#include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 998244353; // constexpr int MOD = 1000000007; constexpr int DY4[]{1, 0, -1, 0}, DX4[]{0, -1, 0, 1}; constexpr int DY8[]{1, 1, 0, -1, -1, -1, 0, 1}; constexpr int DX8[]{0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T& a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T& a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; 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; } }; using ModInt = MInt; template requires requires { typename T::Monoid; typename T::OperatorMonoid; {T::m_id()} -> std::same_as; {T::o_id()} -> std::same_as; {T::m_merge(std::declval(), std::declval())} -> std::same_as; {T::o_merge(std::declval(), std::declval())} -> std::same_as; {T::apply(std::declval(), std::declval())} -> std::same_as; } struct LazySegmentTree { using Monoid = typename T::Monoid; using OperatorMonoid = typename T::OperatorMonoid; explicit LazySegmentTree(const int n) : LazySegmentTree(std::vector(n, T::m_id())) {} explicit LazySegmentTree(const std::vector& a) : n(a.size()), height(std::countr_zero(std::bit_ceil(a.size()))), p2(1 << height) { lazy.assign(p2, T::o_id()); data.assign(p2 << 1, T::m_id()); std::copy(a.begin(), a.end(), data.begin() + p2); for (int i = p2 - 1; i > 0; --i) { data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]); } } void set(int idx, const Monoid val) { idx += p2; for (int i = height; i > 0; --i) { propagate(idx >> i); } data[idx] = val; for (int i = 1; i <= height; ++i) { const int current_idx = idx >> i; data[current_idx] = T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]); } } void apply(int idx, const OperatorMonoid val) { idx += p2; for (int i = height; i > 0; --i) { propagate(idx >> i); } data[idx] = T::apply(data[idx], val); for (int i = 1; i <= height; ++i) { const int current_idx = idx >> i; data[current_idx] = T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]); } } void apply(int left, int right, const OperatorMonoid val) { if (right <= left) [[unlikely]] return; left += p2; right += p2; const int ctz_left = std::countr_zero(static_cast(left)); for (int i = height; i > ctz_left; --i) { propagate(left >> i); } const int ctz_right = std::countr_zero(static_cast(right)); for (int i = height; i > ctz_right; --i) { propagate(right >> i); } for (int l = left, r = right; l < r; l >>= 1, r >>= 1) { if (l & 1) apply_sub(l++, val); if (r & 1) apply_sub(--r, val); } for (int i = left >> (ctz_left + 1); i > 0; i >>= 1) { data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]); } for (int i = right >> (ctz_right + 1); i > 0; i >>= 1) { data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]); } } Monoid get(int left, int right) { if (right <= left) [[unlikely]] return T::m_id(); left += p2; right += p2; const int ctz_left = std::countr_zero(static_cast(left)); for (int i = height; i > ctz_left; --i) { propagate(left >> i); } const int ctz_right = std::countr_zero(static_cast(right)); for (int i = height; i > ctz_right; --i) { propagate(right >> i); } Monoid res_l = T::m_id(), res_r = T::m_id(); for (; left < right; left >>= 1, right >>= 1) { if (left & 1) res_l = T::m_merge(res_l, data[left++]); if (right & 1) res_r = T::m_merge(data[--right], res_r); } return T::m_merge(res_l, res_r); } Monoid operator[](const int idx) { const int node = idx + p2; for (int i = height; i > 0; --i) { propagate(node >> i); } return data[node]; } template int find_right(int left, const G g) { if (left >= n) [[unlikely]] return n; left += p2; for (int i = height; i > 0; --i) { propagate(left >> i); } Monoid val = T::m_id(); do { while (!(left & 1)) left >>= 1; Monoid nxt = T::m_merge(val, data[left]); if (!g(nxt)) { while (left < p2) { propagate(left); left <<= 1; nxt = T::m_merge(val, data[left]); if (g(nxt)) { val = nxt; ++left; } } return left - p2; } val = nxt; ++left; } while (!std::has_single_bit(static_cast(left))); return n; } template int find_left(int right, const G g) { if (right <= 0) [[unlikely]] return -1; right += p2; for (int i = height; i > 0; --i) { propagate((right - 1) >> i); } Monoid val = T::m_id(); do { --right; while (right > 1 && (right & 1)) right >>= 1; Monoid nxt = T::m_merge(data[right], val); if (!g(nxt)) { while (right < p2) { propagate(right); right = (right << 1) + 1; nxt = T::m_merge(data[right], val); if (g(nxt)) { val = nxt; --right; } } return right - p2; } val = nxt; } while (!std::has_single_bit(static_cast(right))); return -1; } private: const int n, height, p2; std::vector data; std::vector lazy; void apply_sub(const int idx, const OperatorMonoid& val) { data[idx] = T::apply(data[idx], val); if (idx < p2) lazy[idx] = T::o_merge(lazy[idx], val); } void propagate(const int idx) { // assert(1 <= idx && idx < p2); apply_sub(idx << 1, lazy[idx]); apply_sub((idx << 1) + 1, lazy[idx]); lazy[idx] = T::o_id(); } }; namespace monoid { template struct RangeMinimumAndUpdateQuery { using Monoid = T; using OperatorMonoid = T; static constexpr Monoid m_id() { return std::numeric_limits::max(); } static constexpr OperatorMonoid o_id() { return std::numeric_limits::max(); } static Monoid m_merge(const Monoid& a, const Monoid& b) { return std::min(a, b); } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return b == o_id() ? a : b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return b == o_id() ? a : b; } }; template struct RangeMaximumAndUpdateQuery { using Monoid = T; using OperatorMonoid = T; static constexpr Monoid m_id() { return std::numeric_limits::lowest(); } static constexpr OperatorMonoid o_id() { return std::numeric_limits::lowest(); } static Monoid m_merge(const Monoid& a, const Monoid& b) { return std::max(a, b); } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return b == o_id() ? a : b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return b == o_id()? a : b; } }; template struct RangeMinimumAndAddQuery { using Monoid = T; using OperatorMonoid = T; static constexpr Monoid m_id() { return Inf; } static constexpr OperatorMonoid o_id() { return 0; } static Monoid m_merge(const Monoid& a, const Monoid& b) { return std::min(a, b); } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return a + b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return a + b; } }; template struct RangeMaximumAndAddQuery { using Monoid = T; using OperatorMonoid = T; static constexpr Monoid m_id() { return -Inf; } static constexpr OperatorMonoid o_id() { return 0; } static Monoid m_merge(const Monoid& a, const Monoid& b) { return std::max(a, b); } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return a + b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return a + b; } }; template struct RangeSumAndUpdateQuery { using Monoid = struct { T sum; int len; }; using OperatorMonoid = T; static std::vector init(const int n) { return std::vector(n, Monoid{0, 1}); } static constexpr Monoid m_id() { return {0, 0}; } static constexpr OperatorMonoid o_id() { return std::numeric_limits::max(); } static Monoid m_merge(const Monoid& a, const Monoid& b) { return Monoid{a.sum + b.sum, a.len + b.len}; } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return b == o_id() ? a : b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return Monoid{b == o_id() ? a.sum : b * a.len, a.len}; } }; template struct RangeSumAndAddQuery { using Monoid = struct { T sum; int len; }; using OperatorMonoid = T; static std::vector init(const int n) { return std::vector(n, Monoid{0, 1}); } static constexpr Monoid m_id() { return {0, 0}; } static constexpr OperatorMonoid o_id() { return 0; } static Monoid m_merge(const Monoid& a, const Monoid& b) { return Monoid{a.sum + b.sum, a.len + b.len}; } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return a + b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return Monoid{a.sum + b * a.len, a.len}; } }; } // namespace monoid int main() { struct S { using Monoid = ModInt; using OperatorMonoid = ModInt; static constexpr Monoid m_id() { return 0; } static constexpr OperatorMonoid o_id() { return 1; } static Monoid m_merge(const Monoid& a, const Monoid& b) { return a + b; } static OperatorMonoid o_merge(const OperatorMonoid& a, const OperatorMonoid& b) { return a * b; } static Monoid apply(const Monoid& a, const OperatorMonoid& b) { return a * b; } }; int n, k; cin >> n >> k; vector a(n + 1), b(n + 1); FOR(i, 1, n + 1) cin >> a[i]; FOR(i, 1, n + 1) cin >> b[i]; LazySegmentTree dp0(n + 1), dp1(n + 1); dp0.set(0, 1); FOR(i, 1, n + 1) { // i番目に失敗 ModInt dp_i0 = 0, dp_i1 = 0; dp_i0 += dp0.get(max(i - k, 0), i) * b[i]; // Kコンボ達成できない if (i > k) dp_i1 += dp0[i - k - 1] * b[i]; // Kコンボ達成できる dp_i1 += dp1.get(max(i - k - 1, 0), i) * b[i]; dp0.apply(0, i, a[i]); dp1.apply(0, i, a[i]); dp0.set(i, dp_i0); dp1.set(i, dp_i1); // REP(j, n + 1) cout << dp0[j] << " \n"[j == n]; // REP(j, n + 1) cout << dp1[j] << " \n"[j == n]; // cout << '\n'; } cout << dp1.get(n - k, n + 1) + dp0[n - k] << '\n'; return 0; }