CSU1977: Bit-reversal Permutation

CSU1977: Bit-reversal Permutation

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 WikipediaDuring 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 :-)InputThe 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

1621 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

比赛中将问题想复杂了，好久也没搞出来。实际上直接按照先0后1的次序，递归地生成顺序序列的二进制数码，同时逆序地计算出这些数码对应的十进制数，直接取出数组中元素就行了。比如我们生成的是 000,001,010,011,100,101,110,111 这八个数，本来是对应十进制 0,1,2,3,4,5,6,7 的，但我们对每一个二进制数逆序地转换进制，例如对 011，按照 0×20+1×21+1×22=6 来计算。这样得出的自然就是位逆序置换后所对应的下标了。取这个下标对应的元素输出即得答案。

#include <iostream>#include <cstdio>#include <cstring>using namespace std;const int maxn = 1e5+10;int sample[maxn*10];int T, n, k;void dfs(int kk, int index){    if(kk == 0){                    // 已产生k位二进制数，并已经知道了它们逆序后对应的十进制数。        if(index == (1<<k)-1)       // 不论如何做位逆序置换，最后一个输出的一定是二进制全为1的数。输出它后换行。            printf("%d\n", sample[index]);        else                        // 否则输出后空格。            printf("%d ", sample[index]);        return;    }    dfs(kk-1, index+0);             // 递归调用的同时进行逆序进制转换。    dfs(kk-1, index+(1<<(k-kk)));}int main(){#ifdef TESTfreopen("test.txt", "r", stdin);#endif // TEST    cin >> T;    while(T--){        memset(sample, 0, sizeof(sample));        scanf("%d", &n);        for(int i = 0; i < n; i++){            scanf("%d", &sample[i]);        }        k = 0;        while(1<<k < n) k++;        // 求出令2^k不小于n的最小k值。        dfs(k, 0);    }    return 0;}/**********************************************************************    Problem: 1977    User: xyJiang    Language: C++    Result: AC    Time:400 ms    Memory:5924 kb**********************************************************************/