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#define PROBLEM "https://judge.yosupo.jp/problem/aplusb" #include "my_template.hpp" #include "mod/modint.hpp" #include "mod/floor_sum_of_linear_polynomial.hpp" using mint = modint998; template <int K1, int K2> void test_1() { FOR(M, 1, 10) { FOR(a, 10) { FOR(b, 10) { array<array<mint, K2 + 1>, K1 + 1> dp{}; FOR(N, 10) { array<array<mint, K2 + 1>, K1 + 1> ans = floor_sum_of_linear_polynomial_nonnegative<mint, K1, K2, u64>(N, a, b, M); assert(dp == ans); mint y = floor(a * N + b, M); FOR(i, K1 + 1) FOR(j, K2 + 1) dp[i][j] += mint(N).pow(i) * y.pow(j); } } } } } template <int K1, int K2> void test_2() { FOR(M, 1, 10) { FOR(a, -5, 6) { FOR(b, -5, 6) { FOR(L, -5, 6) { array<array<mint, K2 + 1>, K1 + 1> dp{}; FOR(R, L, 6) { array<array<mint, K2 + 1>, K1 + 1> ans = floor_sum_of_linear_polynomial<mint, K1, K2, ll>(L, R, a, b, M); assert(dp == ans); mint y = floor(a * R + b, M); FOR(i, K1 + 1) FOR(j, K2 + 1) dp[i][j] += mint(R).pow(i) * y.pow(j); } } } } } } void solve() { int a, b; cin >> a >> b; cout << a + b << "\n"; } signed main() { test_1<0, 0>(); test_1<0, 1>(); test_1<0, 2>(); test_1<1, 0>(); test_1<1, 1>(); test_1<1, 2>(); test_1<2, 0>(); test_1<2, 1>(); test_1<2, 2>(); test_1<10, 10>(); test_2<0, 0>(); test_2<0, 1>(); test_2<0, 2>(); test_2<1, 0>(); test_2<1, 1>(); test_2<1, 2>(); test_2<2, 0>(); test_2<2, 1>(); test_2<2, 2>(); test_2<10, 10>(); solve(); return 0; }
#line 1 "test/1_mytest/floor_sum_of_polynomial.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/aplusb" #line 1 "my_template.hpp" #if defined(LOCAL) #include <my_template_compiled.hpp> #else // https://codeforces.com/blog/entry/96344 #pragma GCC optimize("Ofast,unroll-loops") // いまの CF だとこれ入れると動かない? // #pragma GCC target("avx2,popcnt") #include <bits/stdc++.h> using namespace std; using ll = long long; using u8 = uint8_t; using u16 = uint16_t; using u32 = uint32_t; using u64 = uint64_t; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template <class T> constexpr T infty = 0; template <> constexpr int infty<int> = 1'010'000'000; template <> constexpr ll infty<ll> = 2'020'000'000'000'000'000; template <> constexpr u32 infty<u32> = infty<int>; template <> constexpr u64 infty<u64> = infty<ll>; template <> constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000; template <> constexpr double infty<double> = infty<ll>; template <> constexpr long double infty<long double> = infty<ll>; using pi = pair<ll, ll>; using vi = vector<ll>; template <class T> using vc = vector<T>; template <class T> using vvc = vector<vc<T>>; template <class T> using vvvc = vector<vvc<T>>; template <class T> using vvvvc = vector<vvvc<T>>; template <class T> using vvvvvc = vector<vvvvc<T>>; template <class T> using pq = priority_queue<T>; template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>; #define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__)) #define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_sgn(int x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(ll x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u64 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template <typename T> T kth_bit(int k) { return T(1) << k; } template <typename T> bool has_kth_bit(T x, int k) { return x >> k & 1; } template <typename T> T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template <typename T> T ceil(T x, T y) { return floor(x + y - 1, y); } template <typename T> T bmod(T x, T y) { return x - y * floor(x, y); } template <typename T> pair<T, T> divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template <typename T, typename U> T SUM(const vector<U> &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template <typename T> T POP(deque<T> &que) { T a = que.front(); que.pop_front(); return a; } template <typename T> T POP(pq<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(pqg<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(vc<T> &que) { T