#include #include #include #include #include #include #include #include template struct MInt { unsigned int v; constexpr MInt() : v(0) {} constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static constexpr int get_mod() { return M; } static constexpr void set_mod(const int divisor) { assert(std::cmp_equal(divisor, M)); } static void init(const int x) { inv(x); fact(x); fact_inv(x); } template static MInt inv(const int n) { // assert(0 <= n && n < M && std::gcd(n, M) == 1); static std::vector inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * raw(M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } constexpr MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } constexpr MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } constexpr MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } constexpr MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } constexpr auto operator<=>(const MInt& x) const = default; constexpr MInt& operator++() { if (++v == M) [[unlikely]] v = 0; return *this; } constexpr MInt operator++(int) { const MInt res = *this; ++*this; return res; } constexpr MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } constexpr MInt operator--(int) { const MInt res = *this; --*this; return res; } constexpr MInt operator+() const { return *this; } constexpr MInt operator-() const { return raw(v ? M - v : 0); } constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; } constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; } constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; template struct Mo { explicit Mo(const std::vector& ls, const std::vector& rs, const AddLeft& add_left, const AddRight& add_right, const DelLeft& del_left, const DelRight& del_right) : n(ls.size()), ptr(0), nl(0), nr(0), ls(ls), rs(rs), add_left(add_left), add_right(add_right), del_left(del_left), del_right(del_right) { const int width = ( n == 0 ? 1 : std::max(std::llround(std::ranges::max(rs) / std::sqrt(n)), 1LL)); order.resize(n); std::iota(order.begin(), order.end(), 0); std::sort(order.begin(), order.end(), [&ls, &rs, width](const int a, const int b) -> bool { if (ls[a] / width != ls[b] / width) return ls[a] < ls[b]; return (ls[a] / width) & 1 ? rs[a] < rs[b] : rs[a] > rs[b]; }); } int process() { if (ptr == n) [[unlikely]] return -1; const int id = order[ptr++]; while (ls[id] < nl) { const int idx = --nl; add_left(idx, nl, nr); } while (nr < rs[id]) { const int idx = nr++; add_right(idx, nl, nr); } while (nl < ls[id]) { const int idx = nl++; del_left(idx, nl, nr); } while (rs[id] < nr) { const int idx = --nr; del_right(idx, nl, nr); } return id; } private: const int n; int ptr, nl, nr; std::vector ls, rs, order; AddLeft add_left; AddRight add_right; DelLeft del_left; DelRight del_right; }; template std::vector> multipoint_binomial_prefix_sum( const std::vector& ns, const std::vector& ms) { using ModInt = MInt; assert(ns.size() == ms.size()); if (ns.empty()) [[unlikely]] return {}; assert(T % 2 == 1); const int q = ns.size(); for (int i = 0; i < q; ++i) { assert(0 <= ms[i] && ms[i] <= ns[i]); } ModInt::init(std::ranges::max(ns)); ModInt sum = 1; Mo mo(ms, ns, // S(n, m) -> S(n, m - 1) [&sum](const int, const int m, const int n) { sum -= ModInt::nCk(n, m + 1); }, // S(n, m) -> S(n + 1, m) [&sum](const int, const int m, const int n) { sum += sum - ModInt::nCk(n - 1, m); }, // S(n, m) -> S(n, m + 1) [&sum](const int, const int m, const int n) { sum += ModInt::nCk(n, m); }, // S(n, m) -> S(n - 1, m) [&sum](const int, const int m, const int n) { static const ModInt inv2 = ModInt::inv(2); sum = (sum + ModInt::nCk(n, m)) * inv2; }); std::vector ans(q); for (int i = 0; i < q; ++i) { const int idx = mo.process(); ans[idx] = sum; } return ans; } int main() { constexpr int MOD = 998244353; using ModInt = MInt; int t; std::cin >> t; std::vector n(t), m(t); for (int i = 0; i < t; ++i) { std::cin >> n[i] >> m[i]; --n[i]; --m[i]; } const std::vector sums = multipoint_binomial_prefix_sum(n, m); for (int i = 0; i < t; ++i) { std::cout << (ModInt::raw(2).pow(n[i] + 1) - 1) * sums[i] << '\n'; } return 0; }