PROBLEM LINK:
Div-1 Contest
Div-2 Contest
Div-3 Contest
Practice
Author & Editorialist: Samarth Gupta
Tester: Felipe Mota
DIFFICULTY:
MEDIUM
PREREQUISITES:
Bitwise XOR, FFT/NTT
PROBLEM:
Given an array A, you need to compute sum of bitwise XOR of all subsequences of A of length less than equal to M.
QUICK EXPLANATION:
Let’s precompute the answer for every M. Consider the i^{th} bit. Let there be n_1 ones and n_0 zeros. The answer for subsequences of size M is \sum\limits_{i=1,i+=2}^{n_1} {n_1 \choose i}{n_0 \choose M-i}. This is clearly a convolution and so we can use FFT/NTT to evaluate it for every M and answer the queries in O(1).
EXPLANATION:
We know that in bitwise xor, bits i and j(i != j) are independent and so we can solve the problem for each bit independently. Consider the i^{th} bit from right. Suppose we need to select subsequences of size M. Assuming there are n_1 ones and n_0 zeros, the number of ways to select subsequences of size M such that bitwise XOR is 1 can be written as:
because in the subsequence we need to select odd number of 1's and remaining bits as 0.
The sum of subsequences of size M can be found using:
Clearly we can pre-compute the answer for every M and answer the queries in O(1) using prefix sum arrays. The complexity of this approach is O(N^2*log(max(A_i))+ Q) which is sufficient for subtask 1.
Constructing the Polynomials
Consider the expression for S_{M,i}. We can see this as a convolution and can think of polynomials which would give us the same expression when multiplied.
Consider P(x) = \sum\limits_{i=1,i+=2}^{n_1}{n_1 \choose i}x^i and Q(x) = \sum\limits_{i=0}^{n_0}{n_0 \choose i}x^i. What happens when we multiply P(x) and Q(x)?
Let’s try to find the coefficient of x^M in the polynomial R(x) = P(x)Q(x). We take x^i from P(x) and x^{M-i} from Q(x) for all valid odd i and add them up. We can then see the coefficient of x^M in R(x) is S_{M, i}.
Polynomial multiplication
Naive Multiplication of polynomials P(x) and Q(x) is O(N^2) for a particular bit and so O(N^2*log(max(A_i))) overall which has room for improvement if we do polynomial multiplication using FFT/NTT. Answering the queries is O(1) since we just need to maintain a prefix-sum array.
Overall Complexity: O(N*log(N)*log(max(A_i)) + Q)
SOLUTIONS:
Setter's & Editorialist Solution
#include<bits/stdc++.h>
#define ll long long
#define ull unsigned ll
#define For(i,j,k) for (int i=(int)(j);i<=(int)(k);i++)
#define Rep(i,j,k) for (int i=(int)(j);i>=(int)(k);i--)
using namespace std;
const int mo=998244353;
const int FFTN=1<<18;
#define poly vector<int>
namespace FFT{
int w[FFTN+5],W[FFTN+5],R[FFTN+5];
int power(int x,int y){
int s=1;
for (;y;y/=2,x=1ll*x*x%mo)
if (y&1) s=1ll*s*x%mo;
return s;
}
void FFTinit(){
W[0]=1;
W[1]=power(3,(mo-1)/FFTN);
For(i,2,FFTN) W[i]=1ll*W[i-1]*W[1]%mo;
}
int FFTinit(int n){
int L=1;
for (;L<=n;L<<=1);
For(i,0,L-1) R[i]=(R[i>>1]>>1)|((i&1)?(L>>1):0);
return L;
}
int A[FFTN+5],B[FFTN+5];
ull p[FFTN+5];
void DFT(int *a,int n){
For(i,0,n-1) p[R[i]]=a[i];
for (int d=1;d<n;d<<=1){
int len=FFTN/(d<<1);
for (int i=0,j=0;i<d;i++,j+=len) w[i]=W[j];
for (int i=0;i<n;i+=(d<<1))
for (int j=0;j<d;j++){
int y=p[i+j+d]*w[j]%mo;
p[i+j+d]=p[i+j]+mo-y;
p[i+j]+=y;
}
if (d==1<<15)
For(i,0,n-1) p[i]%=mo;
}
For(i,0,n-1) a[i]=p[i]%mo;
}
void IDFT(int *a,int n){
For(i,0,n-1) p[R[i]]=a[i];
for (int d=1;d<n;d<<=1){
int len=FFTN/(d<<1);
for (int i=0,j=FFTN;i<d;i++,j-=len) w[i]=W[j];
for (int i=0;i<n;i+=(d<<1))
for (int j=0;j<d;j++){
int y=p[i+j+d]*w[j]%mo;
p[i+j+d]=p[i+j]+mo-y;
p[i+j]+=y;
}
if (d==1<<15)
For(i,0,n-1) p[i]%=mo;
}
int val=power(n,mo-2);
For(i,0,n-1) a[i]=p[i]*val%mo;
}
poly Mul(const poly &a,const poly &b){
int sza=a.size()-1,szb=b.size()-1;
poly ans(sza+szb+1);
