UVa 10790 How Many Points of Intersection?(组合数学)

最新推荐文章于 2020-08-02 17:04:36 发布

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最新推荐文章于 2020-08-02 17:04:36 发布

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分类专栏： acm之路--数学组合数学 UVa

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本文介绍了一个计算二维平面上由两行点之间的连线产生的交叉点数量的问题。通过组合数学的方法，给出了具体的计算公式，并提供了一个使用 C 语言实现的示例代码。

10790 - How Many Points of Intersection?

Time limit: 3.000 seconds

http://uva.onlinejudge.org/index.php?option=onlinejudge&page=show_problem&problem=1731

We have two rows. There are a dots on the top row and b dots on the bottom row. We draw line segments connecting every dot on the top row with every dot on the bottom row. The dots are arranged in such a way that the number of internal intersections among the line segments is maximized. To achieve this goal we must not allow more than two line segments to intersect in a point. The intersection points on the top row and the bottom are not included in our count; we can allow more than two line segments to intersect on those two rows. Given the value of a and b, your task is to compute P(a,b), the number of intersections in between the two rows. For example, in the following figure a = 2 and b = 3. This figure illustrates that P(2, 3) = 3.

$\epsfbox{p10790.eps}$

Input

Each line in the input will contain two positive integers a ( 0 < a $\le$ 20000) and b ( 0 < b $\le$ 20000). Input is terminated by a line where both a and b are zero. This case should not be processed. You will need to process at most 1200 sets of inputs.

Output

For each line of input, print in a line the serial of output followed by the value ofP(a, b). Look at the output for sample input for details. You can assume that the output for the test cases will fit in 64-bit signed integers.

Sample Input

Sample Output

Case 1: 1
Case 2: 3
Case 3: 9

题意：

如图，找交点（不存在多于两条线交于一点），此题是这道题的简单版。

思路：

每个点都由两条线上的两个点得到，所以ans=C(a,2) * C(b,2)=ab(a-1)(b-1)/4，最大值（精确）为：3.99960001*10¹⁶

用long long搞定。

完整代码：

#include <cstdio>

int main()
{
	int a, b, t = 1;
	while (scanf("%d%d", &a, &b), a)
		printf("Case %d: %lld\n", t++, (long long)a * b * (a - 1) * (b - 1) / 4);
	return 0;
}