a = que.back(); que.pop_back(); return a; } template <typename F> ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template <typename F> double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template <class T, class S> inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template <class T, class S> inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc<int> s_to_vi(const string &S, char first_char) { vc<int> A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template <typename T, typename U> vector<T> cumsum(vector<U> &A, int off = 1) { int N = A.size(); vector<T> B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template <typename T> vector<int> argsort(const vector<T> &A) { vector<int> ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template <typename T> vc<T> rearrange(const vc<T> &A, const vc<int> &I) { vc<T> B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } template <typename T, typename... Vectors> void concat(vc<T> &first, const Vectors &... others) { vc<T> &res = first; (res.insert(res.end(), others.begin(), others.end()), ...); } #endif #line 3 "test/1_mytest/floor_sum_of_polynomial.test.cpp" #line 2 "mod/modint_common.hpp" struct has_mod_impl { template <class T> static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{}); template <class T> static auto check(...) -> std::false_type; }; template <class T> class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {}; template <typename mint> mint inv(int n) { static const int mod = mint::get_mod(); static vector<mint> dat = {0, 1}; assert(0 <= n); if (n >= mod) n %= mod; while (len(dat) <= n) { int k = len(dat); int q = (mod + k - 1) / k; dat.eb(dat[k * q - mod] * mint::raw(q)); } return dat[n]; } template <typename mint> mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n && n < mod); static vector<mint> dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat))); return dat[n]; } template <typename mint> mint fact_inv(int n) { static vector<mint> dat = {1, 1}; if (n < 0) return mint(0); while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat))); return dat[n]; } template <class mint, class... Ts> mint fact_invs(Ts... xs) { return (mint(1) * ... * fact_inv<mint>(xs)); } template <typename mint, class Head, class... Tail> mint multinomial(Head &&head, Tail &&... tail) { return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...); } template <typename mint> mint C_dense(int n, int k) { assert(n >= 0); if (k < 0 || n < k) return 0; static vvc<mint> C; static int H = 0, W = 0; auto calc = [&](int i, int j) -> mint { if (i == 0) return (j == 0 ? mint(1) : mint(0)); return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0); }; if (W <= k) { FOR(i, H) { C[i].resize(k + 1); FOR(j, W, k + 1) { C[i][j] = calc(i, j); } } W = k + 1; } if (H <= n) { C.resize(n + 1); FOR(i, H, n + 1) { C[i].resize(W); FOR(j, W) { C[i][j] = calc(i, j); } } H = n + 1; } return C[n][k]; } template <typename mint, bool large = false, bool dense = false> mint C(ll n, ll k) { assert(n >= 0); if (k < 0 || n < k) return 0; if constexpr (dense) return C_dense<mint>(n, k); if constexpr (!large) return multinomial<mint>(n, k, n - k); k = min(k, n - k); mint x(1); FOR(i, k) x *= mint(n - i); return x * fact_inv<mint>(k); } template <typename mint, bool large = false> mint C_inv(ll n, ll k) { assert(n >= 0); assert(0 <= k && k <= n); if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k); return mint(1) / C<mint, 1>(n, k); } // [x^d](1-x)^{-n} template <typename mint, bool large = false, bool dense = false> mint C_negative(ll n, ll d) { assert(n >= 0); if (d < 0) return mint(0); if (n == 0) { return (d == 0 ? mint(1) : mint(0)); } return C<mint, large, dense>(n + d - 1, d); } #line 3 "mod/modint.hpp" template <int mod> struct modint { static constexpr u32 umod = u32(mod); static_assert(umod < u32(1) << 31); u32 val; static modint raw(u32 v) { modint x; x.val = v; return x; } constexpr