t(-1));
};
auto power = [&](uint64_t x, uint64_t k) -> uint64_t {
if (k == 0) return 1ULL;
uint64_t res = 1ULL;
while (k) {
if (k & 1) res = mul(res, x);
x = mul(x, x);
k >>= 1;
}
return res;
};
uint64_t res;
if (K == 2) res = sqrtl(N) - 1;
else if (K == 3) res = cbrt(N) - 1;
else res = pow(N, nextafter(1 / double(K), 0));
while (power(res + 1, K) <= N) ++res;
return res;
}
// xor128による乱数生成、周期は2^128-1
unsigned int randInt() {
static unsigned int tx = 123456789, ty=362436069, tz=521288629, tw=88675123;
unsigned int tt = (tx^(tx<<11));
tx = ty; ty = tz; tz = tw;
return ( tw=(tw^(tw>>19))^(tt^(tt>>8)) );
}
int randInt(int minv, int maxv) {
return randInt() % (maxv - minv + 1) + minv;
}
long long randInt(long long minv, long long maxv) {
long long a = randInt(), b = randInt();
return (a * (1LL<<29) + b) % (maxv - minv + 1) + minv;
}
template<class T> void shuffle(vector<T>& vec) {
int n = vec.size();
for (int i = n - 1; i > 0; --i) {
int k = randInt() % (i + 1);
swap(vec[i], vec[k]);
}
}
// int 128
i128 to_integer(const string &s) {
i128 res = 0;
for (auto c : s) {
if (isdigit(c)) res = res * 10 + (c - '0');
}
if (s[0] == '-') res *= -1;
return res;
}
istream& operator >> (istream &is, i128 &x) {
string s;
is >> s;
x = to_integer(s);
return is;
}
ostream& operator << (ostream &os, const i128 &x) {
i128 ax = (x >= 0 ? x : -x);
char buffer[128];
char *d = end(buffer);
do {
--d;
*d = "0123456789"[ax % 10];
ax /= 10;
} while (ax != 0);
if (x < 0) {
--d;
*d = '-';
}
int len = end(buffer) - d;
if (os.rdbuf()->sputn(d, len) != len) {
os.setstate(ios_base::badbit);
}
return os;
}
// safe mod
template<class T_VAL, class T_MOD>
constexpr T_VAL safe_mod(T_VAL a, T_MOD m) {
assert(m > 0);
a %= m;
if (a < 0) a += m;
return a;
}
// mod pow
template<class T_VAL, class T_MOD>
constexpr T_VAL mod_pow(T_VAL a, T_VAL n, T_MOD m) {
T_VAL res = 1;
while (n > 0) {
if (n % 2 == 1) res = res * a % m;
a = a * a % m;
n >>= 1;
}
return res;
}
// mod inv
template<class T_VAL, class T_MOD>
constexpr T_VAL mod_inv(T_VAL a, T_MOD m) {
T_VAL b = m, u = 1, v = 0;
while (b > 0) {
T_VAL t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
u %= m;
if (u < 0) u += m;
return u;
}
// modint
template<int MOD = 998244353, bool PRIME = true> struct Fp {
// inner value
unsigned int val;
// constructor
constexpr Fp() : val(0) { }
template<std::signed_integral T> constexpr Fp(T v) {
long long tmp = (long long)(v % (long long)(get_umod()));
if (tmp < 0) tmp += get_umod();
val = (unsigned int)(tmp);
}
template<std::unsigned_integral T> constexpr Fp(T v) {
val = (unsigned int)(v % get_umod());
}
constexpr long long get() const { return val; }
constexpr static int get_mod() { return MOD; }
constexpr static unsigned int get_umod() { return MOD; }
// arithmetic operators
constexpr Fp operator + () const { return Fp(*this); }
constexpr Fp operator - () const { return Fp() - Fp(*this); }
constexpr Fp operator + (const Fp &r) const { return Fp(*this) += r; }
constexpr Fp operator - (const Fp &r) const { return Fp(*this) -= r; }
constexpr Fp operator * (const Fp &r) const { return Fp(*this) *= r; }
constexpr Fp operator / (const Fp &r) const { return Fp(*this) /= r; }
constexpr Fp& operator += (const Fp &r) {
val += r.val;
if (val >= get_umod()) val -= get_umod();
return *this;
}
constexpr Fp& operator -= (const Fp &r) {
val -= r.val;
if (val >= get_umod()) val += get_umod();
return *this;
}
constexpr Fp& operator *= (const Fp &r) {
unsigned long long tmp = val;
tmp *= r.val;
val = (unsigned int)(tmp % get_umod());
return *this;
}
constexpr Fp& operator /= (const Fp &r) {
return *this = *this * r.inv();
}
constexpr Fp pow(long long n) const {
assert(n >= 0);
Fp res(1), mul(*this);
while (n) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
constexpr Fp inv() const {
if (PRIME) {
assert(val);
return pow(get_umod() - 2);
} else {
assert(val);
return mod_inv((long long)(val), get_umod());
}
}
// other operators
constexpr bool operator == (const Fp &r) const {
return this->val == r.val;
}
constexpr bool operator != (const Fp &r) const {
return this->val != r.val;
}
constexpr bool operator < (const Fp &r) const {
return this->val < r.val;
}
constexpr bool operator > (const Fp &r) const {
return this->val > r.val;
}
constexpr bool operator <= (const Fp &r) const {
return this->val <= r.val;
}
constexpr bool operator >= (const Fp &r) const {
return this->val >= r.val;
}
constexpr Fp& operator ++ () {
++val;
if (val == get_umod()) val = 0;
return *this;
}
constexpr Fp& operator -- () {
if (val == 0) val = get_umod();
--val;
return *this;
}
constexpr Fp operator ++ (int) {
Fp res = *this;
++*this;
return res;
}
constexpr Fp operator -- (int) {
Fp res = *this;
--*this;
return res;
}
friend constexpr istream& operator >> (istream &is, Fp<MOD> &x) {
long long tmp = 1;
is >> tmp;
tmp = tmp % (long long)(get_umod());
if (tmp < 0) tmp += get_umod();
x.val = (unsigned int)(tmp);
return is;
}
friend constexpr ostream& operator << (ostream &os, const Fp<MOD> &x) {
return os << x.val;
}
friend constexpr Fp<MOD> pow(const Fp<MOD> &r, long long n) {
return r.pow(n);
}
friend constexpr Fp<MOD> inv(const Fp<MOD> &r) {
return r.inv();
}
};
// dynamic modint
