HDU 5734 Acperience（数学推导【多校联合】）

http://acm.hdu.edu.cn/showproblem.php?pid=5734
Problem Description
Deep neural networks (DNN) have shown significant improvements in several application domains including computer vision and speech recognition. In computer vision, a particular type of DNN, known as Convolutional Neural Networks (CNN), have demonstrated state-of-the-art results in object recognition and detection.

Convolutional neural networks show reliable results on object recognition and detection that are useful in real world applications. Concurrent to the recent progress in recognition, interesting advancements have been happening in virtual reality (VR by Oculus), augmented reality (AR by HoloLens), and smart wearable devices. Putting these two pieces together, we argue that it is the right time to equip smart portable devices with the power of state-of-the-art recognition systems. However, CNN-based recognition systems need large amounts of memory and computational power. While they perform well on expensive, GPU-based machines, they are often unsuitable for smaller devices like cell phones and embedded electronics.

In order to simplify the networks, Professor Zhang tries to introduce simple, efficient, and accurate approximations to CNNs by binarizing the weights. Professor Zhang needs your help.

More specifically, you are given a weighted vector W=(w1,w2,…,wn). Professor Zhang would like to find a binary vector B=(b1,b2,…,bn) (bi∈{+1,−1}) and a scaling factor α≥0 in such a manner that ∥W−αB∥2 is minimum.

Note that ∥⋅∥ denotes the Euclidean norm (i.e. ∥X∥=x21+⋯+x2n−−−−−−−−−−√, where X=(x1,x2,…,xn)).

Input
There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:

The first line contains an integers n (1≤n≤100000) – the length of the vector. The next line contains n integers: w1,w2,…,wn (−10000≤wi≤10000).

Output
For each test case, output the minimum value of ∥W−αB∥2 as an irreducible fraction “p/q” where p, q are integers, q>0.

Sample Input
3
4
1 2 3 4
4
2 2 2 2
5
5 6 2 3 4

Sample Output
5/1
0/1
10/1

Author
zimpha

本题是一道公式推导的题目，直接按照那个给出的公式往下推。
最后推出那个式子等于x1^2+x2^2+……..+xn^2-n*(average^2)
其中的average为这n个数绝对值的平均数，然后直接除求平均数的话会有精度损失。
然后对这个式子乘以n然后这个式子变为了：n* （x1^2+x2^2+…….+xn^2） - sum*sum 其中sum为这n个数的绝对值的和，然后再求这个式子与n的最大公约数。然后二者分别除以最大公约数就是这个题的结果了。
下面是AC代码：

#include<stdio.h>#include<cmath>#include<algorithm>using namespace std;long long int a[100010];long long int GCD(long long int a,long long int b){    if(b==0) return a;    else return GCD(b,a%b);}int main(){    int t;    scanf("%d",&t);    while(t--)    {        long long int n;        scanf("%I64d",&n);        long long int fenzi=0,fenmu;        long long int sum=0;        for(int i = 0; i < n; i++)        {            scanf("%I64d",&a[i]);            sum+=abs(a[i]);        }        for(int i = 0; i < n; i++)            fenzi+=a[i]*a[i];        fenzi=fenzi*n-sum*sum;        fenmu=n;        if(fenzi==0)        {            printf("0/1\n");            continue;        }        else        {            long long int k=GCD(fenzi,fenmu);            printf("%I64d/%I64d\n",fenzi/k,fenmu/k);            continue;        }    }    return 0;}