You are given two sets of integers: $A$ and $B$. You need to output the sum of elements in the set $C = \{x | x = a \oplus b, a \in A, b \in B\}$ modulo $998244353$, where $\oplus$ denotes the bitwise XOR operation. Each number should be counted only once.
For example, if $A = \{2, 3\}$ and $B = \{2, 3\}$ you should count integer $1$ only once, despite the fact that you can get it as $3 \oplus 2$ and as $2 \oplus 3$. So the answer for this case is equal to $1 + 0 = 1$.
Let’s call a segment $[l; r]$ a set of integers $\{l, l+1, \dots, r\}$.
The set $A$ is given as a union of $n_A$ segments, the set $B$ is given as a union of $n_B$ segments.Input
The first line contains a single integer $n_A$ ($1 \le n_A \le 100$).
The $i$-th of the next $n_A$ lines contains two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le 10^{18}$), describing a segment of values of set $A$.
The next line contains a single integer $n_B$ ($1 \le n_B \le 100$).
The $i$-th of the next $n_B$ lines contains two integers $l_j$ and $r_j$ ($1 \le l_j \le r_j \le 10^{18}$), describing a segment of values of set $B$.
Note that segments in both sets may intersect.Output
Print one integer — the sum of all elements in set $C = \{x | x = a \oplus b, a \in A, b \in B\}$ modulo $998244353$.Examplesinput
2 3 5 5 8 3 1 2 1 9 2 9
output
112
input
1 1 9 2 2 4 2 10
output
120
Note
In the second example, we can discover that the set $C = \{0,1,\dots,15\}$, which means that all numbers between $0$ and $15$ can be represented as $a \oplus b$.
Solution:
#undef _GLIBCXX_DEBUG #include <bits/stdc++.h> using namespace std; template <typename A, typename B> string to_string(pair<A, B> p); template <typename A, typename B, typename C> string to_string(tuple<A, B, C> p); template <typename A, typename B, typename C, typename D> string to_string(tuple<A, B, C, D> p); string to_string(const string& s) { return '"' + s + '"'; } string to_string(const char* s) { return to_string((string) s); } string to_string(bool b) { return (b ? "true" : "false"); } string to_string(vector<bool> v) { bool first = true; string res = "{"; for (int i = 0; i < static_cast<int>(v.size()); i++) { if (!first) { res += ", "; } first = false; res += to_string(v[i]); } res += "}"; return res; } template <size_t N> string to_string(bitset<N> v) { string res = ""; for (size_t i = 0; i < N; i++) { res += static_cast<char>('0' + v[i]); } return res; } template <typename A> string to_string(A v) { bool first = true; string res = "{"; for (const auto &x : v) { if (!first) { res += ", "; } first = false; res += to_string(x); } res += "}"; return res; } template <typename A, typename B> string to_string(pair<A, B> p) { return "(" + to_string(p.first) + ", " + to_string(p.second) + ")"; } template <typename A, typename B, typename C> string to_string(tuple<A, B, C> p) { return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")"; } template <typename A, typename B, typename C, typename D> string to_string(tuple<A, B, C, D> p) { return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")"; } void debug_out() { cerr << endl; } template <typename Head, typename... Tail> void debug_out(Head H, Tail... T) { cerr << " " << to_string(H); debug_out(T...); } #ifdef LOCAL #define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__) #else #define debug(...) 42 #endif template <typename T> T inverse(T a, T m) { T u = 0, v = 1; while (a != 0) { T t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return u; } template <typename T> class Modular { public: using Type = typename decay<decltype(T::value)>::type; constexpr Modular() : value() {} template <typename U> Modular(const U& x) { value = normalize(x); } template <typename U> static Type normalize(const U& x) { Type v; if (-mod() <= x && x < mod()) v = static_cast<Type>(x); else v = static_cast<Type>(x % mod()); if (v < 0) v += mod(); return v; } const Type& operator()() const { return value; } template <typename U> explicit operator U() const { return static_cast<U>(value); } constexpr static Type mod() { return T::value; } Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; } Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; } template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); } template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); } Modular& operator++() { return *this += 1; } Modular& operator--() { return *this -= 1; } Modular operator++(int) { Modular result(*this); *this += 1; return result; } Modular operator--(int) { Modular result(*this); *this -= 1; return result; } Modular operator-() const { return