struct DynamicModint {
using mint = DynamicModint;
// static menber
static int MOD;
// inner value
unsigned int val;
// constructor
DynamicModint() : val(0) { }
template<std::signed_integral T> DynamicModint(T v) {
long long tmp = (long long)(v % (long long)(get_umod()));
if (tmp < 0) tmp += get_umod();
val = (unsigned int)(tmp);
}
template<std::unsigned_integral T> DynamicModint(T v) {
val = (unsigned int)(v % get_umod());
}
long long get() const { return val; }
static int get_mod() { return MOD; }
static unsigned int get_umod() { return MOD; }
static void set_mod(int mod) { MOD = mod; }
// arithmetic operators
mint operator + () const { return mint(*this); }
mint operator - () const { return mint() - mint(*this); }
mint operator + (const mint &r) const { return mint(*this) += r; }
mint operator - (const mint &r) const { return mint(*this) -= r; }
mint operator * (const mint &r) const { return mint(*this) *= r; }
mint operator / (const mint &r) const { return mint(*this) /= r; }
mint& operator += (const mint &r) {
val += r.val;
if (val >= get_umod()) val -= get_umod();
return *this;
}
mint& operator -= (const mint &r) {
val -= r.val;
if (val >= get_umod()) val += get_umod();
return *this;
}
mint& operator *= (const mint &r) {
unsigned long long tmp = val;
tmp *= r.val;
val = (unsigned int)(tmp % get_umod());
return *this;
}
mint& operator /= (const mint &r) {
return *this = *this * r.inv();
}
mint pow(long long n) const {
assert(n >= 0);
mint res(1), mul(*this);
while (n) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
mint inv() const {
assert(val);
return mod_inv((long long)(val), get_umod());
}
// other operators
bool operator == (const mint &r) const {
return this->val == r.val;
}
bool operator != (const mint &r) const {
return this->val != r.val;
}
bool operator < (const mint &r) const {
return this->val < r.val;
}
bool operator > (const mint &r) const {
return this->val > r.val;
}
bool operator <= (const mint &r) const {
return this->val <= r.val;
}
bool operator >= (const mint &r) const {
return this->val >= r.val;
}
mint& operator ++ () {
++val;
if (val == get_umod()) val = 0;
return *this;
}
mint& operator -- () {
if (val == 0) val = get_umod();
--val;
return *this;
}
mint operator ++ (int) {
mint res = *this;
++*this;
return res;
}
mint operator -- (int) {
mint res = *this;
--*this;
return res;
}
friend istream& operator >> (istream &is, mint &x) {
long long tmp = 1;
is >> tmp;
tmp = tmp % (long long)(get_umod());
if (tmp < 0) tmp += get_umod();
x.val = (unsigned int)(tmp);
return is;
}
friend ostream& operator << (ostream &os, const mint &x) {
return os << x.val;
}
friend mint pow(const mint &r, long long n) {
return r.pow(n);
}
friend mint inv(const mint &r) {
return r.inv();
}
};
int DynamicModint::MOD;
// Binomial coefficient
template<class mint> struct BiCoef {
vector<mint> fact_, inv_, finv_;
constexpr BiCoef() {}
constexpr BiCoef(int n) : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
init(n);
}
constexpr void init(int n) {
fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
int MOD = fact_[0].get_mod();
for(int i = 2; i < n; i++){
fact_[i] = fact_[i-1] * i;
inv_[i] = -inv_[MOD%i] * (MOD/i);
finv_[i] = finv_[i-1] * inv_[i];
}
}
constexpr mint com(int n, int k) const {
if (n < k || n < 0 || k < 0) return 0;
return fact_[n] * finv_[k] * finv_[n-k];
}
constexpr mint fact(int n) const {
if (n < 0) return 0;
return fact_[n];
}
constexpr mint inv(int n) const {
if (n < 0) return 0;
return inv_[n];
}
constexpr mint finv(int n) const {
if (n < 0) return 0;
return finv_[n];
}
};
// all inverse
template<class mint> vector<mint> all_inverse(const vector<mint> &v) {
for (auto &&vi : v) assert(vi != mint(0));
int N = (int)v.size();
vector<mint> res(N + 1, mint(1));
for (int i = 0; i < N; i++) res[i + 1] = res[i] * v[i];
mint t = res.back().inv();
res.pop_back();
for (int i = N - 1; i >= 0; i--) res[i] *= t, t *= v[i];
return res;
}
// calc primitive root
constexpr int calc_primitive_root(long long m) {
if (m == 1) return -1;
if (m == 2) return 1;
if (m == 998244353) return 3;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 645922817) return 3;
if (m == 897581057) return 3;
long long divs[20] = {};
divs[0] = 2;
long long cnt = 1;
long long x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (long long i = 3; i * i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) x /= i;
}
}
if (x > 1) divs[cnt++] = x;
for (long long g = 2; ; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (mod_pow(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
//------------------------------//
// NTT
//------------------------------//
// NTT setup
template<class mint, int MOD = mint::get_mod(), int g = calc_primitive_root(mint::get_mod())>
struct ntt_setup {
static constexpr int bsf_constexpr(unsigned int x) {
int i = 0;
while (!(x & (1 << i))) i++;
return i;
};
static constexpr int rank = bsf_constexpr(MOD - 1);
array<mint, rank + 1> root, iroot; // root[i]^(2^i) = 1, root[i] * iroot[i] = 1
array<mint, max(0, rank - 1)> rate2, irate2;
array<mint, max(0, rank - 2)> rate3, irate3;
ntt_setup() {
root[rank] = mint(g).pow((MOD - 1) >> rank);
iroot[rank] = root[rank].inv();
