CF2096H Wonderful XOR Problem（FWT, *）

CF2096H Wonderful XOR Problem

Code

  #include <bits/stdc++.h>

using namespace std;
using ll = long long;
#define sz(a) (int)(a).size()
#define all(a) (a).begin(), (a).end()

template<typename A>
string to_string(A v) {
	string s = "{";
	for (auto x : v) {
		if (sz(s) > 1) {
			s += ", ";
		}
		s += to_string(x);
	}
	return s += "}";
}

void debug_out() {
	cerr << "\n";
}

template<typename T, typename... U>
void debug_out(const T& x, const U&... args) {
	cerr << " " << to_string(x);
	debug_out(args...);
}

#define debug(...) // cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)

constexpr int mod = 998244353;

void reduce(int& x) {
	(x >= mod) ? x -= mod : 0;
}

int exp(int a, int x = mod - 2) {
	int p = 1;
	for (; x; x >>= 1) {
		if (x & 1) {
			p = p * (ll)a % mod;
		}
		a = a * (ll)a % mod;
	}
	return p;
}

struct bixor {
	int a, b;								// a + b * x ^ i

	bixor() : a(1), b(0) {}
	bixor(int a, int b) : a(a), b(b) {}
};

bixor operator*(const bixor& u, const bixor& v) {
	return bixor((u.a * (ll)v.a + u.b * (ll)v.b) % mod, (u.a * (ll)v.b + u.b * (ll)v.a) % mod);
}

int main() {
    // freopen("H.in", "r", stdin);
    ios::sync_with_stdio(false);
    cin.tie(0);
    int tt;
    cin >> tt;
    while (tt--) {
        int n, m;
        cin >> n >> m;
		vector s(m, vector<int>());
		for (int p = 0; p < m; ++p) {
			s[p] = vector<int>(1 << (p + 1));
			s[p][0] = 1;
			for (int i = 1; i < sz(s[p]); ++i) {
				s[p][i] = s[p][i - 1] + (i >> p & 1 ? -1 : 1);
			}
		}
		auto gets = [&](int p, int x) { // might < 0
			return x < 0 ? 0 : s[p][x & ((1 << (p + 1)) - 1)];
		};
        int mask = 1 << m;
		auto fwt = [&](vector<int>& a, bool rev = false) {
			for (int w = 1; w < sz(a); w *= 2) {
				for (int i = 0; i < sz(a); i += w * 2) {
					for (int j = 0; j < w; ++j) {
						auto x = a[i | j], y = a[i | w | j];
						reduce(a[i | j] = x + y);
						reduce(a[i | w | j] = x - y + mod);
					}
				}
			}
			if (!rev) {
				return;
			} 
            int inv = exp(sz(a));
			for (int i = 0; i < sz(a); ++i) {
				a[i] = a[i] * (ll)inv % mod;
			}
		};
        vector S(mask, -1);
        vector<int> L(n), R(n);
        S[0] = 1;
		for (int i = 0; i < n; i++) {
            cin >> L[i] >> R[i], --L[i];
			S[0] = S[0] * (ll)(R[i] - L[i]) % mod;
        }
        debug(S[0]);
		for (int p = 0; p < m; ++p) {
            debug(p);
			int shift = 0;
			int len = 1 << (m - p - 1);
			vector<bixor> D(len);
			for (int i = 0; i < n; ++i) {
				int a = gets(p, R[i]), b = -gets(p, L[i]);
				reduce(a += mod);
				reduce(b += mod);
				int c = R[i] >> (p + 1), d = max(L[i], 0) >> (p + 1);           // L[i] = -1 -> b = 0, shall be alright
				debug(a, c, b, d);
                shift ^= c;
				d ^= c;
                // debug(a, b, d);
				D[d] = D[d] * bixor(a, b);
			}
			vector f(len, vector<int>(2));
			for (int i = 0; i < len; ++i) {
				reduce(f[i][0] = D[i].a + D[i].b);
				reduce(f[i][1] = D[i].a - D[i].b + mod);
				debug(i, D[i].a, D[i].b);
			}
			for (int w = 1; w < len; w *= 2) {
				for (int i = 0; i < len; i += w * 2) {
					for (int j = 0; j < w; ++j) {
						auto x = f[i | j], y = f[i | w | j];
						f[i | j][0] = x[0] * (ll)y[0] % mod;
						f[i | j][1] = x[1] * (ll)y[1] % mod;
						f[i | w | j][0] = x[0] * (ll)y[1] % mod;
						f[i | w | j][1] = x[1] * (ll)y[0] % mod;
					}
				}
			}
            vector<int> g(len);
            for (int i = 0; i < len; ++i) {
                g[i] = f[i][0];
            }
            fwt(g, true);
            for (int i = 0; i < len; ++i) {
                if (i < (i ^ shift)) {
                    swap(g[i], g[i ^ shift]);
                }
            }
            fwt(g);
			for (int i = 0; i < len; ++i) {
				S[i << (p + 1) | 1 << p] = g[i];
			}
		}
        debug(S);
        assert(find(all(S), -1) == S.end());
		fwt(S, true);
		debug(S);
		int res = 0, cur = 1;
		for (int i = 0; i < mask; ++i) {
			res ^= S[i] * (ll)cur % mod;
			reduce(cur += cur);
		}
		cout << res << '\n';
    }
}