hdu 5885 FFT

Posted 2020-08-09 Przz

XM Reserves

Time Limit: 10000/10000 MS (Java/Others)    Memory Limit: 102400/102400 K (Java/Others)
Total Submission(s): 99    Accepted Submission(s): 45

Problem Description

As an eligible Ingress Resistance Agent you should know your power source, the Exotic Matter.
We call it XM, which is the driving force behind all of our actions in Ingress.
XM allows us to construct items through hacking portals, to attack enemy portals, make links and create fields.

We try to collect XM from the ground. XM concentration come from location based services, meaning that areas with a lot of foot traffic have higher amounts versus places that don‘t.
You can collect XM by moving through those areas.
The XM will be automatically harvested by your Scanner when it is within your interaction circle/range.

Alice decides to select a location such that she can collect XM as much as possible.
To simplify the problem, we consider the city as a grid map with size `n*m‘ numbered from (0,0) to (n−1,m−1).
The XM concentration inside the block (i,j) is p(i,j).
The radius of your interaction circle is r.
We can assume that XM of the block (i,j) are located in the centre of this block.
The distance between two blocks is the Euclidean distance between their centres.

Alice stands in the centre of one block and collects the XM.
For each block with the distance d smaller than r to Alice, and whose XM concertation is p(i,j), Alice‘s scanner can collects p(i,j)/(1+d) XM from it.

Help Alice to determine the maximum XM which she can collect once he stands in the centre of one block.

 

 

Input

There are multiple cases.
For each case, the first line consists two integers n,m (1≤n,m≤500) and one float-point number r (0≤r≤300).
Each of the following n line consists m non-negative float-point numbers corresponding to the XM concentrations inside each blocks.

 

 

Output

For each case, output the maximum XM which Alice can collect in one line.
Your answers should be rounded to three decimal places.

 

 

Sample Input

3 3 1

1 3 6

7 9 4

2 8 1

3 3 2

1 3 6

7 9 4

2 8 1

5 5 1.

5 4 3 2 9

1 3 4 3 2

8 9 4 3 2

1 2 3 0 1

2 6 3 4 3 1

 

Sample Output

9.000

24.142

17.956

 

/*
hdu 5885 FFT卷积

problem:
给出一个n×m的格子,距离它小于等于r的那些格子都会贡献p(i,j)/(d+1), d为两个格子的距离. p[i][j]为格子的价值
选择一个格子, 使得贡献和最大.

solve:
参考：https://async.icpc-camp.org/d/558-2016-icpc
因为小于r的格子都会给当前格子贡献, 所以可以考虑先计算出所有格子对<r的其它格子的贡献.那么最后一次遍历就能找出
出最大值.那么会处理出来一个 (n+r+r,m+r+r)的矩阵.
然后就是如何计算, 格子(i,j)按照(x,y)移动后会到格子(i+x,j+y).坐标相加
如果把所有方向处理成一个一维矩阵,那么当前点与它们相乘后的位置就是当前点能到达的目标位置(卷积，坐标相加)
都转换成一维.
0*M+0 0*M+1 0*M+2
0*M+0 0*M+1 0*M+2 1*M+0 ...   (方向)
---->
0*M+0 0*M+1 0*M+2 1*M+0
      0*M+1 0*M+2 1*M+0
求卷积后坐标之间是存在一定关系的.类似于A,B集合中有多少a+b=c那种题的思想.  (新技能get)
(看代码想了很久才懂 TAT.. )
用a[i*M + j]保存p(i,j), b[i*M+j]保存 1/d.
c = a*b, 那么第i*M + j就是i,j的贡献和.