for (int i = rank - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
mint prod = 1, iprod = 1;
for (int i = 0; i < rank - 1; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
prod = 1, iprod = 1;
for (int i = 0; i < rank - 2; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
};
// NTT transformation
template<class mint, int MOD = mint::get_mod()>
void ntt_trans(vector<mint> &v) {
int n = (int)v.size();
int h = ceil_pow2(n);
static const ntt_setup<mint> setup;
int len = 0;
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = v[i + offset];
auto r = v[i + offset + p] * rot;
v[i + offset] = l + r;
v[i + offset + p] = l - r;
}
if (s + 1 != (1 << len)) {
rot *= setup.rate2[bsf(~(unsigned int)(s))];
}
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = setup.root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot, rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * MOD * MOD;
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val * rot.val;
auto a2 = 1ULL * v[i + offset + p * 2].val * rot2.val;
auto a3 = 1ULL * v[i + offset + p * 3].val * rot3.val;
auto tmp = 1ULL * mint(a1 + mod2 - a3).val * imag.val;
auto na2 = mod2 - a2;
v[i + offset] = a0 + a2 + a1 + a3;
v[i + offset + p] = a0 + a2 + (mod2 * 2 - (a1 + a3));
v[i + offset + p * 2] = a0 + na2 + tmp;
v[i + offset + p * 3] = a0 + na2 + (mod2 - tmp);
}
if (s + 1 != (1 << len)) {
rot *= setup.rate3[bsf(~(unsigned int)(s))];
}
}
len += 2;
}
}
}
// NTT inv-transformation
template<class mint, int MOD = mint::get_mod()>
void ntt_trans_inv(vector<mint> &v) {
int n = (int)v.size();
int h = ceil_pow2(n);
static const ntt_setup<mint> setup;
int len = h;
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = v[i + offset];
auto r = v[i + offset + p];
v[i + offset] = l + r;
v[i + offset + p] = (unsigned long long)((long long)(MOD) + l.val - r.val) * irot.val;
}
if (s + 1 != (1 << (len - 1))) {
irot *= setup.irate2[bsf(~(unsigned int)(s))];
}
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = setup.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot, irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val;
auto a2 = 1ULL * v[i + offset + p * 2].val;
auto a3 = 1ULL * v[i + offset + p * 3].val;
auto tmp = 1ULL * mint((MOD + a2 - a3) * iimag.val).val;
v[i + offset] = a0 + a1 + a2 + a3;
v[i + offset + p] = (a0 + (MOD - a1) + tmp) * irot.val;
v[i + offset + p * 2] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val;
v[i + offset + p * 3] = (a0 + (MOD - a1) + (MOD - tmp)) * irot3.val;
}
if (s + 1 != (1 << (len - 2))) {
irot *= setup.irate3[bsf(~(unsigned int)(s))];
}
}
len -= 2;
}
}
mint in = mint(n).inv();
for (int i = 0; i < n; i++) v[i] *= in;
}
// naive convolution
template<class T>
vector<T> sub_convolution_naive(const vector<T> &a, const vector<T> &b) {
int n = (int)a.size(), m = (int)b.size();
vector<T> res(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++) for (int i = 0; i < n; i++) res[i + j] += a[i] * b[j];
} else {
for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
}
return res;
}
// ntt convolution
template<class mint>
vector<mint> sub_convolution_ntt(vector<mint> a, vector<mint> b) {
int MOD = mint::get_mod();
int n = (int)a.size(), m = (int)b.size();
if (!n || !m) return {};
int z = (int)bit_ceil((unsigned int)(n + m - 1));
assert((MOD - 1) % z == 0);
a.resize(z), b.resize(z);
ntt_trans(a), ntt_trans(b);
for (int i = 0; i < z; i++) a[i] *= b[i];
ntt_trans_inv(a);
a.resize(n + m - 1);
return a;
}
// convolution in general mod
template<class mint>
vector<mint> convolution(const vector<mint> &a, const vector<mint> &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m) return {};
if (min(n, m) <= 60) return sub_convolution_naive(std::move(a), std::move(b));
if constexpr (std::is_same_v<mint, Fp<998244353>>) return sub_convolution_ntt(a, b);
static constexpr int MOD0 = 754974721; // 2^24
static constexpr int MOD1 = 167772161; // 2^25
static constexpr int MOD2 = 469762049; // 2^26
using mint0 = Fp<MOD0>;
using mint1 = Fp<MOD1>;
using mint2 = Fp<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
vector<mint0> a0(n, 0), b0(m, 0);
vector<mint1> a1(n, 0), b1(m, 0);
vector<mint2> a2(n, 0), b2(m, 0);
for (int i = 0; i < n; ++i) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
for (int i = 0; i < m; ++i) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
auto c0 = sub_convolution_ntt(std::move(a0), std::move(b0));
auto c1 = sub_convolution_ntt(std::move(a1), std::move(b1));
auto c2 = sub_convolution_ntt(std::move(a2), std::move(b2));
vector<mint> res(n + m - 1);
mint mod0 = MOD0, mod01 = mod0 * MOD1;
for (int i = 0; i < n + m - 1; ++i) {
unsigned int y0 = c0[i].val;
unsigned int y1 = (imod0 * (c1[i] - y0)).val;
unsigned int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val;
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
//------------------------------//
// FPS
//------------------------------//
// Formal Power Series
template<class mint> struct FPS : vector<mint> {
static const int SPARSE_BOARDER = 60;
using vector<mint>::vector;
// constructor
constexpr FPS(const vector<mint> &r) : vector<mint>(r) {}
// core operator
constexpr FPS pre(int siz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), siz));
}
