FFT.cpp

#pragma GCC optimize("Ofast")
#pragma GCC target("avx,avx2,fma")

#include <bits/stdc++.h>
//#include <ext/pb_ds/assoc_container.hpp> //required
//#include <ext/pb_ds/tree_policy.hpp> //required

//using namespace __gnu_pbds; //required 
using namespace std;
//template <typename T> using ordered_set =  tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>; 

// ordered_set <int> s;
// s.find_by_order(k); returns the (k+1)th smallest element
// s.order_of_key(k); returns the number of elements in s strictly less than k

#define pb               push_back
#define mp(x,y)          make_pair(x,y)
#define all(x)           x.begin(), x.end()
#define print(vec,l,r)   for(int i = l; i <= r; i++) cout << vec[i] <<" "; cout << endl;
#define input(vec,N)     for(int i = 0; i < (N); i++) cin >> vec[i];
#define leftmost_bit(x)  (63-__builtin_clzll(x))
#define rightmost_bit(x) __builtin_ctzll(x) // count trailing zeros
#define set_bits(x)      __builtin_popcountll(x) 
#define pow2(i)          (1LL << (i))
#define is_on(x, i)      ((x) & pow2(i)) // state of the ith bit in x
#define set_on(x, i)     ((x) | pow2(i)) // returns integer x with ith bit on
#define set_off(x, i)    ((x) & ~pow2(i)) // returns integer x with ith bit off

#ifdef LOCAL_DEBUG
	#define debug(...) logger(#__VA_ARGS__, __VA_ARGS__)
#else
	#define debug(...) ;
#endif
  
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
// auto dist = uniform_int_distribution<int>(l, r);
// use int a = dist(rng) to get a random number between [l,r] inclusive
template<typename ...Args>
void logger(string vars, Args&&... values) {
    cerr << vars << " = ";
    string delim = "";
    (..., (cerr << delim << values, delim = ", "));
	cerr << endl;
}

typedef long long int ll;
typedef long double ld;

const int MOD = 998244353;
const int MX = 2e5+5;
const ll INF = 1e18; // not too close to LLONG_MAX
const ld PI = acos((ld)-1);
const ld EPS = 1e-8;
const int dx[4] = {1,0,-1,0}, dy[4] = {0,1,0,-1}; // for every grid problem!!

// highly risky #defines
#define int ll // disable when you want to make code a bit faster
#define endl '\n' // disable when dealing with interactive problems

typedef vector<int> vi;
typedef pair<int, int> pii;

template<typename float_t>
struct mycomplex {
	float_t x, y;
 
	mycomplex<float_t>(float_t _x = 0, float_t _y = 0) : x(_x), y(_y) {}
 
	float_t real() const { return x; }
	float_t imag() const { return y; }
 
	void real(float_t _x) { x = _x; }
	void imag(float_t _y) { y = _y; }
 
	mycomplex<float_t>& operator+=(const mycomplex<float_t> &other) { x += other.x; y += other.y; return *this; }
	mycomplex<float_t>& operator-=(const mycomplex<float_t> &other) { x -= other.x; y -= other.y; return *this; }
 
	mycomplex<float_t> operator+(const mycomplex<float_t> &other) const { return mycomplex<float_t>(*this) += other; }
	mycomplex<float_t> operator-(const mycomplex<float_t> &other) const { return mycomplex<float_t>(*this) -= other; }
 
	mycomplex<float_t> operator*(const mycomplex<float_t> &other) const {
		return {x * other.x - y * other.y, x * other.y + other.x * y};
	}
 
	mycomplex<float_t> operator*(float_t mult) const {
		return {x * mult, y * mult};
	}
 
	friend mycomplex<float_t> conj(const mycomplex<float_t> &c) {
		return {c.x, -c.y};
	}
 
	friend ostream& operator<<(ostream &stream, const mycomplex<float_t> &c) {
		return stream << '(' << c.x << ", " << c.y << ')';
	}
};
 
template<typename float_t>
mycomplex<float_t> mypolar(float_t magnitude, float_t angle) {
	return {magnitude * cos(angle), magnitude * sin(angle)};
}
 
namespace FFT {
	using float_t = double;
 
	const float_t ONE = 1;
	const float_t PI = acos(-ONE);
 
	vector<mycomplex<float_t>> roots = {{0, 0}, {1, 0}};
	vector<int> bit_reverse;
 
	bool is_power_of_two(int n) {
		return (n & (n - 1)) == 0;
	}
 
	int round_up_power_two(int n) {
		while (n & (n - 1))
			n = (n | (n - 1)) + 1;
 
		return max(n, 1LL);
	}
 
	// Given n (a power of two), finds k such that n == 1 << k.
	int get_length(int n) {
		assert(is_power_of_two(n));
		return __builtin_ctz(n);
	}
 
