你可以为基数排序制定一个幺半群或半群吗？

Question

这是基数排序的伪代码:

Pseudocode for Radix Sort:
Radix-Sort(A, d)
// Each key in A[1..n] is a d-digit integer. (Digits are
// numbered 1 to d from right to left.)
1. for i = 1 to d do
Use a stable sorting algorithm to sort A on digit i.

这是基数排序的Scala代码:

object RadixSort {
  val WARP_SIZE = 32

  def main(args: Array[String]) = {
    var A = Array(123,432,654,3123,654,2123,543,131,653,123)

    radixSortUintHost(A, 4).foreach(i => println(i))
  }

  // LSB radix sort
  def radixSortUintHost(A: Array[Int], bits: Int): Array[Int] = {
    var a = A
    var b = new Array[Int](a.length)

    var rshift = 0
    var mask = ~(-1 << bits)

    while (mask != 0) {
      val cntArray = new Array[Int](1 << bits)

      for (p <- 0 until a.length) {
        var key = (a(p) & mask) >> rshift
        cntArray(key)+= 1
      }

      for (i <- 1 until cntArray.length)
        cntArray(i) += cntArray(i-1)

      for (p <- a.length-1 to 0 by -1) {
        var key = (a(p) & mask) >> rshift
        cntArray(key)-= 1
        b(cntArray(key)) = a(p)
      }

      val temp = b
      b = a
      a = temp

      mask <<= bits
      rshift += bits
    }

    b
  }
}

这是基数排序的Haskell代码:

import Data.Bits (Bits(testBit, bitSize))
import Data.List (partition)

lsdSort :: (Ord a, Bits a) => [a] -> [a]
lsdSort = fixSort positiveLsdSort

msdSort :: (Ord a, Bits a) => [a] -> [a]
msdSort = fixSort positiveMsdSort

-- Fix a sort that puts negative numbers at the end, like positiveLsdSort and positiveMsdSort
fixSort sorter list = uncurry (flip (++)) (break (< 0) (sorter list))

positiveLsdSort :: (Bits a) => [a] -> [a]
positiveLsdSort list = foldl step list [0..bitSize (head list)] where
step list bit = uncurry (++) (partition (not . flip testBit bit) list)

positiveMsdSort :: (Bits a) => [a] -> [a]
positiveMsdSort list = aux (bitSize (head list) - 1) list where
aux _ [] = []
aux (-1) list = list
aux bit list = aux (bit - 1) lower ++ aux (bit - 1) upper where
    (lower, upper) = partition (not . flip testBit bit) list

我的问题是:你能为基数排序制定一个幺半群或半群吗？

Answer 1

基数排序不变式是使用前 k 位对数据进行排序。如果您想要添加更多已排序或未排序数据的操作，那么您需要的是合并排序功能而不是基数。如果您要添加的是所有记录中的数据位，则可以使用幺半群。

编辑：幺半群的哈特是一种结合运算。我们可以将排序位视为应用偏序的一种方式。您一点一点地注入所有记录的数据。每个位都应用一个部分顺序。这是关联的，您可以合并一些位以获得更复杂的部分顺序。注意顺序很重要，但它仍然是关联的。因此可以被视为一个单子