带通配符单模式串匹配

n≤300000n \le 300000

Luogu P4173 残缺的字符串

Solution

令通配符=0

若字符a,ba,b匹配，则(a−b)2ab=0(a-b)^2ab=0，又因为(a−b)2ab(a-b)^2ab为非负整数，所以只需要∑=0\sum=0

直接把模式串翻转，用ff来判断是否匹配

f[i]=∑j=0i(A[j]−B[i−j])2A[j]B[i−j]=∑j=0iA[j]3B[i−j]−2∑j=0iA[j]2B[i−j]2+∑j=0iA[j]B[i−j]3 \begin{aligned} f[i] &= \sum_{j=0}^{i}(A[j] - B[i - j])^2A[j]B[i - j]\\\\ &= \sum_{j=0}^{i}A[j]^3B[i - j] - 2\sum_{j=0}^{i}A[j]^2B[i - j]^2 + \sum_{j=0}^{i}A[j]B[i - j]^3 \end{aligned}

做三次FFT即可

考虑到FFT常数太大，我们只需要先做6次DFT，最后IDFT一次即可

Code

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#include <bits/stdc++.h>

#define x first
#define y second
#define y1 Y1
#define y2 Y2
#define mp make_pair
#define pb push_back

using namespace std;

typedef long long LL;
typedef pair <int, int> pii;

template <typename T> inline int Chkmax (T &a, T b) { return a < b ? a = b, 1 : 0; }
template <typename T> inline int Chkmin (T &a, T b) { return a > b ? a = b, 1 : 0; }
template <typename T> inline T read ()
{
	T sum = 0, fl = 1; char ch = getchar();
	for (; !isdigit(ch); ch = getchar()) if (ch == '-') fl = -1;
	for (; isdigit(ch); ch = getchar()) sum = (sum << 3) + (sum << 1) + ch - '0';
	return sum * fl;
}

inline void proc_status ()
{
	ifstream t ("/proc/self/status");
	cerr << string (istreambuf_iterator <char> (t), istreambuf_iterator <char> ()) << endl;
}

const int Maxn = 1500000;
const double eps = 0.5, PI = acos(-1.0);

int N, M;
double A[Maxn], B[Maxn], C[Maxn], D[Maxn];
char S[Maxn];

struct Complex 
{
	double x, y;

	inline Complex operator + (const Complex &rhs) const { return (Complex){x + rhs.x, y + rhs.y}; }
	inline Complex operator - (const Complex &rhs) const { return (Complex){x - rhs.x, y - rhs.y}; }
	inline Complex operator * (const Complex &rhs) const { return (Complex){x * rhs.x - y * rhs.y, x * rhs.y + y * rhs.x}; }
	inline Complex operator * (const double &rhs) const { return (Complex){x * rhs, y * rhs}; }
} Ans[Maxn];

namespace Poly
{
	int n, rev[Maxn];
	Complex F[Maxn], G[Maxn];

	inline void init (int N, int M)
	{

		n = 1; while (n < N + M) n <<= 1;
		for (int i = 0; i < n; ++i) rev[i] = (rev[i >> 1] >> 1) + (i & 1 ? (n >> 1) : 0);
	}

	inline void DFT (Complex A[], int flag)
	{
		for (int i = 0; i < n; ++i) if (rev[i] < i) swap(A[i], A[rev[i]]);

		for (int mid = 1; mid < n; mid <<= 1)
		{
			Complex Wn = (Complex){cos(PI / mid), sin(PI / mid) * flag};

			for (int i = 0; i < n; i += (mid << 1))
			{
				Complex W = (Complex){1, 0};
				for (int j = i; j < i + mid; ++j, W = W * Wn)
				{
					Complex a = A[j], b = W * A[j + mid];
					A[j] = a + b, A[j + mid] = a - b;
				}
			}
		}

		if (flag < 0) for (int i = 0; i < n; ++i) A[i].x /= n;
	}

	inline void FFT (double A[], int N, double B[], int M)
	{
/**/	for (int i = 0; i < n; ++i) F[i] = (Complex){i < N ? A[i] : 0, 0};
		for (int i = 0; i < n; ++i) G[i] = (Complex){i < M ? B[i] : 0, 0};

		DFT(F, 1), DFT(G, 1);
		for (int i = 0; i < n; ++i) F[i] = F[i] * G[i];
	}

	inline void Work ()
	{
		init(N, M);
		for (int i = 0; i < N; ++i) C[i] = A[i] * A[i] * A[i];
		for (int i = 0; i < M; ++i) D[i] = B[i];
		FFT (C, N, D, M);
		for (int i = 0; i < n; ++i) Ans[i] = Ans[i] + F[i];

		for (int i = 0; i < N; ++i) C[i] = A[i] * A[i];
		for (int i = 0; i < M; ++i) D[i] = B[i] * B[i];
		FFT (C, N, D, M);
		for (int i = 0; i < n; ++i) Ans[i] = Ans[i] - F[i] * 2.0;

		for (int i = 0; i < N; ++i) C[i] = A[i];
		for (int i = 0; i < M; ++i) D[i] = B[i] * B[i] * B[i];
		FFT (C, N, D, M);
		for (int i = 0; i < n; ++i) Ans[i] = Ans[i] + F[i];

		DFT(Ans, -1);
//		for (int i = 0; i < n; ++i) printf("%.0lf ", Ans[i].x); puts("");
	}
}

vector <int> p;

inline void Solve ()
{
	Poly :: Work ();
	for (int i = N - 1; i < M; ++i) if (fabs(Ans[i].x) < eps) p.pb(i - N + 2);

	cout<<p.size()<<endl;
	for (int i = 0; i < p.size(); ++i) printf("%d ", p[i]);
}

inline void Input ()
{
	N = read<int>(), M = read<int>();
	scanf("%s", S); for (int i = 0; i < N; ++i) if (S[i] != '*') A[i] = S[i] - 'a' + 1;
	scanf("%s", S); for (int i = 0; i < M; ++i) if (S[i] != '*') B[i] = S[i] - 'a' + 1;
	reverse(A, A + N);
}

int main()
{
#ifndef ONLINE_JUDGE
	freopen("A.in", "r", stdin);
	freopen("A.out", "w", stdout);
#endif
	Input();
	Solve();
	return 0;
}

Debug

  • 84L：i < N写成i <= N