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Subset Sum Problem.cpp
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Subset Sum Problem.cpp
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#include<bits/stdc++.h>
using namespace std;
const int N = 3e5 + 9, mod = 998244353;
struct base {
double x, y;
base() { x = y = 0; }
base(double x, double y): x(x), y(y) { }
};
inline base operator + (base a, base b) { return base(a.x + b.x, a.y + b.y); }
inline base operator - (base a, base b) { return base(a.x - b.x, a.y - b.y); }
inline base operator * (base a, base b) { return base(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); }
inline base conj(base a) { return base(a.x, -a.y); }
int lim = 1;
vector<base> roots = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
const double PI = acosl(- 1.0);
void ensure_base(int p) {
if(p <= lim) return;
rev.resize(1 << p);
for(int i = 0; i < (1 << p); i++) rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (p - 1));
roots.resize(1 << p);
while(lim < p) {
double angle = 2 * PI / (1 << (lim + 1));
for(int i = 1 << (lim - 1); i < (1 << lim); i++) {
roots[i << 1] = roots[i];
double angle_i = angle * (2 * i + 1 - (1 << lim));
roots[(i << 1) + 1] = base(cos(angle_i), sin(angle_i));
}
lim++;
}
}
void fft(vector<base> &a, int n = -1) {
if(n == -1) n = a.size();
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = lim - 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++) {
base z = a[i + j + k] * roots[j + k];
a[i + j + k] = a[i + j] - z;
a[i + j] = a[i + j] + z;
}
}
}
}
//eq = 0: 4 FFTs in total
//eq = 1: 3 FFTs in total
vector<int> multiply(vector<int> &a, vector<int> &b, int eq = 0) {
int need = a.size() + b.size() - 1;
int p = 0;
while((1 << p) < need) p++;
ensure_base(p);
int sz = 1 << p;
vector<base> A, B;
if(sz > (int)A.size()) A.resize(sz);
for(int i = 0; i < (int)a.size(); i++) {
int x = (a[i] % mod + mod) % mod;
A[i] = base(x & ((1 << 15) - 1), x >> 15);
}
fill(A.begin() + a.size(), A.begin() + sz, base{0, 0});
fft(A, sz);
if(sz > (int)B.size()) B.resize(sz);
if(eq) copy(A.begin(), A.begin() + sz, B.begin());
else {
for(int i = 0; i < (int)b.size(); i++) {
int x = (b[i] % mod + mod) % mod;
B[i] = base(x & ((1 << 15) - 1), x >> 15);
}
fill(B.begin() + b.size(), B.begin() + sz, base{0, 0});
fft(B, sz);
}
double ratio = 0.25 / sz;
base r2(0, - 1), r3(ratio, 0), r4(0, - ratio), r5(0, 1);
for(int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
base a1 = (A[i] + conj(A[j])), a2 = (A[i] - conj(A[j])) * r2;
base b1 = (B[i] + conj(B[j])) * r3, b2 = (B[i] - conj(B[j])) * r4;
if(i != j) {
base c1 = (A[j] + conj(A[i])), c2 = (A[j] - conj(A[i])) * r2;
base d1 = (B[j] + conj(B[i])) * r3, d2 = (B[j] - conj(B[i])) * r4;
A[i] = c1 * d1 + c2 * d2 * r5;
B[i] = c1 * d2 + c2 * d1;
}
A[j] = a1 * b1 + a2 * b2 * r5;
B[j] = a1 * b2 + a2 * b1;
}
fft(A, sz); fft(B, sz);
vector<int> res(need);
for(int i = 0; i < need; i++) {
long long aa = A[i].x + 0.5;
long long bb = B[i].x + 0.5;
long long cc = A[i].y + 0.5;
res[i] = (aa + ((bb % mod) << 15) + ((cc % mod) << 30))%mod;
}
return res;
}
template <int32_t MOD>
struct modint {
int32_t value;
modint() = default;
modint(int32_t value_) : value(value_) {}
inline modint<MOD> operator + (modint<MOD> other) const { int32_t c = this->value + other.value; return modint<MOD>(c >= MOD ? c - MOD : c); }
inline modint<MOD> operator - (modint<MOD> other) const { int32_t c = this->value - other.value; return modint<MOD>(c < 0 ? c + MOD : c); }
inline modint<MOD> operator * (modint<MOD> other) const { int32_t c = (int64_t)this->value * other.value % MOD; return modint<MOD>(c < 0 ? c + MOD : c); }
inline modint<MOD> & operator += (modint<MOD> other) { this->value += other.value; if (this->value >= MOD) this->value -= MOD; return *this; }
inline modint<MOD> & operator -= (modint<MOD> other) { this->value -= other.value; if (this->value < 0) this->value += MOD; return *this; }
inline modint<MOD> & operator *= (modint<MOD> other) { this->value = (int64_t)this->value * other.value % MOD; if (this->value < 0) this->value += MOD; return *this; }
inline modint<MOD> operator - () const { return modint<MOD>(this->value ? MOD - this->value : 0); }
modint<MOD> pow(uint64_t k) const {
modint<MOD> x = *this, y = 1;
for (; k; k >>= 1) {
if (k & 1) y *= x;
x *= x;
}
return y;
}
modint<MOD> inv() const { return pow(MOD - 2); } // MOD must be a prime
inline modint<MOD> operator / (modint<MOD> other) const { return *this * other.inv(); }
inline modint<MOD> operator /= (modint<MOD> other) { return *this *= other.inv(); }
