algorithm
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cpp_src/math/ModularInclude.hpp
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- Last update: 2023-07-23 15:21:10+09:00
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#include "cpp_src/math/ModularInclude.hpp"
Code
// modint + modular operations + NTT
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
explicit operator int() const { return v; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
friend istream& operator>>(istream& in, M& x) {
ll v_;
in >> v_;
x = M(v_);
return in;
}
bool operator<(const M& r) const { return v < r.v; }
bool operator>(const M& r) const { return v < *this; }
bool operator<=(const M& r) const { return !(r < *this); }
bool operator>=(const M& r) const { return !(*this < r); }
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
static uint get_mod() { return MOD; }
};
// using Mint = ModInt<1000000007>;
using Mint = ModInt<998244353>;
V<Mint> fact, ifact, inv;
VV<Mint> small_comb;
void mod_init() {
const int maxv = 1000010;
const int maxvv = 5000;
fact.resize(maxv);
ifact.resize(maxv);
inv.resize(maxv);
small_comb = make_vec<Mint>(maxvv, maxvv);
fact[0] = 1;
for (int i = 1; i < maxv; ++i) {
fact[i] = fact[i - 1] * i;
}
ifact[maxv - 1] = fact[maxv - 1].inv();
for (int i = maxv - 2; i >= 0; --i) {
ifact[i] = ifact[i + 1] * (i + 1);
}
for (int i = 1; i < maxv; ++i) {
inv[i] = ifact[i] * fact[i - 1];
}
for (int i = 0; i < maxvv; ++i) {
small_comb[i][0] = small_comb[i][i] = 1;
for (int j = 1; j < i; ++j) {
small_comb[i][j] = small_comb[i - 1][j] + small_comb[i - 1][j - 1];
}
}
}
Mint comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
if (n < small_comb.size()) return small_comb[n][r];
return fact[n] * ifact[r] * ifact[n - r];
}
Mint inv_comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
return ifact[n] * fact[r] * fact[n - r];
}
// O(k)
Mint comb_slow(ll n, ll k) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = ifact[k];
for (int i = 0; i < k; ++i) {
res = res * (n - i);
}
return res;
}
// line up
// a 'o' + b 'x'
Mint comb2(int a, int b) {
if (a < 0 || b < 0) return 0;
return comb(a + b, a);
}
// divide a into b groups
Mint nhr(int a, int b) {
if (b == 0) return Mint(a == 0);
return comb(a + b - 1, a);
}
// O(p + log_p n)
Mint lucas(ll n, ll k, int p) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = 1;
while (n > 0) {
res *= comb(n % p, k % p);
n /= p;
k /= p;
}
return res;
}
struct ModPrepare {
ModPrepare() { mod_init(); }
} prep_mod;
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
int v = D(n).inv().v;
if (inv) {
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
if (sz <= 16) {
V<D> ret(s);
for (int i = 0; i < a.size(); i++) {
for (int j = 0; j < b.size(); j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}#line 1 "cpp_src/math/ModularInclude.hpp"
// modint + modular operations + NTT
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
explicit operator int() const { return v; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
friend istream& operator>>(istream& in, M& x) {
ll v_;
in >> v_;
x = M(v_);
return in;
}
bool operator<(const M& r) const { return v < r.v; }
bool operator>(const M& r) const { return v < *this; }
bool operator<=(const M& r) const { return !(r < *this); }
bool operator>=(const M& r) const { return !(*this < r); }
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
static uint get_mod() { return MOD; }
};
// using Mint = ModInt<1000000007>;
using Mint = ModInt<998244353>;
V<Mint> fact, ifact, inv;
VV<Mint> small_comb;
void mod_init() {
const int maxv = 1000010;
const int maxvv = 5000;
fact.resize(maxv);
ifact.resize(maxv);
inv.resize(maxv);
small_comb = make_vec<Mint>(maxvv, maxvv);
fact[0] = 1;
for (int i = 1; i < maxv; ++i) {
fact[i] = fact[i - 1] * i;
}
ifact[maxv - 1] = fact[maxv - 1].inv();
for (int i = maxv - 2; i >= 0; --i) {
ifact[i] = ifact[i + 1] * (i + 1);
}
for (int i = 1; i < maxv; ++i) {
inv[i] = ifact[i] * fact[i - 1];
}
for (int i = 0; i < maxvv; ++i) {
small_comb[i][0] = small_comb[i][i] = 1;
for (int j = 1; j < i; ++j) {
small_comb[i][j] = small_comb[i - 1][j] + small_comb[i - 1][j - 1];
}
}
}
Mint comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
if (n < small_comb.size()) return small_comb[n][r];
return fact[n] * ifact[r] * ifact[n - r];
}
Mint inv_comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
return ifact[n] * fact[r] * fact[n - r];
}
// O(k)
Mint comb_slow(ll n, ll k) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = ifact[k];
for (int i = 0; i < k; ++i) {
res = res * (n - i);
}
return res;
}
// line up
// a 'o' + b 'x'
Mint comb2(int a, int b) {
if (a < 0 || b < 0) return 0;
return comb(a + b, a);
}
// divide a into b groups
Mint nhr(int a, int b) {
if (b == 0) return Mint(a == 0);
return comb(a + b - 1, a);
}
// O(p + log_p n)
Mint lucas(ll n, ll k, int p) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = 1;
while (n > 0) {
res *= comb(n % p, k % p);
n /= p;
k /= p;
}
return res;
}
struct ModPrepare {
ModPrepare() { mod_init(); }
} prep_mod;
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
int v = D(n).inv().v;
if (inv) {
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
if (sz <= 16) {
V<D> ret(s);
for (int i = 0; i < a.size(); i++) {
for (int j = 0; j < b.size(); j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}