[2018 徐州 网络赛|Hard to prepare ] 环形染色问题的公式解法

Posted 2021-10-29 PushyTao

题目来源
After Incident, a feast is usually held in Hakurei Shrine. This time Reimu asked Kokoro to deliver a Nogaku show during the feast. To enjoy the show, every audience has to wear a Nogaku mask, and seat around as a circle.

There are N guests Reimu serves. Kokoro has $2^k$
masks numbered from $0,1,\cdots ,0,1,\cdots ,2^k-1$, and every guest wears one of the masks. The masks have dark power of Dark Nogaku, and to prevent guests from being hurt by the power, two guests seating aside must ensure that if their masks are numbered ii and j , then i XNOR j must be positive. (two guests can wear the same mask). XNOR means $(i^j)$ and every number has kk bits. (1 XNOR 1 = 1, 0 XNOR 0 = 1, 1 XNOR 0 = 0)

You may have seen 《A Summer Day’s dream》, a doujin Animation of Touhou Project. Things go like the anime, Suika activated her ability, and the feast will loop for infinite times. This really troubles Reimu: to not make her customers feel bored, she must prepare enough numbers of different Nogaku scenes. Reimu find that each time the same guest will seat on the same seat, and She just have to prepare a new scene for a specific mask distribution. Two distribution plans are considered different, if any guest wears different masks.

In order to save faiths for Shrine, Reimu have to calculate that to make guests not bored, how many different Nogaku scenes does Reimu and Kokoro have to prepare. Due to the number may be too large, Reimu only want to get the answer modules $1e9+7$ . Reimu did never attend Terakoya, so she doesn’t know how to calculate in module. So Reimu wishes you to help her figure out the answer, and she promises that after you succeed she will give you a balloon as a gift.

Input
First line one number TT , the number of testcases; ($T\le 20$) .

Next T lines each contains two numbers, N and k$(0<N,k\le 1e6)$ .

Output
For each testcase output one line with a single number of scenes Reimu and Kokoro have to prepare, the answer modules $1e9+7$ .

样例输入复制

2
3 1
4 2

样例输出复制

2
84

题目来源
ACM-ICPC 2018 徐州赛区网络预赛

方法1：
可参考

#define Clear(x,val) memset(x,val,sizeof x)
ll a[maxn];
int n,k;
void get() {
	a[0] = 1;
	for(int i=1; i<=1000000; i++) {
		a[i] = (a[i-1] * 2) % mod;
	}
}
ll dfs(ll now) {
	if(now == 1) return a[k] % mod;
	else if(now == 2) return (a[k] * (a[k] - 1)) % mod;
	else {
		ll cur = a[k] * qPow(a[k]-1,now-2) % mod;
		cur = cur * (a[k] - 2) % mod;
		return (cur + dfs(now - 2)) % mod;
	}
}
int main() {
//	cout << (2 ^ 1) << endl;
//	cout << (~(0^1)) << endl;
//	cout << (~(1^0)) << endl;
//	cout << (~(1^1)) << endl;
	get();
	int _ = read;
	while(_ --) {
		n = read,k = read;
		ll ans = dfs(n);
		cout << ans % mod << endl;
	}
	return 0;
}

方法2：

图片来源

	int _ = read;
	while(_ --) {
		n = read,k = read;
		// ll ans = dfs(n);
		// cout << ans % mod << endl;
	    ll tot = qPow(2,k);
        ll ans = qPow(tot-1,n);
        if(n&1) ans += 1;
        else ans += tot - 1;
        cout << (ans + mod) % mod << endl;
    }