constexpr FPS rev() const {
FPS res = *this;
reverse(begin(res), end(res));
return res;
}
constexpr FPS& normalize() {
while (!this->empty() && this->back() == 0) this->pop_back();
return *this;
}
constexpr mint eval(const mint &v) const {
mint res = 0;
for (int i = (int)this->size()-1; i >= 0; --i) {
res *= v;
res += (*this)[i];
}
return res;
}
constexpr int count_terms() const {
int res = 0;
for (int i = 0; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) res++;
return res;
}
// basic operator
constexpr FPS operator - () const noexcept {
FPS res = (*this);
for (int i = 0; i < (int)res.size(); ++i) res[i] = -res[i];
return res;
}
constexpr FPS operator + (const mint &v) const { return FPS(*this) += v; }
constexpr FPS operator + (const FPS &r) const { return FPS(*this) += r; }
constexpr FPS operator - (const mint &v) const { return FPS(*this) -= v; }
constexpr FPS operator - (const FPS &r) const { return FPS(*this) -= r; }
constexpr FPS operator * (const mint &v) const { return FPS(*this) *= v; }
constexpr FPS operator * (const FPS &r) const { return FPS(*this) *= r; }
constexpr FPS operator / (const mint &v) const { return FPS(*this) /= v; }
constexpr FPS operator / (const FPS &r) const { return FPS(*this) /= r; }
constexpr FPS operator % (const FPS &r) const { return FPS(*this) %= r; }
constexpr FPS operator << (int x) const { return FPS(*this) <<= x; }
constexpr FPS operator >> (int x) const { return FPS(*this) >>= x; }
constexpr FPS& operator += (const mint &v) {
if (this->empty()) this->reserve(1), this->resize(1);
(*this)[0] += v;
return *this;
}
constexpr FPS& operator += (const FPS &r) {
if (r.size() > this->size()) this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] += r[i];
return this->normalize();
}
constexpr FPS& operator -= (const mint &v) {
if (this->empty()) this->reserve(1), this->resize(1);
(*this)[0] -= v;
return *this;
}
constexpr FPS& operator -= (const FPS &r) {
if (r.size() > this->size()) this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] -= r[i];
return this->normalize();
}
constexpr FPS& operator *= (const mint &v) {
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= v;
return *this;
}
constexpr FPS& operator *= (const FPS &r) {
return *this = convolution((*this), r);
}
constexpr FPS& operator /= (const mint &v) {
assert(v != 0);
mint iv = v.inv();
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= iv;
return *this;
}
// division, r must be normalized (r.back() must not be 0)
constexpr FPS& operator /= (const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
if (this->size() < r.size()) {
this->clear();
return *this;
}
int need = (int)this->size() - (int)r.size() + 1;
*this = (rev().pre(need) * r.rev().inv(need)).pre(need).rev();
return *this;
}
constexpr FPS& operator %= (const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
FPS q = (*this) / r;
return *this -= q * r;
}
constexpr FPS& operator <<= (int x) {
FPS res(x, 0);
res.insert(res.end(), begin(*this), end(*this));
return *this = res;
}
constexpr FPS& operator >>= (int x) {
FPS res;
res.insert(res.end(), begin(*this) + x, end(*this));
return *this = res;
}
// advanced operation
// df/dx
constexpr FPS diff() const {
int n = (int)this->size();
if (n <= 0) return FPS();
FPS res(n-1);
for (int i = 1; i < n; ++i) res[i-1] = (*this)[i] * i;
return res;
}
// \int f dx
constexpr FPS integral() const {
int n = (int)this->size();
FPS res(n+1, 0);
for (int i = 0; i < n; ++i) res[i+1] = (*this)[i] / (i+1);
return res;
}
// inv(f), f[0] must not be 0
constexpr FPS inv(int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return inv_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>) return inv_ntt_friendly(deg);
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
FPS res({mint(1) / (*this)[0]});
for (int d = 1; d < deg; d <<= 1) {
res = (res + res - res * res * pre(d << 1)).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS inv_ntt_friendly(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
FPS res(deg);
res[0] = mint(1) / (*this)[0];
for (int d = 1; d < deg; d <<= 1) {
FPS g(d * 2), h(d * 2);
mint iv = mint(d * 2).inv();
for (int i = 0; i < min((int)this->size(), d * 2); i++) g[i] = (*this)[i];
for (int i = 0; i < d; i++) h[i] = res[i];
ntt_trans(g), ntt_trans(h);
for (int i = 0; i < d * 2; i++) g[i] *= h[i];
ntt_trans_inv(g);
for (int i = 0; i < d; i++) g[i] = 0;
ntt_trans(g);
for (int i = 0; i < d * 2; i++) g[i] *= h[i];
ntt_trans_inv(g);
for (int i = d; i < min(deg, d * 2); i++) res[i] = -g[i];
}
return res.pre(deg);
}
constexpr FPS inv_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
vector<mint> res(deg);
res[0] = (*this)[0].inv();
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i) break;
r -= val * res[i - k];
}
res[i] = r * res[0];
}
return res;
}
// log(f) = \int f'/f dx, f[0] must be 1
constexpr FPS log(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (count_terms() <= SPARSE_BOARDER) return log_sparse(deg);
if (deg < 0) deg = (int)this->size();
return ((diff() * inv(deg)).pre(deg - 1)).integral();
}
constexpr FPS log_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg), tmp(deg);
for (int i = 0; i < deg - 1; i++) {
mint r = mint(i + 1) * (*this)[i + 1];
