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correct.cpp
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correct.cpp
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#include <iostream>
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <random>
#include <set>
#include <vector>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T> using V = vector<T>;
template <class T> using VV = V<V<T>>;
struct Random {
private:
// Use xoshiro256**
// Refereces: http://xoshiro.di.unimi.it/xoshiro256starstar.c
static uint64_t rotl(const uint64_t x, int k) {
return (x << k) | (x >> (64 - k));
}
std::array<uint64_t, 4> s;
uint64_t next() {
const uint64_t result_starstar = rotl(s[1] * 5, 7) * 9;
const uint64_t t = s[1] << 17;
s[2] ^= s[0];
s[3] ^= s[1];
s[1] ^= s[2];
s[0] ^= s[3];
s[2] ^= t;
s[3] = rotl(s[3], 45);
return result_starstar;
}
// random choice from [0, upper]
uint64_t next(uint64_t upper) {
if (!(upper & (upper + 1))) {
// b = 00..0011..11
return next() & upper;
}
int lg = 63 - __builtin_clzll(upper);
uint64_t mask = (lg == 63) ? ~0ULL : (1ULL << (lg + 1)) - 1;
while (true) {
uint64_t r = next() & mask;
if (r <= upper)
return r;
}
}
public:
Random(uint64_t seed = 0) {
// Use splitmix64
// Reference: http://xoshiro.di.unimi.it/splitmix64.c
for (int i = 0; i < 4; i++) {
uint64_t z = (seed += 0x9e3779b97f4a7c15);
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
s[i] = z ^ (z >> 31);
}
}
// random choice from [lower, upper]
template <class T>
T uniform(T lower, T upper) {
assert(lower <= upper);
return T(lower + next(uint64_t(upper - lower)));
}
bool uniform_bool() { return uniform(0, 1) == 1; }
double uniform01() {
uint64_t v = next(1ULL << 63);
return double(v) / (1ULL << 63);
}
// generate random lower string that length = n
std::string lower_string(size_t n) {
std::string res = "";
for (size_t i = 0; i < n; i++) {
res += uniform('a', 'z');
}
return res;
}
// random shuffle
template <class Iter>
void shuffle(Iter first, Iter last) {
int len = 0;
// Reference and edit:
// cpprefjp - C++日本語リファレンス
// (https://cpprefjp.github.io/reference/algorithm/shuffle.html)
for (auto it = first + 1; it != last; it++) {
len++;
int j = uniform(0, len - 1);
if (j != len - 1)
iter_swap(it, first + j);
}
}
// generate random permutation that length = n
template <class T>
std::vector<T> perm(size_t n) {
std::vector<T> idx(n);
std::iota(idx.begin(), idx.end(), T(0));
shuffle(idx.begin(), idx.end());
return idx;
}
template <class T>
std::vector<T> choice(size_t n, T lower, T upper) {
assert(n <= upper - lower + 1);
std::set<T> res;
while (res.size() < n) res.insert(uniform(lower, upper));
return {res.begin(), res.end()};
}
} global_gen;
template <uint MD> struct ModInt {
using M = ModInt;
const static M G;
uint v;
ModInt(ll _v = 0) { set_v(_v % MD + MD); }
M& set_v(uint _v) {
v = (_v < MD) ? _v : _v - MD;
return *this;
}
explicit operator bool() const { return v != 0; }
M operator-() const { return M() - *this; }
M operator+(const M& r) const { return M().set_v(v + r.v); }
M operator-(const M& r) const { return M().set_v(v + MD - r.v); }
M operator*(const M& r) const { return M().set_v(ull(v) * r.v % MD); }
M operator/(const M& r) const { return *this * r.inv(); }