modint() : val(0) {} constexpr modint(u32 x) : val(x % umod) {} constexpr modint(u64 x) : val(x % umod) {} constexpr modint(u128 x) : val(x % umod) {} constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){}; bool operator<(const modint &other) const { return val < other.val; } modint &operator+=(const modint &p) { if ((val += p.val) >= umod) val -= umod; return *this; } modint &operator-=(const modint &p) { if ((val += umod - p.val) >= umod) val -= umod; return *this; } modint &operator*=(const modint &p) { val = u64(val) * p.val % umod; return *this; } modint &operator/=(const modint &p) { *this *= p.inverse(); return *this; } modint operator-() const { return modint::raw(val ? mod - val : u32(0)); } modint operator+(const modint &p) const { return modint(*this) += p; } modint operator-(const modint &p) const { return modint(*this) -= p; } modint operator*(const modint &p) const { return modint(*this) *= p; } modint operator/(const modint &p) const { return modint(*this) /= p; } bool operator==(const modint &p) const { return val == p.val; } bool operator!=(const modint &p) const { return val != p.val; } modint inverse() const { int a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return modint(u); } modint pow(ll n) const { assert(n >= 0); modint ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } static constexpr int get_mod() { return mod; } // (n, r), r は 1 の 2^n 乗根 static constexpr pair<int, int> ntt_info() { if (mod == 120586241) return {20, 74066978}; if (mod == 167772161) return {25, 17}; if (mod == 469762049) return {26, 30}; if (mod == 754974721) return {24, 362}; if (mod == 880803841) return {23, 211}; if (mod == 943718401) return {22, 663003469}; if (mod == 998244353) return {23, 31}; if (mod == 1004535809) return {21, 582313106}; if (mod == 1012924417) return {21, 368093570}; return {-1, -1}; } static constexpr bool can_ntt() { return ntt_info().fi != -1; } }; #ifdef FASTIO template <int mod> void rd(modint<mod> &x) { fastio::rd(x.val); x.val %= mod; // assert(0 <= x.val && x.val < mod); } template <int mod> void wt(modint<mod> x) { fastio::wt(x.val); } #endif using modint107 = modint<1000000007>; using modint998 = modint<998244353>; #line 1 "mod/floor_sum_of_linear_polynomial.hpp" #line 2 "alg/monoid_pow.hpp" // chat gpt template <typename U, typename Arg1, typename Arg2> struct has_power_method { private: // ヘルパー関数の実装 template <typename V, typename A1, typename A2> static auto check(int) -> decltype(std::declval<V>().power(std::declval<A1>(), std::declval<A2>()), std::true_type{}); template <typename, typename, typename> static auto check(...) -> std::false_type; public: // メソッドの有無を表す型 static constexpr bool value = decltype(check<U, Arg1, Arg2>(0))::value; }; template <typename Monoid> typename Monoid::X monoid_pow(typename Monoid::X x, ll exp) { using X = typename Monoid::X; if constexpr (has_power_method<Monoid, X, ll>::value) { return Monoid::power(x, exp); } else { assert(exp >= 0); X res = Monoid::unit(); while (exp) { if (exp & 1) res = Monoid::op(res, x); x = Monoid::op(x, x); exp >>= 1; } return res; } } #line 2 "mod/floor_monoid_product.hpp" // https://yukicoder.me/submissions/883884 // https://qoj.ac/contest/1411/problem/7620 // U は範囲内で ax+b がオーバーフローしない程度 // yyy x yyyy x ... yyy x yyy (x を N 個) // k 個目の x までに floor(ak+b,m) 個の y がある // my<=ax+b における lattice path における辺の列と見なせる template <typename Monoid, typename X, typename U> X floor_monoid_product(X x, X y, U N, U a, U b, U m) { U c = (a * N + b) / m; X pre = Monoid::unit(), suf = Monoid::unit(); while (1) { const U p = a / m, q = b / m; a %= m, b %= m; x = Monoid::op(x, monoid_pow<Monoid>(y, p)); pre = Monoid::op(pre, monoid_pow<Monoid>(y, q)); c -= (p * N + q); if (c == 0) break; const U d = (m * c - b - 1) / a + 1; suf = Monoid::op(y, Monoid::op(monoid_pow<Monoid>(x, N - d), suf)); b = m - b - 1 + a, N = c - 1, c = d; swap(m, a), swap(x, y); } x = monoid_pow<Monoid>(x, N); return