for (auto &&[k, val] : dat) {
if (k > i) break;
r -= val * tmp[i - k];
}
tmp[i] = r;
res[i + 1] = r * bc.inv(i + 1);
}
return res;
}
// exp(f), f[0] must be 0
constexpr FPS exp(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
if (count_terms() <= SPARSE_BOARDER) return exp_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>) return exp_ntt_friendly(deg);
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
FPS res(1, 1);
for (int d = 1; d < deg; d <<= 1) {
res = res * (pre(d << 1) - res.log(d << 1) + 1).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS exp_ntt_friendly(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
FPS fiv;
fiv.reserve(deg + 1);
fiv.emplace_back(mint(0));
fiv.emplace_back(mint(1));
auto inplace_integral = [&](FPS &F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while ((int)fiv.size() <= n) {
int i = fiv.size();
fiv.emplace_back((-fiv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for (int i = 1; i <= n; i++) F[i] *= fiv[i];
};
auto inplace_diff = [](FPS &F) -> void {
if (F.empty()) return;
F.erase(begin(F));
mint coef = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coef;
coef++;
}
};
FPS b{1, (1 < (int)this->size() ? (*this)[1] : 0)}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m <<= 1) {
auto y = b;
y.resize(m * 2);
ntt_trans(y);
z1 = z2;
FPS z(m);
for (int i = 0; i < m; i++) z[i] = y[i] * z1[i];
ntt_trans_inv(z);
fill(begin(z), begin(z) + m / 2, mint(0));
ntt_trans(z);
for (int i = 0; i < m; i++) z[i] *= -z1[i];
ntt_trans_inv(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(m * 2);
ntt_trans(z2);
FPS x(begin(*this), begin(*this) + min((int)this->size(), m));
inplace_diff(x);
x.emplace_back(mint(0));
ntt_trans(x);
for (int i = 0; i < m; i++) x[i] *= y[i];
ntt_trans_inv(x);
x -= b.diff();
x.resize(m * 2);
for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = mint(0);
ntt_trans(x);
for (int i = 0; i < m * 2; i++) x[i] *= z2[i];
ntt_trans_inv(x);
x.pop_back();
inplace_integral(x);
for (int i = m; i < min((int)this->size(), m * 2); i++) x[i] += (*this)[i];
fill(begin(x), begin(x) + m, mint(0));
ntt_trans(x);
for (int i = 0; i < m * 2; i++) x[i] *= y[i];
ntt_trans_inv(x);
b.insert(end(b), begin(x) + m, end(x));
}
return FPS(begin(b), begin(b) + deg);
}
constexpr FPS exp_sparse(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i - 1, (*this)[i] * i);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i - 1) break;
r += val * res[i - k - 1];
}
res[i] = r * bc.inv(i);
}
return res;
}
// pow(f) = exp(e * log f)
constexpr FPS pow(long long e, int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return pow_sparse(e, deg);
assert(e >= 0);
if (deg < 0) deg = (int)this->size();
if (deg == 0) return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0) ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e) return FPS(deg, 0);
mint k = (*this)[ord];
FPS res = ((((*this) >> ord) / k).log(deg) * e).exp(deg) * mint(k).pow(e) << (e * ord);
res.resize(deg);
return res;
}
constexpr FPS pow_sparse(long long e, int deg = -1) const {
assert(e >= 0);
if (deg < 0) deg = (int)this->size();
if (deg == 0) return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0) ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e) return FPS(deg, 0);
if ((*this)[0] == 1) return pow_sparse_constant1(e, deg);
auto f = (*this);
rotate(f.begin(), f.begin() + ord, f.end());
mint con = f[0], icon = f[0].inv();
for (int i = 0; i < deg; i++) f[i] *= icon;
auto res = f.pow_sparse_constant1(e, deg);
int ord2 = e * ord;
rotate(res.begin(), res.begin() + (deg - ord2), res.end());
fill(res.begin(), res.begin() + ord2, mint(0));
mint pw = con.pow(e);
for (int i = ord2; i < deg; i++) res[i] *= pw;
return res;
}
constexpr FPS pow_sparse_constant1(mint e, int deg = -1) const {
assert((int)this->size() > 0 && (*this)[0] == 1);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 0; i < deg - 1; i++) {
mint &r = res[i + 1];
for (auto &&[k, val] : dat) {
if (k > i + 1) break;
mint t = val * res[i - k + 1];
r += t * (mint(k) * e - mint(i - k + 1));
}
r *= bc.inv(i + 1);
}
return res;
}
// sqrt(f)
constexpr FPS sqrt(int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return sqrt_sparse(deg);
if (deg < 0) deg = (int)this->size();
if ((int)this->size() == 0) return FPS(deg, 0);
if ((*this)[0] == mint(0)) {
for (int i = 1; i < (int)this->size(); i++) {
if ((*this)[i] != mint(0)) {
if (i & 1) return FPS();
if (deg - i / 2 <= 0) return FPS(deg, 0);
auto res = ((*this) >> i).sqrt(deg - i / 2);
if (res.empty()) return FPS();
res = res << (i / 2);
if ((int)res.size() < deg) res.resize(deg, mint(0));
return res;
}
}
return FPS(deg, 0);
}
long long sqr = mod_sqrt<long long>((*this)[0].val, mint::get_mod());
if (sqr == -1) return FPS();
assert((*this)[0].val == sqr * sqr % mint::get_mod());
FPS res = {mint(sqr)};
mint iv2 = mint(2).inv();
for (int d = 1; d < deg; d <<= 1) {
res = (res + pre(d << 1) * res.inv(d << 1)).pre(d << 1) * iv2;
}
res.resize(deg);
return res;
}
constexpr FPS sqrt_sparse(int deg) const {
if (deg < 0) deg = (int)this->size();
if ((int)this->size() == 0) return FPS(deg, 0);
if ((*this)[0] == mint(0)) {
for (int i = 1; i < (int)this->size(); i++) {
if ((*this)[i] != mint(0)) {