	// Rearranges the indices to be sorted by lowest bit first, then second lowest, etc., rather than highest bit first.
	// This makes even-odd div-conquer much easier.
	template<typename mycomplex_array>
	void bit_reorder(int n, mycomplex_array &&values) {
		if ((int) bit_reverse.size() != n) {
			bit_reverse.assign(n, 0);
			int length = get_length(n);
 
			for (int i = 0; i < n; i++)
				bit_reverse[i] = (bit_reverse[i >> 1] >> 1) | ((i & 1) << (length - 1));
		}
 
		for (int i = 0; i < n; i++)
			if (i < bit_reverse[i])
				swap(values[i], values[bit_reverse[i]]);
	}
 
	void prepare_roots(int n) {
		if ((int) roots.size() >= n)
			return;
 
		int length = get_length(roots.size());
		roots.resize(n);
 
		// The roots array is set up such that for a given power of two n >= 2, roots[n / 2] through roots[n - 1] are
		// the first half of the n-th roots of unity.
		while (1 << length < n) {
			float_t min_angle = 2 * PI / (1 << (length + 1));
 
			for (int i = 0; i < 1 << (length - 1); i++) {
				int index = (1 << (length - 1)) + i;
				roots[2 * index] = roots[index];
				roots[2 * index + 1] = mypolar(ONE, min_angle * (2 * i + 1));
			}
 
			length++;
		}
	}
 
	template<typename mycomplex_array>
	void fft_iterative(int N, mycomplex_array &&values) {
		assert(is_power_of_two(N));
		prepare_roots(N);
		bit_reorder(N, values);
 
		for (int n = 1; n < N; n *= 2)
			for (int start = 0; start < N; start += 2 * n)
				for (int i = 0; i < n; i++) {
					const mycomplex<float_t> &even = values[start + i];
					mycomplex<float_t> odd = values[start + n + i] * roots[n + i];
					values[start + n + i] = even - odd;
					values[start + i] = even + odd;
				}
	}
 
	inline mycomplex<float_t> extract(int N, const vector<mycomplex<float_t>> &values, int index, int side) {
		if (side == -1) {
			// Return the product of 0 and 1.
			int other = (N - index) & (N - 1);
			return (conj(values[other] * values[other]) - values[index] * values[index]) * mycomplex<float_t>(0, 0.25);
		}
 
		int other = (N - index) & (N - 1);
		int sign = side == 0 ? +1 : -1;
		mycomplex<float_t> multiplier = side == 0 ? mycomplex<float_t>(0.5, 0) : mycomplex<float_t>(0, -0.5);
		return multiplier * mycomplex<float_t>(values[index].real() + values[other].real() * sign,
											 values[index].imag() - values[other].imag() * sign);
	}
 
	void invert_fft(int N, vector<mycomplex<float_t>> &values) {
		assert(N >= 2);
 
		for (int i = 0; i < N; i++)
			values[i] = conj(values[i]) * (ONE / N);
 
		for (int i = 0; i < N / 2; i++) {
			mycomplex<float_t> first = values[i] + values[N / 2 + i];
			mycomplex<float_t> second = (values[i] - values[N / 2 + i]) * roots[N / 2 + i];
			values[i] = first + second * mycomplex<float_t>(0, 1);
		}
 
		fft_iterative(N / 2, values);
 
		for (int i = N - 1; i >= 0; i--)
			values[i] = i % 2 == 0 ? values[i / 2].real() : values[i / 2].imag();
	}
 
	const int FFT_CUTOFF = 150;
	const double SPLIT_CUTOFF = 2e15;
	const int SPLIT_BASE = 1 << 15;
 
	template<typename T_out, typename T_in>
	vector<T_out> square(const vector<T_in> &input) {
		if (input.empty())
			return {0};
 
		int n = input.size();
 