if (sza<=30||szb<=30){
For(i,0,sza) For(j,0,szb)
ans[i+j]=(ans[i+j]+1ll*a[i]*b[j])%mo;
return ans;
}
int L=FFTinit(sza+szb);
For(i,0,L-1) A[i]=(i<=sza?a[i]:0);
For(i,0,L-1) B[i]=(i<=szb?b[i]:0);
DFT(A,L); DFT(B,L);
For(i,0,L-1) A[i]=1ll*A[i]*B[i]%mo;
IDFT(A,L);
For(i,0,sza+szb) ans[i]=A[i];
return ans;
}
}
using FFT::Mul;
using FFT::power;
#define mxn 200005
int fact[mxn];
int inv[mxn];
void pre()
{
fact[0] = 1;
int i;
For(i,1,mxn-1)
fact[i] = fact[i-1]*1ll*i%mo;
inv[mxn-1] = power(fact[mxn-1], mo-2);
Rep(i, mxn-2, 0)
inv[i] = inv[i+1]*1ll*(i+1)%mo;
}
int nCr(int n, int r)
{
if(n < 0 || r < 0 || n < r)
return 0;
int ans = fact[n]*1ll*inv[r]%mo;
ans = ans*1ll*inv[n-r]%mo;
return ans;
}
poly compute(int n, poly &arr)
{
poly ans(n+1);
For(bit, 0, 29)
{
int k = 0;
For(i, 0, n-1)
if((arr[i] >> bit)&1)
k++;
poly ones(k+1), zero_odd(n-k+1);
for(int i=1;i<=k;i+=2) // ones(x) = C(k, 1)*x + C(k, 3)*x^3 + C(k, 5)*x^5....
ones[i] = nCr(k, i);
For(i, 0, (n-k)) // zero_odd(x) = C(n-k, 0) + C(n-k, 1)*x+....
zero_odd[i] = nCr(n-k, i);
poly odd = Mul(ones, zero_odd);
For(i, 1, n)
ans[i] = (ans[i] + odd[i]*1ll*(1 << bit)%mo)%mo;
}
return ans;
}
int main(){
ios_base::sync_with_stdio(false);
cin.tie(0);
FFT::FFTinit();
pre();
int n;
cin >> n;
poly arr(n);
For(i, 0 ,n-1)
cin >> arr[i];
poly ans = compute(n, arr);
For(i, 1, n)
ans[i] = (ans[i] + ans[i-1])%mo;
int q;
cin >> q;
while(q--)
{
int m;
cin >> m;
cout << ans[m] << '\n';
}
}
Tester's Solution
#include <bits/stdc++.h>
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#include <utility>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
struct barrett {
unsigned int _m;
unsigned long long im;
barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
unsigned int umod() const { return _m; }
unsigned int mul(unsigned int a, unsigned int b) const {
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(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 (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
} // namespace internal
} // namespace atcoder
#include <cassert>
#include <numeric>
#include <type_traits>
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = internal::barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
dynamic_modint(T v) {
long long x = (long long)(v % (long long)(mod()));
if (x < 0) x += mod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
dynamic_modint(T v) {
_v = (unsigned int)(v % mod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
auto eg = internal::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static internal::barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
int bsf(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
static constexpr int g = internal::primitive_root<mint::mod()>;
int n = int(a.size());
int h = internal::ceil_pow2(n);
static bool first = true;
static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
if (first) {
first = false;
mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mint::mod() - 1);
mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
mint now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_e[i] = es[i] * now;
now *= ies[i];
}
}
for (int ph = 1; ph <= h; ph++) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint now = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * now;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
now *= sum_e[bsf(~(unsigned int)(s))];
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
static constexpr int g = internal::primitive_root<mint::mod()>;
int n = int(a.size());
int h = internal::ceil_pow2(n);
static bool first = true;
static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
if (first) {
first = false;
mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mint::mod() - 1);
mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
mint now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_ie[i] = ies[i] * now;
now *= es[i];
}
}
for (int ph = h; ph >= 1; ph--) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint inow = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(mint::mod() + l.val() - r.val()) *
inow.val();
}
inow *= sum_ie[bsf(~(unsigned int)(s))];
}
}
}
} // namespace internal
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 60) {
if (n < m) {
std::swap(n, m);
std::swap(a, b);
}
std::vector<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 z = 1 << internal::ceil_pow2(n + m - 1);
a.resize(z);
internal::butterfly(a);
b.resize(z);
internal::butterfly(b);
for (int i = 0; i < z; i++) {
a[i] *= b[i];
}
internal::butterfly_inv(a);
a.resize(n + m - 1);
mint iz = mint(z).inv();
for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
return a;
}
template <unsigned int mod = 998244353,
class T,
std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
using mint = static_modint<mod>;
std::vector<mint> a2(n), b2(m);
for (int i = 0; i < n; i++) {
a2[i] = mint(a[i]);
}
for (int i = 0; i < m; i++) {
b2[i] = mint(b[i]);
}
auto c2 = convolution(move(a2), move(b2));
std::vector<T> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
c[i] = c2[i].val();
}
return c;
}
std::vector<long long> convolution_ll(const std::vector<long long>& a,
const std::vector<long long>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
static constexpr unsigned long long MOD1 = 754974721; // 2^24
static constexpr unsigned long long MOD2 = 167772161; // 2^25
static constexpr unsigned long long MOD3 = 469762049; // 2^26
static constexpr unsigned long long M2M3 = MOD2 * MOD3;
static constexpr unsigned long long M1M3 = MOD1 * MOD3;
static constexpr unsigned long long M1M2 = MOD1 * MOD2;
static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
static constexpr unsigned long long i1 =
internal::inv_gcd(MOD2 * MOD3, MOD1).second;
static constexpr unsigned long long i2 =
internal::inv_gcd(MOD1 * MOD3, MOD2).second;
static constexpr unsigned long long i3 =
internal::inv_gcd(MOD1 * MOD2, MOD3).second;
auto c1 = convolution<MOD1>(a, b);
auto c2 = convolution<MOD2>(a, b);
auto c3 = convolution<MOD3>(a, b);
std::vector<long long> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
unsigned long long x = 0;
x += (c1[i] * i1) % MOD1 * M2M3;
x += (c2[i] * i2) % MOD2 * M1M3;
x += (c3[i] * i3) % MOD3 * M1M2;
long long diff =
c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
if (diff < 0) diff += MOD1;
static constexpr unsigned long long offset[5] = {
0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
x -= offset[diff % 5];
c[i] = x;
}
return c;
}
} // namespace atcoder
using namespace std;
using mint = atcoder::modint998244353;
int main(){
ios::sync_with_stdio(false);
cin.tie(0);
const int mod = 998244353;
int n;
cin >> n;
vector<int> a(n);
vector<mint> ans(n + 1);
for(int i = 0; i < n; i++)
cin >> a[i];
vector<mint> fat(n + 1, 1), ifat(n + 1);
for(int i = 1; i <= n; i++)
fat[i] = fat[i - 1] * i;
for(int i = 0; i <= n; i++)
ifat[i] = mint(1) / fat[i];
auto ncr = [&](int n, int r){
if(n < r) return mint(0);
return fat[n] * ifat[r] * ifat[n - r];
};
for(int b = 0; b < 30; b++){
int c[2] = {};
for(int i = 0; i < n; i++){
c[(a[i]>>b)&1]++;
}
vector<int> A(n + 1, 0), B(n + 1, 0);
for(int i = 0; i <= c[0]; i++) A[i] = ncr(c[0], i).val();
for(int i = 1; i <= c[1]; i += 2) B[i] = ncr(c[1], i).val();
auto res = atcoder::convolution(A, B);
for(int i = 0; i <= n; i++){
ans[i] += mint(2).pow(b) * res[i];
}
}
for(int i = 1; i <= n; i++)
ans[i] += ans[i - 1];
int q;
cin >> q;
while(q--){
int x;
cin >> x;
cout << ans[x].val() << '\n';
}
return 0;
}