1 条题解
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#include <bits/stdc++.h> #define lg2 std::__lg using std::cin; using std::cout; typedef long long ll; const int N = 530000, mod = 998244353, iv2 = (mod + 1) / 2, root = 31; typedef int vec[N], *pvec; vec inv, fact, finv; inline int max(const int x, const int y) {return x < y ? y : x;} inline void sub(int &x, const int y) {x -= y, x += x >> 31 & mod;} inline int & reduce(int &x) {return x += x >> 31 & mod;} inline int & neg(int &x) {return x = (!x - 1) & (mod - x);} ll PowerMod(ll a, int n, ll c = 1) {for (; n; n >>= 1, a = a * a % mod) if (n & 1) c = c * a % mod; return c;} void init() { int i; for (inv[1] = 1, i = 2; i < N; ++i) inv[i] = ll(mod - mod / i) * inv[mod % i] % mod; for (*finv = *fact = i = 1; i < N; ++i) fact[i] = (ll)fact[i - 1] * i % mod, finv[i] = (ll)finv[i - 1] * inv[i] % mod; } namespace Poly { int l, n; vec rev, x, y; void NTT_init(int len) { if (l == len) return; n = 1 << (l = len); ll g = PowerMod(root, 1 << (23 - l)); *x = 1, *rev = 0; for (int i = 1; i < n; ++i) x[i] = x[i - 1] * g % mod, rev[i] = rev[i >> 1] >> 1 | (i & 1) << (l - 1); } void DNTT(int *d, int *t) { int i, *j, *k, len = 1, delta = n, R; for (i = 0; i < n; ++i) t[rev[i]] = d[i]; for (i = 0; i < l; ++i) { delta >>= 1; for (k = x, j = y; j < y + len; k += delta, ++j) *j = *k; for (j = t; j < t + n; j += len << 1) for (k = j; k < j + len; ++k) R = (ll)y[k - j] * k[len] % mod, k[len] = (*k - R < 0 ? *k - R + mod : *k - R), *k = (*k + R >= mod ? *k + R - mod : *k + R); len <<= 1; } } vec B1, B2, B3, B4, B5, B6, B7; void Mul(int deg, pvec a, pvec b, pvec c) { if (!deg) {*c = (ll)*a * *b % mod; return;} NTT_init(lg2(deg) + 1); int i; ll iv = mod - (mod - 1) / n; DNTT(a, c), DNTT(b, B1); for (i = 0; i < n; ++i) B1[i] = (ll)B1[i] * c[i] % mod; DNTT(B1, c), std::reverse(c + 1, c + n); for (i = 0; i < n; ++i) c[i] = c[i] * iv % mod; } void Diff(int deg, pvec a, pvec b) {for (int i = 1; i <= deg; ++i) b[i - 1] = (ll)a[i] * i % mod;} void Exp(int deg, pvec a, pvec b) { int i, len; ll iv = iv2; pvec c = B7; assert(!*a); if (*b = 1, deg <= 1) return; if (b[1] = a[1], deg == 2) return; memset(b + 2, 0, i = 8 << lg2(deg - 1)), memset(c, 0, i), memset(B1, 0, i), *c = 1, neg(c[1] = b[1]); for (len = 1; 1 << len < deg; ++len) { NTT_init(len + 1), iv = (iv >> 1) + iv2; DNTT(c, B2), DNTT(b, B3); for (i = 0; i < n; ++i) B4[i] = (ll)B3[i] * B2[i] % mod; DNTT(B4, B5); for (i = n >> 1; i < n; ++i) B5[i] = B5[n - i] * iv % mod; memset(B5, 0, n << 1), DNTT(B5, B4); for (i = 0; i < n; ++i) B4[i] = (ll)B4[i] * B2[i] % mod; DNTT(B4, B5); for (i = n >> 1; i < n; ++i) B5[i] = B5[n - i] * (mod - iv) % mod; memcpy(B5, c, n << 1), DNTT(B5, B6); Diff(n >> 1, b, B1), DNTT(B1, B4); for (i = 0; i < n; ++i) B4[i] = (ll)B4[i] * B6[i] % mod; DNTT(B4, B6); for (i = n >> 1; i < n; ++i) reduce(B5[i] = (a[i] - B6[n - i + 1] * iv % mod * inv[i]) % mod); memset(B5, 0, n << 1), DNTT(B5, B4); for (i = 0; i < n; ++i) B4[i] = (ll)B4[i] * B3[i] % mod; DNTT(B4, B5); for (i = n >> 1; i < n; ++i) b[i] = B5[n - i] * iv % mod; if (2 << len >= deg) return; DNTT(b, B3); for (i = 0; i < n; ++i) B3[i] = (ll)B3[i] * B2[i] % mod; DNTT(B3, B4); for (i = n >> 1; i < n; ++i) B4[i] = B4[n - i] * iv % mod; memset(B4, 0, n << 1), DNTT(B4, B3); for (i = 0; i < n; ++i) B3[i] = (ll)B3[i] * B2[i] % mod; DNTT(B3, B4); for (i = n >> 1; i < n; ++i) c[i] = B4[n - i] * (mod - iv) % mod; } } } namespace maker { int