#ifdef NEAL
		// Sanity check to make sure I'm not forgetting to split.
		double max_value = *max_element(input.begin(), input.end());
		assert(n * max_value * max_value < SPLIT_CUTOFF);
#endif
 
		// Brute force when n is small enough.
		if (n < 1.5 * FFT_CUTOFF) {
			vector<T_out> result(2 * n - 1);
 
			for (int i = 0; i < n; i++) {
				result[2 * i] += (T_out) input[i] * input[i];
 
				for (int j = i + 1; j < n; j++)
					result[i + j] += (T_out) 2 * input[i] * input[j];
			}
 
			return result;
		}
 
		int N = round_up_power_two(n);
		assert(N >= 2);
		prepare_roots(2 * N);
		vector<mycomplex<float_t>> values(N, 0);
 
		for (int i = 0; i < n; i += 2)
			values[i / 2] = mycomplex<float_t>(input[i], i + 1 < n ? input[i + 1] : 0);
 
		fft_iterative(N, values);
 
		for (int i = 0; i <= N / 2; i++) {
			int j = (N - i) & (N - 1);
			mycomplex<float_t> even = extract(N, values, i, 0);
			mycomplex<float_t> odd = extract(N, values, i, 1);
			mycomplex<float_t> aux = even * even + odd * odd * roots[N + i] * roots[N + i];
			mycomplex<float_t> tmp = even * odd;
			values[i] = aux - mycomplex<float_t>(0, 2) * tmp;
			values[j] = conj(aux) - mycomplex<float_t>(0, 2) * conj(tmp);
		}
 
		for (int i = 0; i < N; i++)
			values[i] = conj(values[i]) * (ONE / N);
 
		fft_iterative(N, values);
		vector<T_out> result(2 * n - 1);
 
		for (int i = 0; i < (int) result.size(); i++) {
			float_t value = i % 2 == 0 ? values[i / 2].real() : values[i / 2].imag();
			result[i] = is_integral<T_out>::value ? round(value) : value;
		}
 
		return result;
	}
 
	template<typename T_out, typename T_in>
	vector<T_out> multiply(const vector<T_in> &left, const vector<T_in> &right) {
		if (left.empty() || right.empty())
			return {0};
 
		if (left == right)
			return square<T_out>(left);
 
		int n = left.size();
		int m = right.size();
 
#ifdef NEAL
		// Sanity check to make sure I'm not forgetting to split.
		double max_left = *max_element(left.begin(), left.end());
		double max_right = *max_element(right.begin(), right.end());
		assert(max(n, m) * max_left * max_right < SPLIT_CUTOFF);
#endif
 
		// Brute force when either n or m is small enough.
		if (min(n, m) < FFT_CUTOFF) {
			vector<T_out> result(n + m - 1, 0);
 
			for (int i = 0; i < n; i++)
				for (int j = 0; j < m; j++)
					result[i + j] += (T_out) left[i] * right[j];
 
			return result;
		}
 
		int N = round_up_power_two(max(n, m));
		vector<mycomplex<float_t>> values(N, 0);
 
		for (int i = 0; i < n; i++)
			values[i].real(left[i]);
 
		for (int i = 0; i < m; i++)
			values[i].imag(right[i]);
 
		fft_iterative(N, values);
 
		for (int i = 0; i <= N / 2; i++) {
			int j = (N - i) & (N - 1);
			mycomplex<float_t> product_i = extract(N, values, i, -1);
			values[i] = product_i;
			values[j] = conj(product_i);
		}
 
		invert_fft(N, values);
		vector<T_out> result(max(n, m), 0);
 
		for (int i = 0; i < (int) result.size(); i++)
			result[i] = is_integral<T_out>::value ? round(values[i].real()) : values[i].real();
 
		return result;
	}
 
	template<typename T>
	vector<T> mod_multiply(const vector<T> &left, const vector<T> &right, T mod, bool split) {
		if (left.empty() || right.empty())
			return {0};
 
		int n = left.size();
		int m = right.size();
 
		for (int i = 0; i < n; i++)
			assert(0 <= left[i] && left[i] < mod);
 
		for (int i = 0; i < m; i++)
			assert(0 <= right[i] && right[i] < mod);
 
#ifdef NEAL
		// Sanity check to make sure I'm not forgetting to split.
		assert(split || (double) max(n, m) * mod * mod < SPLIT_CUTOFF);
#endif
 
