如何有效地转置2D位矩阵

时间：2017-01-21 10:40:56

标签： java c arrays performance bit-manipulation

我一直在绊倒这个问题（例如在this question中）。给定基本整数类型阵列形式的2D位矩阵/板/阵列，例如，一个<html ng-app="angularTypeahead"> 的数组。为简单起见，我们可以假设一个方阵，例如，在64位long的平台上有64 long个值的数组。

让long x[i]成为输入数组。为0 <= i < 64计算数组y[i]，以便：

0 <= i <= 64

此处(x[i] >> j) & 1 == (y[j] >> i) & 1 是x >> i按x位的按位右移，i是按位的，而&是x[i]的值数组i中的第1个位置。

如何实现最有效地将数组x映射到数组x的函数？

主要是我正在寻找非破坏性方法，这使得输入数组y保持不变。

实施语言

使用的编程语言应该对整数类型进行数组和按位运算。许多语言都满足这些要求。 C / C ++和Java解决方案看起来非常相似，所以让我们选择这些语言。

2 个答案:

答案 0 :(得分：4)

这似乎是问题Bitwise transpose of 8 bytes的概括。这个问题只是大约8x8换位，所以你要问的是有点不同。但是，您的问题也会在本书的第7.3节（{3}}中得到解答（您可能能够在Google图书上看到Hacker's Delight）。在那里呈现的代码显然源于the relevant pages。

Guy Steele仅包含本书针对Hacker's Delight website和8x8案例的源代码，但后者简单地概括为您的64x64案例：

#include <stdint.h>

void
transpose64(uint64_t a[64]) {
  int j, k;
  uint64_t m, t;

  for (j = 32, m = 0x00000000FFFFFFFF; j; j >>= 1, m ^= m << j) {
    for (k = 0; k < 64; k = ((k | j) + 1) & ~j) {
      t = (a[k] ^ (a[k | j] >> j)) & m;
      a[k] ^= t;
      a[k | j] ^= (t << j);
    }
  }
}

这种方法的工作原理是该函数连续交换较小的位块，从32x32块开始（不将位移到那些块中），然后在那些32x32块内交换相应的块16x16块等。保存块大小的变量是j。因此，外循环j连续取值32,16,8,4,2和1，这意味着外循环运行六次。内部循环遍历 half 位线，即变量k中给定位等于零的行。当j为32时，那些是0-31行，当j为16时，那些是0-15和32-47等等。循环的内部部分一起运行6 * 32 = 192次在这个内部部分内部发生的是掩码m确定应该交换的比特，在t xor或那些比特被计算，并且xor-ed比特列表用于适当地更新两个地方的位。

本书（和网站）也有这个代码的一个版本，其中这些循环都已展开，并且掩码m未计算，但只是分配。我想这取决于寄存器的数量和指令缓存的大小是否有所改善？

为了测试这是否有效，假设我们定义了一些位模式，比如说：

uint64_t logo[] = {
0b0000000000000000000000000000000000000000000100000000000000000000,
0b0000000000000000000000000000000000000000011100000000000000000000,
0b0000000000000000000000000000000000000000111110000000000000000000,
0b0000000000000000000000000000000000000001111111000000000000000000,
0b0000000000000000000000000000000000000000111111100000000000000000,
0b0000000000000000000000000000000000000000111111100000000000000000,
0b0000000000000000000000000000000000000000011111110000000000000000,
0b0000000000000000000000000000000000000000001111111000000000000000,
0b0000000000000000000000000000000000000000001111111100000000000000,
0b0000000000000000000000000000000010000000000111111100000000000000,
0b0000000000000000000000000000000011100000000011111110000000000000,
0b0000000000000000000000000000000111110000000001111111000000000000,
0b0000000000000000000000000000001111111000000001111111100000000000,
0b0000000000000000000000000000011111111100000000111111100000000000,
0b0000000000000000000000000000001111111110000000011111110000000000,
0b0000000000000000000000000000000011111111100000001111111000000000,
0b0000000000000000000000000000000001111111110000001111111100000000,
0b0000000000000000000000000000000000111111111000000111111100000000,
0b0000000000000000000000000000000000011111111100000011111110000000,
0b0000000000000000000000000000000000001111111110000001111111000000,
0b0000000000000000000000000000000000000011111111100001111111100000,
0b0000000000000000000000001100000000000001111111110000111111100000,
0b0000000000000000000000001111000000000000111111111000011111110000,
0b0000000000000000000000011111110000000000011111111100001111100000,
0b0000000000000000000000011111111100000000001111111110001111000000,
0b0000000000000000000000111111111111000000000011111111100110000000,
0b0000000000000000000000011111111111110000000001111111110000000000,
0b0000000000000000000000000111111111111100000000111111111000000000,
0b0000000000000000000000000001111111111111100000011111110000000000,
0b0000000000000000000000000000011111111111111000001111100000000000,
0b0000000000000000000000000000000111111111111110000011000000000000,
0b0000000000000000000000000000000001111111111111100000000000000000,
0b0000000000000000000000000000000000001111111111111000000000000000,
0b0000000000000000000000000000000000000011111111111100000000000000,
0b0000000000000000000111000000000000000000111111111100000000000000,
0b0000000000000000000111111110000000000000001111111000000000000000,
0b0000000000000000000111111111111100000000000011111000000000000000,
0b0000000000000000000111111111111111110000000000110000000000000000,
0b0000000000000000001111111111111111111111100000000000000000000000,
0b0000000000000000001111111111111111111111111111000000000000000000,
0b0000000000000000000000011111111111111111111111100000000000000000,
0b0000001111110000000000000001111111111111111111100000111111000000,
0b0000001111110000000000000000000011111111111111100000111111000000,
0b0000001111110000000000000000000000000111111111100000111111000000,
0b0000001111110000000000000000000000000000001111000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
};

