Does there exist a stable sorting algorithm that can sort a binary array in O(n) time complexity and O(1) auxiliary space complexity?

user_11436382 0
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I know how to write an algorithm that sorts a binary array in O(n) time and O(1) auxiliary space complexity, but it doesn't seem to be stable.

void binarySort(int arr[], int arr_size)
{
  int i;
  int count_zeros=0;
  for (i=0 ; i<arr_size ; i++)
  {
    if(arr[i]==0)
      count_zeros++;
  }
    int j;
    for(j=0 ; j<count_zeros ; j++)
      arr[j]=0;
    int k;
    for(k=j ; k<arr_size ; k++)
      arr[k]=1;
}

Consider I want to perform binary radix sort using the stable sort algorithm as a subroutine. Here is an example in which we sort we sort according to the least significant bit.

Input: 11, 101, 1000, 1010, 111, 110, 10

Output: 1000, 1010, 110, 10, 11, 101, 111

Notice in the example above, the order of the binary numbers with the same LSBs is preserved.

Is there any way to edit the above algorithm or make a new algorithm such that the sort algorithm is stable while preserving it's O(n) time complexity and O(1) auxiliary space complexity?

From Wikipedia: Stable sorting algorithms maintain the relative order of records with equal keys (i.e. values). That is, a sorting algorithm is stable if whenever there are two records R and S with the same key and with R appearing before S in the original list, R will appear before S in the sorted list. For those who say that stability doesn't matter, read this post.

space-complexity time-complexity sorting algorithm
1 Answers

I think I may have come up with a stable sort algorithm that has O(n) time complexity and O(1) auxiliary space complexity. I would appreciate it if someone could verify that it satisfies all the properties. I would also like to see any other possible approaches to this problem. You can try running it here.

void binarySort(int arr[], int arr_length)
{
  int i, count=1;
  for(i=0 ; i<arr_length ; i++)
  {
    if(arr[i]==1)
    {
      arr[i]=count;
      count++;
    }
  }
  int num0s=arr_length-count+1;
  int j=0;
  int k=0;
  while(j<num0s)
  {
    if(arr[k]==0)
    {
      int temp=arr[k];
      arr[k]=arr[j];
      arr[j]=temp;
      j++;
    }
    k++;
  }
  int a;
  for(a=0 ; a<arr_length-num0s ; )
  {
    if(arr[num0s+a]!=a+1)
    {
      int temp=arr[num0s+a];
      arr[num0s+a]=arr[num0s+temp-1];
      arr[num0s+temp-1]=temp;
    }
    else
    {
      arr[num0s+a]=1;
      a++;
    }
  }
}

For those who are downvoting, I would request them to please explain what is wrong with this algorithm. From what I see, it fulfills all the requirements set.

user_114363820
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