M& operator+=(const M& r) { return *this = *this + r; }
M& operator-=(const M& r) { return *this = *this - r; }
M& operator*=(const M& r) { return *this = *this * r; }
M& operator/=(const M& r) { return *this = *this / r; }
bool operator==(const M& r) const { return v == r.v; }
M pow(ll n) const {
M x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
M inv() const { return pow(MD - 2); }
friend ostream& operator<<(ostream& os, const M& r) { return os << r.v; }
};
using Mint = ModInt<998244353>;
template<> const Mint Mint::G = Mint(3);
template <class Mint> void nft(bool type, V<Mint>& a) {
int n = int(a.size()), s = 0;
while ((1 << s) < n) s++;
assert(1 << s == n);
static V<Mint> ep, iep;
while (int(ep.size()) <= s) {
ep.push_back(Mint::G.pow(Mint(-1).v / (1 << ep.size())));
iep.push_back(ep.back().inv());
}
V<Mint> b(n);
for (int i = 1; i <= s; i++) {
int w = 1 << (s - i);
Mint base = type ? iep[i] : ep[i], now = 1;
for (int y = 0; y < n / 2; y += w) {
for (int x = 0; x < w; x++) {
auto l = a[y << 1 | x];
auto r = now * a[y << 1 | x | w];
b[y | x] = l + r;
b[y | x | n >> 1] = l - r;
}
now *= base;
}
swap(a, b);
}
}
template <class Mint> V<Mint> multiply(const V<Mint>& a, const V<Mint>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (min(n, m) <= 8) {
V<Mint> ans(n + m - 1);
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++) ans[i + j] += a[i] * b[j];
return ans;
}
int lg = 0;
while ((1 << lg) < n + m - 1) lg++;
int z = 1 << lg;
auto a2 = a, b2 = b;
a2.resize(z);
b2.resize(z);
nft(false, a2);
nft(false, b2);
for (int i = 0; i < z; i++) a2[i] *= b2[i];
nft(true, a2);
a2.resize(n + m - 1);
Mint iz = Mint(z).inv();
for (int i = 0; i < n + m - 1; i++) a2[i] *= iz;
return a2;
}
template <class D> struct Poly {
V<D> v;
Poly(const V<D>& _v = {}) : v(_v) { shrink(); }
void shrink() {
while (v.size() && !v.back()) v.pop_back();
}
int size() const { return int(v.size()); }
D freq(int p) const { return (p < size()) ? v[p] : D(0); }
Poly operator+(const Poly& r) const {
auto n = max(size(), r.size());
V<D> res(n);
for (int i = 0; i < n; i++) res[i] = freq(i) + r.freq(i);
return res;
}
Poly operator-(const Poly& r) const {
int n = max(size(), r.size());
V<D> res(n);
for (int i = 0; i < n; i++) res[i] = freq(i) - r.freq(i);
return res;
}
Poly operator*(const Poly& r) const { return {multiply(v, r.v)}; }
Poly operator*(const D& r) const {
int n = size();
V<D> res(n);
for (int i = 0; i < n; i++) res[i] = v[i] * r;
return res;
}
Poly operator/(const D &r) const{
return *this * r.inv();
}
Poly operator/(const Poly& r) const {
if (size() < r.size()) return {{}};
int n = size() - r.size() + 1;
return (rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
Poly operator%(const Poly& r) const { return *this - *this / r * r; }
Poly operator<<(int s) const {
V<D> res(size() + s);
for (int i = 0; i < size(); i++) res[i + s] = v[i];
return res;
}
Poly operator>>(int s) const {
if (size() <= s) return Poly();
V<D> res(size() - s);
for (int i = 0; i < size() - s; i++) res[i] = v[i + s];
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D &r) {return *this = *this/r;}
Poly& operator%=(const Poly& r) { return *this = *this % r; }
Poly& operator<<=(const size_t& n) { return *this = *this << n; }