Monoid::op(Monoid::op(pre, x), suf); } #line 1 "alg/monoid/monoid_for_floor_sum.hpp" // sum i^k1floor^k2: floor path で (x,y) から x 方向に進むときに x^k1y^k2 を足す template <typename T, int K1, int K2> struct Monoid_for_floor_sum { using ARR = array<array<T, K2 + 1>, K1 + 1>; struct Data { ARR dp; T dx, dy; }; using value_type = Data; using X = value_type; static X op(X a, X b) { static constexpr int n = max(K1, K2); static T comb[n + 1][n + 1]; if (comb[0][0] != T(1)) { comb[0][0] = T(1); FOR(i, n) FOR(j, i + 1) { comb[i + 1][j] += comb[i][j], comb[i + 1][j + 1] += comb[i][j]; } } array<T, K1 + 1> pow_x; array<T, K2 + 1> pow_y; pow_x[0] = 1, pow_y[0] = 1; FOR(i, K1) pow_x[i + 1] = pow_x[i] * a.dx; FOR(i, K2) pow_y[i + 1] = pow_y[i] * a.dy; // +dy FOR(i, K1 + 1) { FOR_R(j, K2 + 1) { T x = b.dp[i][j]; FOR(k, j + 1, K2 + 1) b.dp[i][k] += comb[k][j] * pow_y[k - j] * x; } } // +dx FOR(j, K2 + 1) { FOR_R(i, K1 + 1) { FOR(k, i, K1 + 1) a.dp[k][j] += comb[k][i] * pow_x[k - i] * b.dp[i][j]; } } a.dx += b.dx, a.dy += b.dy; return a; } static X to_x() { X x = unit(); x.dp[0][0] = 1, x.dx = 1; return x; } static X to_y() { X x = unit(); x.dy = 1; return x; } static constexpr X unit() { return {ARR{}, T(0), T(0)}; } static constexpr bool commute = 0; }; #line 4 "mod/floor_sum_of_linear_polynomial.hpp" // 全部非負, T は答, U は ax+b がオーバーフローしない template <typename T, int K1, int K2, typename U> array<array<T, K2 + 1>, K1 + 1> floor_sum_of_linear_polynomial_nonnegative(U N, U a, U b, U mod) { static_assert(is_same_v<U, u64> || is_same_v<U, u128>); assert(a == 0 || N < (U(-1) - b) / a); using Mono = Monoid_for_floor_sum<T, K1, K2>; auto x = floor_monoid_product<Mono>(Mono::to_x(), Mono::to_y(), N, a, b, mod); return x.dp; }; // sum_{L<=x<R} x^i floor(ax+b,mod)^j // a+bx が I, U でオーバーフローしない template <typename T, int K1, int K2, typename I> array<array<T, K2 + 1>, K1 + 1> floor_sum_of_linear_polynomial(I L, I R, I a, I b, I mod) { static_assert(is_same_v<I, ll> || is_same_v<I, i128>); assert(L <= R && mod > 0); if (a < 0) { auto ANS = floor_sum_of_linear_polynomial<T, K1, K2, I>(-R + 1, -L + 1, -a, b, mod); FOR(i, K1 + 1) { if (i % 2 == 1) { FOR(j, K2 + 1) ANS[i][j] = -ANS[i][j]; } } return ANS; } assert(a >= 0); I ADD_X = L; I N = R - L; b += a * L; I ADD_Y = floor<I>(b, mod); b -= ADD_Y * mod; assert(a >= 0 && b >= 0); using Mono = Monoid_for_floor_sum<T, K1, K2>; using Data = typename Mono::Data; using U = std::conditional_t<is_same_v<I, ll>, i128, u128>; Data A = floor_monoid_product<Mono, Data, U>(Mono::to_x(), Mono::to_y(), N, a, b, mod); Data offset = Mono::unit(); offset.dx = T(ADD_X), offset.dy = T(ADD_Y); A = Mono::op(offset, A); return A.dp; }; #line 6 "test/1_mytest/floor_sum_of_polynomial.test.cpp" using mint = modint998; template <int K1, int K2> void test_1() { FOR(M, 1, 10) { FOR(a, 10) { FOR(b, 10) { array<array<mint, K2 + 1>, K1 + 1> dp{}; FOR(N, 10) { array<array<mint, K2 + 1>, K1 + 1> ans = floor_sum_of_linear_polynomial_nonnegative<mint, K1, K2, u64>(N, a, b, M); assert(dp == ans); mint y = floor(a * N + b, M); FOR(i, K1 + 1) FOR(j, K2 + 1) dp[i][j] += mint(N).pow(i) * y.pow(j); } } } } } template <int K1, int K2> void test_2() { FOR(M, 1, 10) { FOR(a, -5, 6) { FOR(b, -5, 6) { FOR(L, -5, 6) { array<array<mint, K2 + 1>, K1 + 1> dp{}; FOR(R, L, 6) { array<array<mint, K2 + 1>, K1 + 1> ans = floor_sum_of_linear_polynomial<mint, K1, K2, ll>(L, R, a, b, M); assert(dp == ans); mint y = floor(a * R + b, M); FOR(i, K1 + 1) FOR(j, K2 + 1) dp[i][j] += mint(R).pow(i) * y.pow(j); } } } } } } void solve() { int a, b; cin >> a >> b; cout << a + b << "\n"; } signed main() { test_1<0, 0>(); test_1<0, 1>(); test_1<0, 2>(); test_1<1, 0>(); test_1<1, 1>(); test_1<1, 2>(); test_1<2, 0>(); test_1<2, 1>(); test_1<2, 2>(); test_1<10, 10>(); test_2<0, 0>(); test_2<0, 1>(); test_2<0, 2>(); test_2<1, 0>(); test_2<1, 1>(); test_2<1, 2>(); test_2<2, 0>(); test_2<2, 1>(); test_2<2, 2>(); test_2<10, 10>(); solve(); return 0; }