if (i & 1) return FPS();
if (deg - i / 2 <= 0) return FPS(deg, 0);
auto res = ((*this) >> i).sqrt_sparse(deg - i / 2);
if (res.empty()) return FPS();
res = res << (i / 2);
if ((int)res.size() < deg) res.resize(deg, mint(0));
return res;
}
}
return FPS(deg, 0);
}
mint con = (*this)[0], icon = con.inv();
long long sqr = mod_sqrt<long long>(con.val, mint::get_mod());
if (sqr == -1) return FPS();
assert(con.val == sqr * sqr % mint::get_mod());
auto res = (*this) * icon;
return res.sqrt_sparse_constant1(deg) * sqr;
}
constexpr FPS sqrt_sparse_constant1(int deg) const {
return pow_sparse_constant1(mint(2).inv(), deg);
}
// friend operators
friend constexpr FPS diff(const FPS &f) { return f.diff(); }
friend constexpr FPS integral(const FPS &f) { return f.integral(); }
friend constexpr FPS inv(const FPS &f, int deg = -1) { return f.inv(deg); }
friend constexpr FPS log(const FPS &f, int deg = -1) { return f.log(deg); }
friend constexpr FPS exp(const FPS &f, int deg = -1) { return f.exp(deg); }
friend constexpr FPS pow(const FPS &f, long long e, int deg = -1) { return f.pow(e, deg); }
friend constexpr FPS sqrt(const FPS &f, int deg = -1) { return f.sqrt(deg); }
};
//------------------------------//
// Polynomial Algorithms
//------------------------------//
// find f(x)^n mod g(x)
template<class mint, class T_VAL = long long>
FPS<mint> mod_pow(const FPS<mint> &f, T_VAL e, const FPS<mint> &mod) {
assert(!mod.empty());
auto iv = mod.rev().inv();
auto calc_quo = [&](const FPS<mint> &pol) -> FPS<mint> {
if (pol.size() < mod.size()) return FPS<mint>();
int deg = (int)pol.size() - (int)mod.size() + 1;
return (pol.rev().pre(deg) * iv.pre(deg)).pre(deg).rev();
};
FPS<mint> res{1}, b(f);
while (e) {
if (e & 1) res *= b, res -= calc_quo(res) * mod;
b *= b;
b -= calc_quo(b) * mod;
e >>= 1;
assert(b.size() + 1 <= mod.size());
assert(res.size() + 1 <= mod.size());
}
return res;
}
// middle product
// c[i] = sum_j a[i+j]b[j] (deg(a) = n, deg(b) = m -> deg(c) = n-m)
template<class mint>
FPS<mint> middle_product(const FPS<mint> &a, const FPS<mint> &b) {
assert(a.size() >= b.size());
if (b.empty()) return FPS<mint>((int)a.size() - (int)b.size() + 1);
int N = 1;
while (N < (int)a.size()) N <<= 1;
FPS<mint> fa(N), fb(N);
copy(a.begin(), a.end(), fa.begin());
copy(b.rbegin(), b.rend(), fb.begin());
fa *= fb;
fa.resize(a.size());
fa.erase(fa.begin(), fa.begin() + (int)b.size() - 1);
return fa;
}
// multipoint evaluation, polynomial interpolation
template<class mint> struct SubproductTree {
// inner data
int num_points, siz;
vector<FPS<mint>> tree;
// constructor
SubproductTree() {}
SubproductTree(const vector<mint> &x) {
num_points = (int)x.size();
siz = 1;
while (siz < num_points) siz <<= 1;
tree.resize(siz * 2);
for (int i = 0; i < siz; i++) tree[siz + i] = {1, (i < num_points ? -x[i] : 0)};
for (int i = siz - 1; i >= 1; i--) tree[i] = tree[i * 2] * tree[i * 2 + 1];
}
// multipoint evaluation
vector<mint> eval(FPS<mint> f) {
int N = (int)f.size();
if (N == 0) return vector<mint>(num_points, mint(0));
f.resize(N * 2 - 1);
vector<FPS<mint>> g(siz * 2);
g[1] = tree[1];
g[1].resize(N);
g[1] = inv(g[1]);
g[1] = middle_product(f, g[1]);
g[1].resize(siz);
for (int i = 1; i < siz; i++) {
g[i * 2] = middle_product(g[i], tree[i * 2 + 1]);
g[i * 2 + 1] = middle_product(g[i], tree[i * 2]);
}
vector<mint> res(num_points);
for (int i = 0; i < num_points; i++) res[i] = g[siz + i][0];
return res;
}
// polynomial interpolation
FPS<mint> interpolate(const vector<mint> &y) {
assert((int)y.size() == num_points);
vector<mint> p(num_points);
for (int i = 0; i < num_points; i++) p[i] = tree[1][num_points - i - 1] * (i + 1);
p = eval(p);
vector<FPS<mint>> t(siz * 2);
for (int i = 0; i < siz; i++) t[siz + i] = {(i < num_points ? y[i] / p[i] : 0)};
for (int i = siz - 1; i >= 1; i--) {
t[i] = t[i * 2] * tree[i * 2 + 1];
auto rt = t[i * 2 + 1] * tree[i * 2];
for (int k = 0; k < (int)t[i].size(); k++) t[i][k] += rt[k];
}
t[1].resize(num_points);
reverse(t[1].begin(), t[1].end());
return t[1];
}
};
// multipoint evaluation, polynomial interpolation
template<class mint>
vector<mint> multipoint_eval(const FPS<mint> &f, const vector<mint> &x) {
if (x.empty()) return {};
SubproductTree<mint> st(x);
return st.eval(f);
}
template<class mint>
FPS<mint> interpolate(const vector<mint> &x, const vector<mint> &y) {
assert(x.size() == y.size());
if (x.empty()) return {};
SubproductTree<mint> st(x);
return st.interpolate(y);
}
// polynomial interpolation (case: geometric sequence)
// y[i] = f(ar^i) -> find f
template<class mint>
vector<mint> interpolate(const mint &a, const mint &r, const FPS<mint> &y) {
int N = (int)y.size();
if (N == 0) return FPS<mint>();
if (N == 1) return {y[0]};
auto Y = y;
mint ir = r.inv(), ia = a.inv();
FPS<mint> po(N + N - 1, 1), po2(N + N - 1, 1), ipo(N + N - 1, 1), ipo2(N + N - 1, 1);
for (int i = 0; i < N + N - 2; i++) po[i + 1] = po[i] * r, po2[i + 1] = po2[i] * po[i];
for (int i = 0; i < N; i++) ipo[i + 1] = ipo[i] * ir, ipo2[i + 1] = ipo2[i] * ipo[i];
vector<mint> S(N, mint(1));
for (int i = 1; i < N; i++) S[i] = S[i - 1] * (mint(1) - po[i]);
vector<mint> iS = all_inverse(S);
mint sn = S[N - 1] * (mint(1) - po[N]);