		// Brute force when either n or m is small enough. Brute force up to higher values when split = true.
		if (min(n, m) < (split ? 2 : 1) * FFT_CUTOFF) {
			const uint64_t U64_BOUND = numeric_limits<uint64_t>::max() - (uint64_t) mod * mod;
			vector<uint64_t> result(n + m - 1, 0);
 
			for (int i = 0; i < n; i++)
				for (int j = 0; j < m; j++) {
					result[i + j] += (uint64_t) left[i] * right[j];
 
					if (result[i + j] > U64_BOUND)
						result[i + j] %= mod;
				}
 
			for (uint64_t &x : result)
				if (x >= (uint64_t) mod)
					x %= mod;
 
			return vector<T>(result.begin(), result.end());
		}
 
		if (!split) {
			const vector<uint64_t> &product = multiply<uint64_t>(left, right);
			vector<T> result(n + m - 1, 0);
 
			for (int i = 0; i < (int) result.size(); i++)
				result[i] = product[i] % mod;
 
			return result;
		}
 
		int N = round_up_power_two(n + m - 1);
		vector<mycomplex<float_t>> left_fft(N, 0), right_fft(N, 0);
 
		for (int i = 0; i < n; i++) {
			left_fft[i].real(left[i] % SPLIT_BASE);
			left_fft[i].imag(left[i] / SPLIT_BASE);
		}
 
		fft_iterative(N, left_fft);
 
		if (left == right) {
			copy(left_fft.begin(), left_fft.end(), right_fft.begin());
		} else {
			for (int i = 0; i < m; i++) {
				right_fft[i].real(right[i] % SPLIT_BASE);
				right_fft[i].imag(right[i] / SPLIT_BASE);
			}
 
			fft_iterative(N, right_fft);
		}
 
		vector<mycomplex<float_t>> product(N);
		vector<T> result(n + m - 1, 0);
 
		for (int exponent = 0; exponent <= 2; exponent++) {
			uint64_t multiplier = 1;
 
			for (int k = 0; k < exponent; k++)
				multiplier = multiplier * SPLIT_BASE % mod;
 
			fill(product.begin(), product.end(), 0);
 
			for (int x = 0; x < 2; x++)
				for (int y = 0; y < 2; y++)
					if (x + y == exponent)
						for (int i = 0; i < N; i++)
							product[i] += extract(N, left_fft, i, x) * extract(N, right_fft, i, y);
 
			invert_fft(N, product);
 
			for (int i = 0; i < (int) result.size(); i++) {
				uint64_t value = round(product[i].real());
				result[i] = (result[i] + value % mod * multiplier) % mod;
			}
		}
 
		return result;
	}
 
	template<typename T>
	vector<T> mod_power(const vector<T> &v, int exponent, T mod, bool split) {
		assert(exponent >= 0);
		vector<T> result = {1};
 
		if (exponent == 0)
			return result;
 
		for (int k = 31 - __builtin_clz(exponent); k >= 0; k--) {
			result = mod_multiply(result, result, mod, split);
 
			if (exponent >> k & 1)
				result = mod_multiply(result, v, mod, split);
		}
 
		return result;
	}
}

// Credits: https://codeforces.com/profile/neal
// neal_wu's Iterative FFT: https://codeforces.com/contest/1334/submission/76217102
// FFT template was shamelessely copied from Ashishgup1's repo. Ashishgup Orz
// https://github.com/Ashishgup1/Competitive-Coding/blob/master/FFT%20(Iterative).cpp

void solve(){
    // documentation
	vi left, right; // assume left and right are the two polynomials of size N to multiply
	vi ans = FFT::mod_multiply<int>(left, right, MOD, true);
	// usually the MOD is 998244353 because its NTT friendly
	// ans is calculated mod in time O(NlogN)

	// Reference: https://www.codechef.com/viewsolution/50533932
	// I used divide and conquer above to multiply M polyonimals whose sum of sizes is N
	// time complexity for that is O(N*logN*logM)
}


clock_t startTime;
double getCurrentTime() {
	return (double)(clock() - startTime) / CLOCKS_PER_SEC;
}

signed main(){
 	ios_base::sync_with_stdio(false);
    cin.tie(NULL);
	//startTime = clock();
	// mt19937_64 rnd(time(NULL));

	int T = 1;
	cin >> T;

	while(T--){
		solve();
	}
	
	//cerr << getCurrentTime() << endl;
	return 0;
}