题目大意:给定长度为$n-1$的数组$g_{[1,n)}$,求$f_{[0,n)}$,要求:
$$
f_i=\sum_{j=1}^if_{i-j}g_j\\
f_0=1
$$
题解:直接求复杂度是$O(n^2)$,明显不可以通过此题
分治$FFT$,可以用$CDQ$分治,先求出$f_{[l,mid)}$,可以发现这部分对区间的$f_{[mid,r)}$的贡献是$f_{[l,mid)}*g_{[0,r-l)}$,卷出来加到对应位置就行了,复杂度$O(n\log_2^2n)$
卡点:无
C++ Code:
#include <algorithm> #include <cstdio> #include <cctype> namespace std { struct istream { #define M (1 << 21 | 3) char buf[M], *ch = buf - 1; inline istream() { #ifndef ONLINE_JUDGE freopen("input.txt", "r", stdin); #endif fread(buf, 1, M, stdin); } inline istream& operator >> (int &x) { while (isspace(*++ch)); for (x = *ch & 15; isdigit(*++ch); ) x = x * 10 + (*ch & 15); return *this; } #undef M } cin; struct ostream { #define M (1 << 21 | 3) char buf[M], *ch = buf - 1; int w; inline ostream& operator << (int x) { if (!x) { *++ch = '0'; return *this; } for (w = 1; w <= x; w *= 10); for (w /= 10; w; w /= 10) *++ch = (x / w) ^ 48, x %= w; return *this; } inline ostream& operator << (const char x) {*++ch = x; return *this;} inline ~ostream() { #ifndef ONLINE_JUDGE freopen("output.txt", "w", stdout); #endif fwrite(buf, 1, ch - buf + 1, stdout); } #undef M } cout; } #define maxn 131072 | 3 const int mod = 998244353, G = 3; namespace Math { inline int pw(int base, int p) { static int res; for (res = 1; p; p >>= 1, base = static_cast<long long> (base) * base % mod) if (p & 1) res = static_cast<long long> (res) * base % mod; return res; } inline int inv(int x) {return pw(x, mod - 2);} } int n; int f[maxn], g[maxn]; namespace Poly { #define N 131072 | 3 int s, lim, ilim, rev[N]; int Wn[N + 1]; inline void reduce(int &x) {x += x >> 31 & mod;} inline void clear(register int *l, const int *r) { if (l >= r) return ; while (l != r) *l++ = 0; } inline void init(const int n) { s = -1, lim = 1; while (lim <= n) lim <<= 1, s++; ilim = Math::inv(lim); for (int i = 1; i < lim; i++) rev[i] = rev[i >> 1] >> 1 | (i & 1) << s; const int t = Math::pw(G, (mod - 1) / lim); *Wn = 1; for (register int *i = Wn; i != Wn + lim; ++i) *(i + 1) = static_cast<long long> (*i) * t % mod; } inline void NTT(int *A, const int op = 1) { for (register int i = 1; i < lim; i++) if (i < rev[i]) std::swap(A[i], A[rev[i]]); for (register int mid = 1; mid < lim; mid <<= 1) { const int t = lim / mid >> 1; for (register int i = 0; i < lim; i += mid << 1) { for (register int j = 0; j < mid; j++) { const int W = op ? Wn[t * j] : Wn[lim - t * j]; const int X = A[i + j], Y = static_cast<long long> (A[i + j + mid]) * W % mod; reduce(A[i + j] += Y - mod), reduce(A[i + j + mid] = X - Y); } } } if (!op) for (int i = 0; i < lim; i++) A[i] = static_cast<long long> (A[i]) * ilim % mod; } int A[N], B[N]; void CDQ_NTT(const int l, const int r) { if (r - l < 2) return ; const int mid = l + r >> 1; CDQ_NTT(l, mid); init(r - l); std::copy(f + l, f + mid, A); clear(A + mid - l, A + lim); std::copy(g, g + r - l, B); clear(B + r - l, B + lim); NTT(A), NTT(B); for (int i = 0; i < lim; i++) A[i] = static_cast<long long> (A[i]) * B[i] % mod; NTT(A, 0); for (int i = mid; i < r; i++) reduce(f[i] += A[i - l] - mod); CDQ_NTT(mid, r); } #undef N } int main() { std::cin >> n; for (int i = 1; i < n; i++) std::cin >> g[i]; *f = 1; Poly::CDQ_NTT(0, n); for (int i = 0; i < n; i++) std::cout << f[i] << ' '; std::cout << '\n'; return 0; }
来源:https://www.cnblogs.com/Memory-of-winter/p/10127918.html