for (int i = 0; i < N; i++) {
Y[i] = Y[i] * po2[N - i - 1] * ipo2[N - 1] * iS[i] * iS[N - i - 1];
if (i & 1) Y[i] = -Y[i];
}
for (int i = 0; i < N; i++) Y[i] *= ipo2[i];
FPS<mint> f = middle_product(po2, Y);
for (int i = 0; i < N; i++) f[i] *= ipo2[i];
FPS<mint> g(N, mint(1));
for (int i = 1; i < N; i++) {
g[i] = po2[i] * sn * iS[i] * iS[N - i];
if (i & 1) g[i] = -g[i];
}
f = f * g;
f.resize(N);
reverse(f.begin(), f.end());
mint p = 1;
for (int i = 0; i < N; i++) f[i] *= p, p *= ia;
return f;
}
//------------------------------//
// Matrix
//------------------------------//
// matrix
template<class mint> struct MintMatrix {
// inner value
vector<vector<mint>> val;
// constructors
MintMatrix(int H, int W) : val(H, vector<mint>(W)) {}
MintMatrix(int H, int W, mint x) : val(H, vector<mint>(W, x)) {}
MintMatrix(const MintMatrix &mat) : val(mat.val) {}
void init(int H, int W, mint x) {
val.assign(H, vector<mint>(W, x));
}
void resize(int H, int W) {
val.resize(H);
for (int i = 0; i < H; ++i) val[i].resize(W);
}
// getter and debugger
constexpr int height() const {
return (int)val.size();
}
constexpr int width() const {
return (height() > 0 ? (int)val[0].size() : 0);
}
vector<mint>& operator [] (int i) { return val[i]; }
constexpr vector<mint>& operator [] (int i) const { return val[i]; }
friend constexpr ostream& operator << (ostream &os, const MintMatrix<mint> &mat) {
os << endl;
for (int i = 0; i < mat.height(); ++i) {
for (int j = 0; j < mat.width(); ++j) {
if (j) os << ", ";
os << mat.val[i][j];
}
os << endl;
}
return os;
}
// comparison operators
constexpr bool operator == (const MintMatrix &r) const {
return this->val == r.val;
}
constexpr bool operator != (const MintMatrix &r) const {
return this->val != r.val;
}
// arithmetic operators
constexpr MintMatrix& operator += (const MintMatrix &r) {
assert(height() == r.height());
assert(width() == r.width());
for (int i = 0; i < height(); ++i) {
for (int j = 0; j < width(); ++j) {
val[i][j] = val[i][j] + r.val[i][j];
}
}
return *this;
}
constexpr MintMatrix& operator -= (const MintMatrix &r) {
assert(height() == r.height());
assert(width() == r.width());
for (int i = 0; i < height(); ++i) {
for (int j = 0; j < width(); ++j) {
val[i][j] = val[i][j] - r.val[i][j];
}
}
return *this;
}
constexpr MintMatrix& operator *= (const mint &v) {
for (int i = 0; i < height(); ++i)
for (int j = 0; j < width(); ++j)
val[i][j] = val[i][j] * v;
return *this;
}
constexpr MintMatrix& operator *= (const MintMatrix &r) {
assert(width() == r.height());
MintMatrix<mint> res(height(), r.width());
for (int i = 0; i < height(); ++i)
for (int j = 0; j < r.width(); ++j)
for (int k = 0; k < width(); ++k)
res[i][j] = res[i][j] + val[i][k] * r.val[k][j];
return (*this) = res;
}
constexpr MintMatrix operator + () const { return MintMatrix(*this); }
constexpr MintMatrix operator - () const { return MintMatrix(*this) *= mint(-1); }
constexpr MintMatrix operator + (const MintMatrix &r) const { return MintMatrix(*this) += r; }
constexpr MintMatrix operator - (const MintMatrix &r) const { return MintMatrix(*this) -= r; }
constexpr MintMatrix operator * (const mint &v) const { return MintMatrix(*this) *= v; }
constexpr MintMatrix operator * (const MintMatrix &r) const { return MintMatrix(*this) *= r; }
constexpr vector<mint> operator * (const vector<mint> &v) const {
assert(width() == v.size());
vector<mint> res(height(), mint(0));
for (int i = 0; i < height(); i++)
for (int j = 0; j < width(); j++)
res[i] += val[i][j] * v[j];
return res;
}
// pow
constexpr MintMatrix pow(long long n) const {
assert(height() == width());
MintMatrix<mint> res(height(), width()), mul(*this);
for (int row = 0; row < height(); ++row) res[row][row] = mint(1);
while (n > 0) {
if (n & 1) res = res * mul;
mul = mul * mul;
n >>= 1;
}
return res;
}
friend constexpr MintMatrix<mint> pow(const MintMatrix<mint> &mat, long long n) {
return mat.pow(n);
}
// gauss-jordan
constexpr int find_pivot(int cur_rank, int col) const {
int pivot = -1;
for (int row = cur_rank; row < height(); ++row) {
if (val[row][col] != 0) {
pivot = row;
break;
}
}
return pivot;
}
constexpr void sweep(int cur_rank, int col, int pivot) {
swap(val[pivot], val[cur_rank]);
auto ifac = val[cur_rank][col].inv();
for (int col2 = 0; col2 < width(); ++col2) {
val[cur_rank][col2] *= ifac;
}
for (int row = 0; row < height(); ++row) {
if (row != cur_rank && val[row][col] != 0) {
auto fac = val[row][col];
for (int col2 = 0; col2 < width(); ++col2) {
val[row][col2] -= val[cur_rank][col2] * fac;
}
}
}
}
constexpr int gauss_jordan(int not_sweep_width = 0) {
int rank = 0;
for (int col = 0; col < width(); ++col) {
if (col == width() - not_sweep_width) break;
int pivot = find_pivot(rank, col);
if (pivot == -1) continue;
sweep(rank++, col, pivot);
}
return rank;
}
constexpr int gauss_jordan(int not_sweep_width, vector<int> &core) {
core.clear();
int rank = 0;
for (int col = 0; col < width(); ++col) {
if (col == width() - not_sweep_width) break;
int pivot = find_pivot(rank, col);
if (pivot == -1) continue;
core.push_back(col);
sweep(rank++, col, pivot);
}
return rank;
}
friend constexpr int gauss_jordan(MintMatrix<mint> &mat, int not_sweep_width = 0) {