Modular(-value); } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) { #ifdef _WIN32 uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value); uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m; asm( "divl %4; \n\t" : "=a" (d), "=d" (m) : "d" (xh), "a" (xl), "r" (mod()) ); value = m; #else value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value)); #endif return *this; } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) { int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod()); value = normalize(value * rhs.value - q * mod()); return *this; } template <typename U = T> typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) { value = normalize(value * rhs.value); return *this; } Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); } template <typename U> friend const Modular<U>& abs(const Modular<U>& v) { return v; } template <typename U> friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend std::istream& operator>>(std::istream& stream, Modular<U>& number); private: Type value; }; template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; } template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); } template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; } template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; } template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template<typename T, typename U> Modular<T> power(const Modular<T>& a, const U& b) { assert(b >= 0); Modular<T> x = a, res = 1; U p = b; while (p > 0) { if (p & 1) res *= x; x *= x; p >>= 1; } return res; } template <typename T> bool IsZero(const Modular<T>& number) { return number() == 0; } template <typename T> string to_string(const Modular<T>& number) { return to_string(number()); } template <typename T> std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) { return stream << number(); } template <typename T> std::istream& operator>>(std::istream& stream, Modular<T>& number) { typename common_type<typename Modular<T>::Type, int64_t>::type x; stream >> x; number.value = Modular<T>::normalize(x); return stream; } /* using ModType = int; struct VarMod { static ModType value; }; ModType VarMod::value; ModType& md = VarMod::value; using Mint = Modular<VarMod>; */ constexpr int md = 998244353; using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>; int main() { ios::sync_with_stdio(false); cin.tie(0); const int BITS = 61; vector<vector<vector<long long>>> all(2, vector<vector<long long>>(BITS)); vector<vector<vector<long long>>> any(2, vector<vector<long long>>(BITS)); for (int rot = 0; rot < 2; rot++) { auto Add = [&](long long from, long long to) { int bits = (from == to ? 0 : 64 - __builtin_clzll(from ^ to)); assert(to - from + 1 == (1LL << bits)); all[rot][bits].push_back(from); }; int n; cin >> n; for (int i = 0; i < n; i++) { long long L, R; cin >> L >> R; while ((L | (L - 1)) <= R) { long long new_L = (L | (L - 1)) + 1; Add(L, new_L - 1); L = new_L; } while (L <= (R & (R + 1))) { long long new_R = (R & (R + 1)) - 1; Add(new_R + 1, R); R = new_R; } assert(L == R + 1); } for (int k = 0; k < BITS; k++) { sort(all[rot][k].begin(), all[rot][k].end()); all[rot][k].resize(unique(all[rot][k].begin(), all[rot][k].end()) - all[rot][k].begin()); sort(any[rot][k].begin(), any[rot][k].end()); any[rot][k].resize(unique(any[rot][k].begin(), any[rot][k].end()) - any[rot][k].begin()); if (k < BITS - 1) { for (long long x : all[rot][k]) { any[rot][k + 1].push_back(x & (~(1LL << k))); } for (long long x : any[rot][k]) { any[rot][k + 1].push_back(x & (~(1LL << k))); } } } } vector<pair<int, int>> trie(1, make_pair(-1, -1)); vector<bool> finish(1, false); Mint ans = 0; for (int k = BITS - 1; k >= 0; k--) { vector<long long> v; auto Do = [&](long long x, long long y) { long long z = x ^ y; v.push_back(z); }; for (long long x : all[0][k]) { for (long long y : all[1][k]) { Do(x, y); } } for (long long x : all[0][k]) { for (long long y : any[1][k]) { Do(x, y); } } for (long long x : any[0][k]) { for (long long y : all[1][k]) { Do(x, y); } } sort(v.begin(), v.end()); v.resize(unique(v.begin(), v.end()) - v.begin()); for (long long x : v) { int t = 0; bool good = true; for (int b = BITS - 1; b >= k; b--) { int& to = ((x & (1LL << b)) ? trie[t].first : trie[t].second); if (to == -1) { to = (int) trie.size(); trie.emplace_back(-1, -1); finish.push_back(false); } t = ((x & (1LL << b)) ? trie[t].first : trie[t].second); if (finish[t]) { good = false; break; } } if (good) { finish[t] = true; ans += (Mint(x) + Mint(x + (1LL << k) - 1)) * Mint(1LL << k) / Mint(2); } } } cout << ans << '\n'; return 0; }