然后我们调用transpose32函数并打印结果位模式：

#include <stdio.h>

void
printbits(uint64_t a[64]) {
  int i, j;

  for (i = 0; i < 64; i++) {
    for (j = 63; j >= 0; j--)
      printf("%c", (a[i] >> j) & 1 ? '1' : '0');
    printf("\n");
  }
}

int
main() {
  transpose64(logo);
  printbits(logo);
  return 0;
}

然后这将作为输出：

0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000011000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000100000000011111000000011111100000111111
0000000000000000000000011110000000011111100000011111100000111111
0000000000000000000001111110000000011111100000011111100000111111
0000000000000000000001111111000000011111100000011111100000111111
0000000000000000000000111111000000011111100000011111100000111111
0000000000000000000000111111100000001111110000011111100000111111
0000000000000000000000011111100000001111110000011111100000111111
0000000000000100000000011111110000001111110000011111100000111111
0000000000001110000000001111110000001111110000011111100000111111
0000000000011110000000001111111000001111110000011111100000111111
0000000001111111000000000111111000000111111000011111100000111111
0000000000111111100000000111111100000111111000011111100000111111
0000000000111111110000000011111100000111111000011111100000111111
0000000000011111111000000011111100000111111000011111100000111111
0000000000001111111100000001111110000011111000011111100000111111
0000000000000111111100000001111110000011111100011111100000111111
0000000000000011111110000000111111000011111100011111100000111111
0001000000000001111111000000111111000011111100011111100000111111
0011110000000001111111100000111111100011111100011111100000111111
0111111000000000111111110000011111100001111100011111100000111111
0111111110000000011111111000011111110001111110011111100000111111
1111111111000000001111111000001111110001111110011111100000111111
0011111111100000000111111100001111111001111110011111100000111111
0001111111111000000011111110000111111001111110011111100000111111
0000111111111100000011111111000111111100111100000000000000111111
0000001111111110000001111111100011111100000000000000000000111111
0000000111111111100000111111110011111000000000000000000000111111
0000000011111111110000011111110001100000000000000000000000111111
0000000000111111111000001111111000000000000000000000000000111111
0000000000011111111110000111111000000000000000000000000000111111
0000000000001111111111000111110000000000011111111111111111111111
0000000000000011111111100011100000000000011111111111111111111111
0000000000000001111111111001000000000000011111111111111111111111
0000000000000000111111111100000000000000011111111111111111111111
0000000000000000001111111100000000000000011111111111111111111111
0000000000000000000111111000000000000000011111111111111111111111
0000000000000000000011110000000000000000000000000000000000000000
0000000000000000000000100000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000

正如我们所希望的那样，这很好地翻转了。

编辑：

这实际上并不是您要求的，因为您要求提供此代码的非 - 破坏性版本。您可以通过将32x32块的第一次交换从x转换为y来实现此目的。例如，您可能会执行以下操作：

void
non_destructive_transpose64(uint64_t x[64], uint64_t y[64]) {
  int j, k;
  uint64_t m, t;

  for (k = 0; k < 64; k += 2) {
    ((uint32_t *) y)[k] = ((uint32_t *) x)[k ^ 64 + 1];
    ((uint32_t *) y)[k + 1] = ((uint32_t *) x)[k + 1];
  }
  for (; k < 128; k += 2) {
    ((uint32_t *) y)[k] = ((uint32_t *) x)[k];
    ((uint32_t *) y)[k + 1] = ((uint32_t *) x)[k ^ 64];
  }
  for (j = 16, m = 0x0000FFFF0000FFFF; j; j >>= 1, m ^= m << j) {
    for (k = 0; k < 64; k = ((k | j) + 1) & ~j) {
      t = (y[k] ^ (y[k | j] >> j)) & m;
      y[k] ^= t;
      y[k | j] ^= (t << j);
    }
  }
}

与其他版本的代码不同，无论架构的字节顺序如何， 都无法正常工作。此外，我知道C标准不允许您以uint64_t的数组的形式访问uint32_t数组。但是，我喜欢它，当你这样做时，移动块周围循环的第一次迭代不需要移位或xors。

答案 1 :(得分：0)

在C ++ 8x8矩阵中将是这样，但是您可以轻松地对其进行更改以使其更通用（不仅仅是8x8）。 另外，我将main包含在1个测试向量中只是为了获得一种感觉：

#include <iostream>
#include <string>
#include <vector>

std::vector<long> rotate(std::vector<long>& v) {
    std::vector<long> temp = { 0,0,0,0,0,0,0,0 };
    for (unsigned int i = 0; i<8; i++) {
        int number = v[i];
        for (unsigned int j = 0; j<8; j++) {
            int z = (number & (1 << (7-j)));
            if (z != 0) {
                temp[j] |= (1 << (7 - i));
            }
        }
    }
    return temp;
}



int main()
{
    std::vector<long> v = { 0, 1, 2, 3, 4, 5, 6, 7 };
    std::vector<long> rotated = rotate(v);
    for (unsigned int i = 0; i<8; i++) {
        std::cout << rotated.at(i) << " ";
    }
    return 0;
}

因此，如果Java需要它，则可以轻松转换它，因为Java也提供位运算符。

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