inline bool operator == (modint<MOD> other) const { return value == other.value; }
inline bool operator != (modint<MOD> other) const { return value != other.value; }
inline bool operator < (modint<MOD> other) const { return value < other.value; }
inline bool operator > (modint<MOD> other) const { return value > other.value; }
};
template <int32_t MOD> modint<MOD> operator * (int64_t value, modint<MOD> n) { return modint<MOD>(value) * n; }
template <int32_t MOD> modint<MOD> operator * (int32_t value, modint<MOD> n) { return modint<MOD>(value % MOD) * n; }
template <int32_t MOD> ostream & operator << (ostream & out, modint<MOD> n) { return out << n.value; }
using mint = modint<mod>;
struct poly {
vector<mint> a;
inline void normalize() {
while((int)a.size() && a.back() == 0) a.pop_back();
}
template<class...Args> poly(Args...args): a(args...) { }
poly(const initializer_list<mint> &x): a(x.begin(), x.end()) { }
int size() const { return (int)a.size(); }
inline mint coef(const int i) const { return (i < a.size() && i >= 0) ? a[i]: mint(0); }
mint operator[](const int i) const { return (i < a.size() && i >= 0) ? a[i]: mint(0); } //Beware!! p[i] = k won't change the value of p.a[i]
bool is_zero() const {
for (int i = 0; i < size(); i++) if (a[i] != 0) return 0;
return 1;
}
poly operator + (const poly &x) const {
int n = max(size(), x.size());
vector<mint> ans(n);
for(int i = 0; i < n; i++) ans[i] = coef(i) + x.coef(i);
while ((int)ans.size() && ans.back() == 0) ans.pop_back();
return ans;
}
poly operator - (const poly &x) const {
int n = max(size(), x.size());
vector<mint> ans(n);
for(int i = 0; i < n; i++) ans[i] = coef(i) - x.coef(i);
while ((int)ans.size() && ans.back() == 0) ans.pop_back();
return ans;
}
poly operator * (const poly& b) const {
if(is_zero() || b.is_zero()) return {};
vector<int> A, B;
for(auto x: a) A.push_back(x.value);
for(auto x: b.a) B.push_back(x.value);
auto res = multiply(A, B, (A == B));
vector<mint> ans;
for(auto x: res) ans.push_back(mint(x));
while ((int)ans.size() && ans.back() == 0) ans.pop_back();
return ans;
}
poly operator * (const mint& x) const {
int n = size();
vector<mint> ans(n);
for(int i = 0; i < n; i++) ans[i] = a[i] * x;
return ans;
}
poly operator / (const mint &x) const{ return (*this) * x.inv(); }
poly& operator += (const poly &x) { return *this = (*this) + x; }
poly& operator -= (const poly &x) { return *this = (*this) - x; }
poly& operator *= (const poly &x) { return *this = (*this) * x; }
poly& operator *= (const mint &x) { return *this = (*this) * x; }
poly& operator /= (const mint &x) { return *this = (*this) / x; }
poly mod_xk(int k) const { return {a.begin(), a.begin() + min(k, size())}; } //modulo by x^k
poly mul_xk(int k) const { // multiply by x^k
poly ans(*this);
ans.a.insert(ans.a.begin(), k, 0);
return ans;
}
poly div_xk(int k) const { // divide by x^k
return vector<mint>(a.begin() + min(k, (int)a.size()), a.end());
}
poly substr(int l, int r) const { // return mod_xk(r).div_xk(l)
l = min(l, size());
r = min(r, size());
return vector<mint>(a.begin() + l, a.begin() + r);
}
poly differentiate() const {
int n = size(); vector<mint> ans(n);
for(int i = 1; i < size(); i++) ans[i - 1] = coef(i) * i;
return ans;
}
poly integrate() const {
int n = size(); vector<mint> ans(n + 1);
for(int i = 0; i < size(); i++) ans[i + 1] = coef(i) / (i + 1);
return ans;
}
poly inverse(int n) const { // 1 / p(x) % x^n, O(nlogn)
assert(!is_zero()); assert(a[0] != 0);
poly ans{mint(1) / a[0]};
for(int i = 1; i < n; i *= 2) {
ans = (ans * mint(2) - ans * ans * mod_xk(2 * i)).mod_xk(2 * i);
}
return ans.mod_xk(n);
}
poly log(int n) const { //ln p(x) mod x^n
assert(a[0] == 1);
return (differentiate().mod_xk(n) * inverse(n)).integrate().mod_xk(n);
}
poly exp(int n) const { //e ^ p(x) mod x^n
if(is_zero()) return {1};
assert(a[0] == 0);
poly ans({1});
int i = 1;
while(i < n) {
poly C = ans.log(2 * i).div_xk(i) - substr(i, 2 * i);
ans -= (ans * C).mod_xk(i).mul_xk(i);
i *= 2;
}
return ans.mod_xk(n);
}
};
// number of subsets of an array of n elements having sum equal to k for each k from 1 to m
int32_t main() {
ios_base::sync_with_stdio(0);
cin.tie(0);
int n, m; cin >> n >> m;
vector<int> a(m + 1, 0);
for (int i = 0; i < n; i++) {
int k; cin >> k; // k >= 1, handle [k = 0] separately
if (k <= m) a[k]++;
}
poly p(m + 1, 0);
for (int i = 1; i <= m; i++) {
for (int j = 1; i * j <= m; j++) {
if (j & 1) p.a[i * j] += mint(a[i]) / j;
else p.a[i * j] -= mint(a[i]) / j;
}
}
p = p.exp(m + 1);
for (int i = 1; i <= m; i++) cout << p[i] << ' '; cout << '\n'; // check for m = 0
return 0;
}
// https://judge.yosupo.jp/problem/sharp_p_subset_sum