HDU 5734 Acperience（數學推導【多校聯合】）

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

本題是一道公式推導的題目，直接按照那個給出的公式往下推。

最後推出那個式子等于x1^2+x2^2+……..+xn^2-n*(average^2)

其中的average為這n個數絕對值的平均數，然後直接除求平均數的話會有精度損失。

然後對這個式子乘以n然後這個式子變為了：n* （x1^2+x2^2+…….+xn^2） - sum*sum 其中sum為這n個數的絕對值的和，然後再求這個式子與n的最大公約數。然後二者分别除以最大公約數就是這個題的結果了。

下面是AC代碼：

#include<stdio.h>
#include<cmath>
#include<algorithm>
using namespace std;

long long int a[];
long long int GCD(long long int a,long long int b)
{
    if(b==) 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=,fenmu;
        long long int sum=;
        for(int i = ; i < n; i++)
        {
            scanf("%I64d",&a[i]);
            sum+=abs(a[i]);
        }
        for(int i = ; i < n; i++)
            fenzi+=a[i]*a[i];
        fenzi=fenzi*n-sum*sum;
        fenmu=n;
        if(fenzi==)
        {
            printf("0/1\n");
            continue;
        }
        else
        {
            long long int k=GCD(fenzi,fenmu);
            printf("%I64d/%I64d\n",fenzi/k,fenmu/k);
            continue;
        }
    }
    return ;
}