Poly& operator>>=(const size_t& n) { return *this = *this >> n; }
Poly pre(int le) const {
return {{v.begin(), v.begin() + min(size(), le)}};
}
Poly rev(int n = -1) const {
V<D> res = v;
if (n != -1) res.resize(n);
reverse(res.begin(), res.end());
return res;
}
Poly pow_mod(ll n, const Poly& mod) {
Poly x = *this, r = {{1}};
while (n) {
if (n & 1) r = r * x % mod;
x = x * x % mod;
n >>= 1;
}
return r;
}
friend ostream& operator<<(ostream& os, const Poly& p) {
if (p.size() == 0) return os << "0";
for (auto i = 0; i < p.size(); i++) {
if (p.v[i]) {
os << p.v[i] << "x^" << i;
if (i != p.size() - 1) os << "+";
}
}
return os;
}
};
template <class Mint> struct MultiEval {
using NP = MultiEval*;
NP l, r;
V<Mint> que;
int sz;
Poly<Mint> mul;
MultiEval(const V<Mint>& _que, int off, int _sz) : sz(_sz) {
if (sz <= 100) {
que = {_que.begin() + off, _que.begin() + off + sz};
mul = {{1}};
for (auto x : que) mul *= {{-x, 1}};
return;
}
l = new MultiEval(_que, off, sz / 2);
r = new MultiEval(_que, off + sz / 2, sz - sz / 2);
mul = l->mul * r->mul;
}
MultiEval(const V<Mint>& _que) : MultiEval(_que, 0, int(_que.size())) {}
void query(const Poly<Mint>& _pol, V<Mint>& res) const {
if (sz <= 100) {
for (auto x : que) {
Mint sm = 0, base = 1;
for (int i = 0; i < _pol.size(); i++) {
sm += base * _pol.freq(i);
base *= x;
}
res.push_back(sm);
}
return;
}
auto pol = _pol % mul;
l->query(pol, res);
r->query(pol, res);
}
V<Mint> query(const Poly<Mint>& pol) const {
V<Mint> res;
query(pol, res);
return res;
}
};
template <class Mint> Poly<Mint> berlekamp_massey(const V<Mint>& s) {
int n = int(s.size());
V<Mint> b = {Mint(-1)}, c = {Mint(-1)};
Mint y = Mint(1);
for (int ed = 1; ed <= n; ed++) {
int l = int(c.size()), m = int(b.size());
Mint x = 0;
for (int i = 0; i < l; i++) {
x += c[i] * s[ed - l + i];
}
b.push_back(0);
m++;
if (!x) continue;
Mint freq = x / y;
if (l < m) {
// use b
auto tmp = c;
c.insert(begin(c), m - l, Mint(0));
for (int i = 0; i < m; i++) {
c[m - 1 - i] -= freq * b[m - 1 - i];
}
b = tmp;
y = x;
} else {
// use c
for (int i = 0; i < m; i++) {
c[l - 1 - i] -= freq * b[m - 1 - i];
}
}
}
return c;
}
template <class E, class Mint = decltype(E().f)>
Mint sparse_det(const VV<E>& g) {
int n = int(g.size());
if (n == 0) return 1;
auto rand_v = [&]() {
V<Mint> res(n);
for (int i = 0; i < n; i++) {
res[i] = Mint(global_gen.uniform<int>(1, Mint(-1).v));
}
return res;
};
V<Mint> c = rand_v(), l = rand_v(), r = rand_v();
// l * mat * r
V<Mint> buf(2 * n);
for (int fe = 0; fe < 2 * n; fe++) {
for (int i = 0; i < n; i++) {
buf[fe] += l[i] * r[i];
}
for (int i = 0; i < n; i++) {
r[i] *= c[i];
}
V<Mint> tmp(n);
for (int i = 0; i < n; i++) {
for (auto e : g[i]) {
tmp[i] += r[e.to] * e.f;
}
}
r = tmp;
}
auto u = berlekamp_massey(buf);
if (!u.freq(0)) return 0;
if (u.size() != n + 1) return sparse_det(g);
auto acdet = u.freq(0) * Mint(-1);
if (n % 2) acdet *= Mint(-1);
assert(acdet);
Mint cdet = 1;
for (int i = 0; i < n; i++) cdet *= c[i];
return acdet / cdet;
}
int main() {
int n, k;
scanf("%d %d", &n, &k);
struct E { int to; Mint f; };
VV<E> edges(n);
for (int i = 0; i < k; i++) {
int a, b; Mint c;
scanf("%d %d %d", &a, &b, &c.v);
edges[a].push_back({b, c});
}
auto answer = sparse_det(edges);
printf("%d\n", answer.v);
return 0;
}