1977: Bit-reversal Permutation(递归)

1977: Bit-reversal Permutation
Submit Page Summary Time Limit: 1 Sec Memory Limit: 128 Mb Submitted: 7 Solved: 3
Description
A fast Fourier transform (FFT) algorithm computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IFFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O(n2), which arises if one simply applies the definition of DFT, to O(nlogn), where n is the data size.

                                                                                                                                                                               ——From Wikipedia

During this summer holiday, csuxushu feels so bored to learn FFT. Because FFT is a complicated algorithm, he need to apply a bit-reversal permutation to a sequence first before DFT which is a part of FFT.

In applied mathematics, a bit-reversal permutation is a permutation of a sequence of n items, where n = 2^k is a power of two. It is defined by indexing the elements of the sequence by the numbers from 0 to n − 1 and then reversing the binary representations of each of these numbers (padded so that each of these binary numbers has length exactly k). Each item is then mapped to the new position given by this reversed value.

Because all fellows in CSU(California State University ) can apply FFT, NTT or even FWT, it is a shame that he even doesn’t know how to take the first step. As one of the best computer programmer in CSU, can you help him?

You may think this problem is too hard to solve. In fact, it is a piece of cake to you. Remember to see the hint :-)

Input
The first line of the input gives the number of test cases T(T≤10); T test cases follow.Each test case contains a number sequence.
In each case, the first line is a number N(1≤N≤10^5), the number of elements in the following sequence.The second line is the sequence.Its length may not be exactly a power of two, so you can append some zeros to make it the minimal power of two larger than or equal to N.

Output
For each test case, output the sequence from input in bit-reversal order.

Sample Input
1
6
21 58 96 12 45 65
Sample Output
21 45 96 0 58 65 12 0
Hint
Bit-reverse Order
中文提示：可以看到，我们最终处理的系数从左至右的编号的二进制形式分别为000,100,010,110,001,101,011,111，若将其二进制反序，可得000,001,010,011,100,101,110,111，这些反序的二进制编码是从小到大排列的。也就是说，我们可以按照每个下标的二进制编码来确定处理系数的顺序。这种方法就称为位逆序置换(Bit-reversal permutation)。

Source
2017年8月月赛

Author
徐戍

有图画很好理解，原题地址：http://acm.csu.edu.cn/csuoj/problemset/problem?pid=1977

题意：

模拟FFT,FWT,NTT这类运算的核心分治思想，偶数下标放左区间，奇数放右区间，注意因为不是2的整次幂需要补到2的整次幂，所以空间要开两倍

可以直接模拟也可以先对下标bit进行reverse然后根据结果去sort

#include<cstdio>#include<cstring>#include<algorithm>#include<iostream>using namespace std;typedef long long LL;const int maxn=1e6+5;int n,m,a[maxn],b[maxn]; inline void divd(int l,int r){  if(l>=r)return ;  int mid=(l+r)>>1;  int L=l-1,R=mid;  for(int i=l;i<=r;++i)  if(i&1)b[++L]=a[i];  else b[++R]=a[i];  for(int i=l;i<=r;++i)a[i]=b[i];  divd(l,mid);  divd(mid+1,r);}int main(){  int T;  scanf("%d",&T);  while(T--)  {    scanf("%d",&n);    for(int i=1;i<=n;++i)scanf("%d",a+i);    for(m=1;m<n;m<<=1);    for(int i=n+1;i<=m;++i)a[i]=0;    divd(1,m);    for(int i=1;i<m;++i)printf("%d ",a[i]);    printf("%d\n",a[m]);  }  return 0;}