return mat.gauss_jordan(not_sweep_width);
}
// find one solution
friend constexpr int linear_equation
(const MintMatrix<mint> &mat, const vector<mint> &b, vector<mint> &res) {
// extend
MintMatrix<mint> A(mat.height(), mat.width() + 1);
for (int i = 0; i < mat.height(); ++i) {
for (int j = 0; j < mat.width(); ++j) A[i][j] = mat.val[i][j];
A[i].back() = b[i];
}
int rank = A.gauss_jordan(1);
// check if it has no solution
for (int row = rank; row < mat.height(); ++row) if (A[row].back() != 0) return -1;
// answer
res.assign(mat.width(), 0);
for (int i = 0; i < rank; ++i) res[i] = A[i].back();
return rank;
}
friend constexpr int linear_equation(const MintMatrix<mint> &mat, const vector<mint> &b) {
vector<mint> res;
return linear_equation(mat, b, res);
}
// find all solutions
friend int linear_equation
(const MintMatrix<mint> &mat, const vector<mint> &b, vector<mint> &res, vector<vector<mint>> &zeros) {
// extend
MintMatrix<mint> A(mat.height(), mat.width() + 1);
for (int i = 0; i < mat.height(); ++i) {
for (int j = 0; j < mat.width(); ++j) A[i][j] = mat.val[i][j];
A[i].back() = b[i];
}
vector<int> core;
int rank = A.gauss_jordan(1, core);
// check if it has no solution
for (int row = rank; row < mat.height(); ++row) {
if (A[row].back() != 0) return -1;
}
// construct the core solution
res.assign(mat.width(), mint(0));
for (int i = 0; i < (int)core.size(); i++) res[core[i]] = A[i].back();
// construct the all solutions
zeros.clear();
vector<bool> use(mat.width(), 0);
for (auto c : core) use[c] = true;
for (int j = 0; j < mat.width(); j++) {
if (use[j]) continue;
vector<mint> zero(mat.width(), mint(0));
zero[j] = mint(1);
for (int i = 0; i < (int)core.size(); i++) zero[core[i]] = -A[i][j];
zeros.push_back(zero);
}
return rank;
}
// determinant
constexpr mint det() const {
MintMatrix<mint> A(*this);
int rank = 0;
mint res = 1;
for (int col = 0; col < width(); ++col) {
int pivot = A.find_pivot(rank, col);
if (pivot == -1) return mint(0);
res *= A[pivot][rank];
A.sweep(rank++, col, pivot);
}
return res;
}
friend constexpr mint det(const MintMatrix<mint> &mat) {
return mat.det();
}
};
// characteristic_polynomial
// find f(x) = det(xI - B), for N x N mint matrix B, O(N^3)
template<class mint>
void hessenberg_reduction(MintMatrix<mint> &M) {
assert(M.height() == M.width());
int N = (int)M.height();
for (int r = 0; r < N - 2; r++) {
int pivot = -1;
for (int h = r + 1; h < N; h++) {
if (M[h][r] != 0) {
pivot = h;
break;
}
}
if (pivot == -1) continue;
for (int i = 0; i < N; i++) swap(M[r + 1][i], M[pivot][i]);
for (int i = 0; i < N; i++) swap(M[i][r + 1], M[i][pivot]);
mint ir = M[r + 1][r].inv();
for (int i = r + 2; i < N; i++) {
mint ir2 = M[i][r] * ir;
for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * ir2;
for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * ir2;
}
}
}
template<class mint>
FPS<mint> calc_characteristic_polynomial(MintMatrix<mint> B) {
assert(B.height() == B.width());
hessenberg_reduction(B);
int N = (int)B.height();
vector<FPS<mint>> res(N + 1);
res[0] = {mint(1)};
for (int i = 0; i < N; i++) {
res[i + 1].assign(i + 2, mint(0));
for (int j = 0; j < i + 1; j++) res[i + 1][j + 1] += res[i][j];
for (int j = 0; j < i + 1; j++) res[i + 1][j] -= res[i][j] * B[i][i];
mint beta = 1;
for (int j = i - 1; j >= 0; j--) {
beta *= B[j + 1][j];
mint beta2 = -B[j][i] * beta;
for (int k = 0; k < j + 1; k++) res[i + 1][k] += beta2 * res[j][k];
}
}
return res[N];
}
// find f(x) = det(M1x + M0), for given N x N matrix M0, M1, in O(N^3)
template<class mint>
FPS<mint> calc_det_linear_expression(MintMatrix<mint> M0, MintMatrix<mint> M1) {
int N = (int)M0.height();
assert(M0.width() == N && M1.height() == N && M1.width() == N);
int con = 0;
mint invAB = 1;
for (int p = 0; p < N; p++) {
int pivot = -1;
for (int r = p; r < N; r++) {
if (M1[r][p] != mint(0)) {
pivot = r;
break;
}
}
if (pivot == -1) {
con++;
if (con > N) return FPS<mint>(N + 1, mint(0));
for (int r = 0; r < p; r++) {
mint v = M1[r][p];
M1[r][p] = 0;
for (int i = 0; i < N; i++) M0[i][p] -= v * M0[i][r];
}
for (int i = 0; i < N; i++) swap(M0[i][p], M1[i][p]);
p--;
continue;
}
if (pivot != p) {
swap(M1[p], M1[pivot]);
swap(M0[p], M0[pivot]);
invAB *= -1;
}
mint v = M1[p][p], iv = v.inv();
invAB *= v;
for (int c = 0; c < N; c++) M0[p][c] *= iv, M1[p][c] *= iv;
for (int r = 0; r < N; r++) {
if (r == p) continue;
mint v = M1[r][p];
for (int c = 0; c < N; c++) M0[r][c] -= M0[p][c] * v, M1[r][c] -= M1[p][c] * v;
}
}
for (int r = 0; r < M0.height(); r++) for (int c = 0; c < M0.width(); c++) M0[r][c] *= -1;
auto pol = calc_characteristic_polynomial(M0);
for (auto &x : pol) x *= invAB;
pol.erase(pol.begin(), pol.begin() + con);
pol.resize(N + 1);
return pol;
}
//------------------------------//
// Examples
//------------------------------//
// yukicoder No.1907 DETERMINATION
void yukicoder_1907() {
using mint = Fp<>;
int N;
cin >> N;
MintMatrix<mint> M0(N, N), M1(N, N);
for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) cin >> M0[i][j];
for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) cin >> M1[i][j];
auto res = calc_det_linear_expression(M0, M1);
for (int i = 0; i < res.size(); i++) cout << res[i] << '\n';
}
int main() {
yukicoder_1907();
}