n, lim, a[9]; vec c, f, F, E[5], EI; vec C1, C2, C3, C4, C5; void solve(int L, int w) { int i, $, R = L + (1 << w), M; ll iv; if (!w) { if (L) { int d, c = (ll(E[1][L] - E[0][L] + mod) * inv[L] + EI[L] + ll(L == *a - 1) * finv[*a]) % mod; for ($ = 2; $ < n; ++$) if (d = L - *a + $[a] + 1, d >= 0) c = (c + (ll)$[E][d] * finv[*a - $[a]]) % mod; f[L] = c % mod, F[L] = (ll)L * f[L] % mod; for ($ = 0; $ < n; ++$) $[E][L] = ((ll)$[E][L] * inv[L] + (ll)$[a] * f[L]) % mod; sub(EI[L], E[0][L]); } else for ($ = 0; $ < n; ++$) $[E][0] = 1; return; } solve(L, w - 1); if ((M = (1 << (w - 1)) + L) > lim) return; Poly::NTT_init(w), iv = mod - (mod - 1) / Poly::n; if (L) { memcpy(C2, F, 4 << w), memcpy(C3, F + L, 2 << w), memset(C3 + (1 << (w - 1)), 0, 2 << w), Poly::DNTT(C2, C1), Poly::DNTT(C3, C2); for ($ = 0; $ < n; ++$) { memcpy(C4, $[E], 4 << w), memcpy(C5, $[E] + L, 2 << w), memset(C5 + (1 << (w - 1)), 0, 2 << w), Poly::DNTT(C4, C3), Poly::DNTT(C5, C4); for (i = 0; i < Poly::n; ++i) C3[i] = ((ll)C1[i] * C4[i] + (ll)C2[i] * C3[i]) % mod; Poly::DNTT(C3, C5); for (i = M; i < R; ++i) $[E][i] = ($[E][i] + C5[Poly::n - (i - L)] * iv % mod * $[a]) % mod; if (!$) { Poly::DNTT(c, C3); for (i = 0; i < Poly::n; ++i) C3[i] = (ll)C3[i] * C4[i] % mod; Poly::DNTT(C3, C4); for (i = M; i < R; ++i) EI[i] = (EI[i] + C4[Poly::n - (i - L)] * iv) % mod; } } } else { memcpy(C2, F, 2 << w), memset(C2 + M, 0, 2 << w), Poly::DNTT(C2, C1); for ($ = 0; $ < n; ++$) { memcpy(C3, $[E], 2 << w), memset(C3 + M, 0, 2 << w), Poly::DNTT(C3, C2); for (i = 0; i < Poly::n; ++i) C3[i] = (ll)C1[i] * C2[i] % mod; Poly::DNTT(C3, C4); for (i = M; i < R; ++i) $[E][i] = ($[E][i] + C4[Poly::n - i] * iv % mod * $[a]) % mod; if (!$) { Poly::DNTT(c, C3); for (i = 0; i < Poly::n; ++i) C2[i] = (ll)C2[i] * C3[i] % mod; Poly::DNTT(C2, C3); for (i = M; i < R; ++i) EI[i] = (EI[i] + C3[Poly::n - i] * iv) % mod; } } } solve(M, w - 1); } int main(int _n, int *_a, int _lim, pvec result) { int i; n = _n, *a = _a[n - 1], lim = _lim; for (i = 1; i < n; ++i) a[i] = *a - _a[i - 1]; for (i = 0; i < lim; ++i) c[i] = mod - finv[i + 1]; memset(f, 0, 1 << (lg2(lim) + 3)), memset(F, 0, lim << 2), memset(EI, 0, lim << 2); for (i = 0; i < 5; ++i) memset(E[i], 0, lim << 2); solve(0, 19); for (i = 0; i < lim; ++i) f[i] = (ll)f[i] * lim % mod; Poly::Exp(lim, f, F); for (i = 1; i <= lim; ++i) result[i] = (ll)F[lim - i] * finv[i - 1] % mod * fact[lim - 1] % mod; return 0; } } int R, C, n, q, nr, nc; int ri[9], ci[9]; vec rf, cf, f, qa, qb; int main() { int i, qi, ans = 0; init(); std::ios::sync_with_stdio(false), cin.tie(NULL); cin >> R >> C >> q >> nr >> nc, qi = PowerMod(q, mod - 2); for (i = 0; i < nr; ++i) cin >> ri[i]; for (i = 0; i < nc; ++i) cin >> ci[i]; maker::main(nr, ri, R, rf), maker::main(nc, ci, C, cf); n = R + C, qa[1] = *qa = 1; for (i = 2; i <= n; ++i) qa[i] = (ll)qa[i - 1] * q % mod; for (i = 2; i <= n; ++i) qa[i] = (ll)qa[i] * qa[i - 1] % mod; n = max(R, C), qb[1] = *qb = 1; for (i = 2; i <= n; ++i) qb[i] = (ll)qb[i - 1] * qi % mod; for (i = 2; i <= n; ++i) qb[i] = (ll)qb[i] * qb[i - 1] % mod; for (i = 1; i <= R; ++i) rf[i - 1] = (ll)rf[i] * qb[i] % mod; for (i = 1; i <= C; ++i) cf[i - 1] = (ll)cf[i] * qb[i] % mod; rf[R] = cf[C] = 0; Poly::Mul(R + C - 2, rf, cf, f); for (i = 2; i <= R + C; ++i) ans = (ans + (ll)f[i - 2] * qa[i]) % mod; cout << ans << '\n'; return 0; }
- 1
信息
- ID
- 2235
- 时间
- 5000ms
- 内存
- 512MiB
- 难度
- 10
- 标签
- 递交数
- 4
- 已通过
- 4
- 上传者