2019湖北省省赛 D题题解

最新推荐文章于 2024-12-04 17:03:31 发布

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最新推荐文章于 2024-12-04 17:03:31 发布

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解析了湖北省网络赛D题“Dandelion”的算法题解，探讨了种子从起点到终点的不同路径数量计算，利用费马小定理求逆元，通过组合数学中的C(n+m-1,m)和C(n+m-1,m-1)公式解决x<y约束条件下的路径计数问题。

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湖北省网络赛D题题解

D. Dandelion

time limit per test

1.0 s

memory limit per test

256 MB

input

standard input

output

standard output

"O dandelion, yellow as gold,

What do you do all day?"

"I just wait here in the tall green grass

Till the children come to play."

"O dandelion, yellow as gold,

What do you do all night?"

"I wait and wait till the cool dews fall

And my hair grows long and white."

"And what do you do when your hair is white

And the children come to play?"

"They take me up in their dimpled hands

And blow my hair away!"

Spring is here and the dandelion is blooming. There is a small dandelion seed taking off. It will float upwards when there is no wind, and when the wind blows up it will be blown to the right.

In other words, if a dandelion seed is currently at point (x,y)(x,y), the next second it will only appear in either point (x,y+1)(x,y+1) or point (x+1,y)(x+1,y). All points (x,y)(x,y) on the path must satisfy the constraint that x is less than y(i.e x<yx<y.

Now, there is a dandelion seed at point (0,0)(0,0), please find the number of ways(mod 109+7109+7) to reach the point (m,n)(m,n). 0<m<n≤1000000<m<n≤100000.

Input

The first line has a integer TT (T<10T<10), it means the number of the case.

The following lines of input will be two integers: mm and nn.

Output

For each case, print an integer in a single line to represent the answer.

Example

input

Copy

output

Copy

5
14
894965608

思路：

最开始我们选择打表出10*10 的数据结果

如下图：

假设不存在 x < y 不能通过的情况即所有的路都能走

得到下图（同时我也根据上图找到了规律……在已知题解的情况下找规律）

官方题解说的是：

用费马小定理求逆元，然后直接套公式。
ans = C(n+m-1,m) - C(n+m-1,m-1)

C(n+m-1,m) 表示的是不存在 x < y 不能走的情况

C(n+m-1,m-1) 表示的是考虑不能走的情况

所以两者相减

以下是官方给的标程：

#include <bits/stdc++.h>
using namespace std;
const int maxn = 500005, mod = 1e9 + 7;
int mi[maxn];
inline int quick(int a, int b) {
	int res = a, ans = 1;
	for (int i = 0; i < 31; i ++) {
		if (1 << i & b) ans = (long long)ans * res % mod;
		res = (long long)res * res % mod;
	}
	return ans;
}
int C(int n, int m) {
	int tmp = (long long)mi[n - m] * mi[m] % mod;
	int ans = (long long)mi[n] * quick(tmp, mod - 2) % mod;
	return ans;
}
int main() {
	int n, m;
	mi[0] = 1;
	for (int i = 1; i <= 200000; i ++) mi[i] = (long long)mi[i - 1] * i % mod;
	int t;
	scanf("%d",&t);
	while (t--) {
		scanf("%d%d", &m, &n);
		printf("%d\n", (C(n + m - 1, m) - C(n + m - 1, m - 1) + mod) % mod);
	}
	return 0;
}

据说这题是原题啊~~~

稍后放上原址……

http://www.51nod.com/Blog/Index.html#!#blogId=12

Alone的题解也来一发

https://blog.youkuaiyun.com/qq_41279172/article/details/89165744