[CH3401]石头游戏

Description

[CH3401]石头游戏

Solution

因为\(gcd(1,2,3,4,5,6)=60\)，所以每\(60s\)内的变换是固定的，可以用\(60\)个矩阵表示，其中\(\forall x\in[1,60]\),令\(A[x][0][0]=1\)，作为一个源源不断的常数来源
\(F[i]\)为行从\(0\)到\(0\)，列从\(0\)到\(n\ast m\)的矩阵，表示第\(i\)秒的状态，其中\(F[0][0]=1\)
使\(t=60*p+q\)，令\(A\)为\(60\)个矩阵相乘，\(F[0]\ast A^p*\prod^{q}_{i=1}A[i]\)则为最终矩阵
在这个矩阵里找最大值即可

Code

#include<iostream>
#include<cstdio>
#include<cstring>
using namespace std;
#define ll long long
#define num(x, y) ((x - 1) * m + y)
struct Mat {
	ll x, y;
	ll val[65][65];
} f[61];
Mat operator * (const Mat &a, const Mat &b) {
	Mat c;
	c.x = a.x;
	c.y = b.y;
	for (register ll i = 0; i <= c.x; i++)
		for (register ll j = 0; j <= c.y; j++) {
			c.val[i][j] = 0;
			for (register ll k = 0; k <= a.y; k++)
				c.val[i][j] += a.val[i][k] * b.val[k][j];
		}
	return c;
}
ll n, m, t, act, X, Y, ans;
ll c[101], Len[101];
char str[101], s[101][101];
Mat a, HSH;
inline ll read() {
	ll s = 0, w = 1;
	char c = getchar();
	for (; !isdigit(c); c = getchar()) if (c == '-') w = -1;
	for (; isdigit(c); c = getchar()) s = (s << 1) + (s << 3) + (c ^ 48);
	return s * w;
}
Mat Pow(Mat a, int k) {
	Mat ans;
	ans.x = 64, ans.y = 64;
	for (register ll i = 0; i <= 64; i++)
		for (register ll j = 0; j <= 64; j++)
			if (i == j) ans.val[i][j] = 1;
			else ans.val[i][j] = 0;
	while (k) {
		if (k & 1) ans = ans * a;
		a = a * a;
		k >>= 1;
	}
	return ans;
}
int main() {
	n = read(), m = read(), t = read(), act = read();
	for (register ll i = 1; i <= n; i++) {
		scanf("%s", str + 1);
		for (register ll j = 1; j <= m; j++)
			c[num(i, j)] = (str[j] ^ 48);
	}
	for (register ll i = 0; i < act; i++) {
		scanf("%s", s[i] + 1);
		Len[i] = strlen(s[i] + 1);
	}
	a.x = 0, a.y = 64;
	a.val[0][0] = 1;
	for (register ll k = 1; k <= 60; k++) {
		f[k].x = 64;
		f[k].y = 64;
		f[k].val[0][0] = 1;
		for (register ll i = 1; i <= n; i++)
			for (register ll j = 1; j <= m; j++) {
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '0') f[k].val[0][num(i, j)] = 0, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '1') f[k].val[0][num(i, j)] = 1, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '2') f[k].val[0][num(i, j)] = 2, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '3') f[k].val[0][num(i, j)] = 3, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '4') f[k].val[0][num(i, j)] = 4, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '5') f[k].val[0][num(i, j)] = 5, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '6') f[k].val[0][num(i, j)] = 6, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '7') f[k].val[0][num(i, j)] = 7, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '8') f[k].val[0][num(i, j)] = 8, f[k].val[num(i, j)][num(i, j)] = 1;
				if (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == '9') f[k].val[0][num(i, j)] = 9, f[k].val[num(i, j)][num(i, j)] = 1;
				if (i > 1 && (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == 'N')) f[k].val[num(i, j)][num(i - 1, j)] = 1;
				if (j > 1 && (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == 'W')) f[k].val[num(i, j)][num(i, j - 1)] = 1;
				if (i < n && (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == 'S')) f[k].val[num(i, j)][num(i + 1, j)] = 1;
				if (j < m && (s[c[num(i, j)]][(k - 1) % (Len[c[num(i, j)]]) + 1] == 'E')) f[k].val[num(i, j)][num(i, j + 1)] = 1;
			}
	}
	HSH.x = 64, HSH.y = 64;
	for (register ll i = 0; i <= 64; i++)
			HSH.val[i][i] = 1;
	for (register ll i = 1; i <= 60; i++)
		HSH	= HSH * f[i];
	X = t / 60;
	Y = t % 60;
	if (X) a = a * Pow(HSH, X);
	for (register ll i = 1; i <= Y; i++)
		a = a * f[i];
	for (register ll i = 1; i <= 64; i++)
		ans = max(ans, a.val[0][i]);
	printf("%lld", ans);
	return 0;
}