HDU Problem 5363 Key set 【快速幂】

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本文探讨了如何解决计数KeySet问题，该问题是求解一个由1到n的整数构成的集合中，所有非空子集里元素之和为偶数的数量。通过数学分析给出了解决方案，并提供了具体的C语言实现代码。

Key Set

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 1856 Accepted Submission(s): 972

Problem Description

soda has a set

S with

n integers

{1,2,…,n} . A set is called key set if the sum of integers in the set is an even number. He wants to know how many nonempty subsets of

S are key set.

Input

There are multiple test cases. The first line of input contains an integer

(1≤T≤105) , indicating the number of test cases. For each test case:

The first line contains an integer

(1≤n≤109) , the number of integers in the set.

Output

For each test case, output the number of key sets modulo 1000000007.

Sample Input

Sample Output

Author

zimpha@zju

Source

2015 Multi-University Training Contest 6

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对于一个含有n个元素的非空集合，他的非空子集有2^n-1个。因为题目中的集合里的元素是从1~n的，所以和为奇数的子集个数比和为偶数的子集多一个，也就是2^(n-1)-1个。

#include <stdio.h>
__int64 pow(__int64 x, __int64 y) {
    __int64 res = 1;
    __int64 base = x;
    y -= 1;
    while (y) {
        if (y&1) res = base*res%1000000007;
        base = base*base%1000000007;
        y >>= 1;
    }
    return res;
}
int main() {
    int t;
    __int64 n;
    scanf("%d", &t);
    while (t--) {
        scanf("%I64d", &n);
        printf("%I64d\n", pow(2, n) - 1);
    }
    return 0;
}