本来最开始用的m但是发现有问题,因为一行处理后有m+2*r个点,换成M就好了....
hhh-2016-09-21 21:10:30
*/
#pragma comment(linker,"/STACK:124000000,124000000")
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cstdio>
#include <cstring>
#include <vector>
#include <map>
#include <math.h>
#define lson  i<<1
#define rson  i<<1|1
#define ll long long
#define clr(a,b) memset(a,b,sizeof(a))
#define key_val ch[ch[root][1]][0]
using namespace std;
const int maxn = 1e6 + 1000;
const int inf = 0x3f3f3f3f;
const ll mod = 1e9 + 7;
const double eps = 1e-7;
template<class T> void read(T&num)
{
    char CH;
    bool F=false;
    for(CH=getchar(); CH<‘0‘||CH>‘9‘; F= CH==‘-‘,CH=getchar());
    for(num=0; CH>=‘0‘&&CH<=‘9‘; num=num*10+CH-‘0‘,CH=getchar());
    F && (num=-num);
}
int stk[70], tp;
template<class T> inline void print(T p)
{
    if(!p)
    {
        puts("0");
        return;
    }
    while(p) stk[++ tp] = p%10, p/=10;
    while(tp) putchar(stk[tp--] + ‘0‘);
    putchar(‘\n‘);
}

const double PI = acos(-1.0);

struct Complex
{
    double x,y;
    Complex(double _x = 0.0,double _y = 0.0)
    {
        x = _x;
        y = _y;
    }
    Complex operator-(const Complex &b)const
    {
        return Complex(x-b.x,y-b.y);
    }
    Complex operator+(const Complex &b)const
    {
        return Complex(x+b.x,y+b.y);
    }
    Complex operator*(const Complex &b)const
    {
        return Complex(x*b.x-y*b.y,x*b.y+y*b.x);
    }
};

void change(Complex y[],int len)
{
    int i,j,k;
    for(i = 1,j = len/2; i < len-1; i++)
    {
        if(i < j) swap(y[i],y[j]);
        k = len/2;
        while(j >= k)
        {
            j-=k;
            k/=2;
        }
        if(j < k) j+=k;
    }
}

void fft(Complex y[],int len,int on)
{
    change(y,len);
    for(int h = 2; h <= len; h <<= 1)
    {
        Complex wn(cos(-on*2*PI/h),sin(-on*2*PI/h));
        for(int j = 0; j < len; j+=h)
        {
            Complex w(1,0);
            for(int k = j; k < j+h/2; k++)
            {
                Complex u = y[k];
                Complex t = w*y[k+h/2];
                y[k] = u+ t;
                y[k+h/2] = u-t;
                w = w*wn;
            }
        }
    }
    if(on == -1)
    {
        for(int i = 0; i < len; i++)
            y[i].x /= len;
    }
}

double dis(int a,int b)
{
    return sqrt(a*a + b*b);
}

Complex a[2*maxn];
Complex b[2*maxn];

int main()
{
    int k;
    int n,m,limit;
    double x,r;
    while(scanf("%d%d%lf",&n,&m,&r) != EOF)
    {
        limit =0, k = 1;
        while(r >= limit)
        {
            limit ++ ;
        }
        limit -= 1;
        int M = m +limit + limit;
        while(k < (n+r+r)*(m+r+r))
           k <<= 1;

        for(int i = 0;i < k;i++)
        {
            a[i] = b[i] = 0;
        }
        for(int i = 0;i < n;i++)
        {
            for(int j = 0;j < m;j++)
            {
                read(x);
                a[i*M + j] = x;
            }
        }
        fft(a,k,1);
        for(int i = -limit;i <= limit;i++)
        {
            for(int j = -limit;j <= limit;j++)
            {
                double len = dis(i,j);
                if( r > len)
                b[(i+limit) * M + j + limit] = 1.0/(len+1);

            }
        }

        fft(b,k,1);
        for(int i = 0;i < k;i++)
            a[i] = a[i] * b[i];
        fft(a,k,-1);
        double ans = 0;
        for(int i = 0;i < n;i++)
        {
            for(int j = 0;j < m;j++)
            {
                ans = max(ans,a[(i + limit) * M + j + limit].x);
            }
        }
        printf("%.3f\n",ans);
    }
    return 0;
}