Quadsort: Introduction to a new stable sorting algorithm faster than quicksort

Intro

This document describes a stable non-recursive merge sort named quadsort.

The quad swap

At the core of quadsort is the quad swap. Traditionally most sorting algorithms have been designed using the binary swap where two variables are sorted using a third temporary variable. This typically looks as following.

if (val[0] > val[1])
{
    tmp[0] = val[0];
    val[0] = val[1];
    val[1] = tmp[0];
}

Instead the quad swap sorts four variables using four temporary variables. During the first stage the four variables are partially sorted in the four temporary variables, in the second stage they are fully sorted back to the original four variables.

A --> A,B --> A,C
   B --> A,B --> A,B,C,D
   C --> C,D --> A,B,C,D
   D --> C,D --> B,D

This allows for the distribution visualised above. The quad swap elimates 2 wasteful swap operations and allows to make several deductions about the final ordering.

A --> A,B --> A? --> C? --> A,C
   B --> A,B --> B? --> D? --> A,C
   C --> C,D --> C? --> A? --> B,D
   D --> C,D --> D? --> B? --> B,D

The next step is checking if the four temporary variables are in order or in reverse-order. In the visualization above in-order is checked first and reverse-order is checked second. If both checks fail the final arrangement is known and two checks remain to determine the final order.

This eliminates 1 wasteful comparisons for in order sequences while creating an additional comparison for random sequences. However, in the real world we are rarely comparing truly random data, so in any instance where data is more likely to be orderly than disorderly this shift in probability will give an advantage.

By changing the comparison order probability can be shifted to give random data the advantage. Regardless, there should be a slight overall advantage due to the elimination of wasteful swapping. In C this looks as following:

if (val[0] > val[1])
{
    tmp[0] = val[1];
    tmp[1] = val[0];
}
else
{
    tmp[0] = val[0];
    tmp[1] = val[1];
}

if (val[2] > val[3])
{
    tmp[2] = val[3];
    tmp[3] = val[2];
}
else
{
    tmp[2] = val[2];
    tmp[3] = val[3];
}

if (tmp[1] <= tmp[2])
{
    val[0] = tmp[0];
    val[1] = tmp[1];
    val[2] = tmp[2];
    val[3] = tmp[3];
}
else if (tmp[0] > tmp[3])
{
    val[0] = tmp[2];
    val[1] = tmp[3];
    val[2] = tmp[0];
    val[3] = tmp[1];
}
else
{
   if (tmp[0] <= tmp[2])
   {
       val[0] = tmp[0];
       val[1] = tmp[2];
   }
   else
   {
       val[0] = tmp[2];
       val[1] = tmp[0];
   }

   if (tmp[1] <= tmp[3])
   {
       val[2] = tmp[1];
       val[3] = tmp[3];
   }
   else
   {
       val[2] = tmp[3];
       val[3] = tmp[1];
   }
}

In the case the array cannot be perfectly divided by 4 the tail, existing of 1-3 elements, is sorted using the traditional swap.

quadsort

In the first stage of quadsort the quad swap is used to pre-sort the array into sorted 4-element blocks as described above.

The second stage still uses an approach similar to the quad swap to detect in order and reverse-order arrangements, but as it's sorting blocks of 4 or more the final step needs to be handled like the traditional merge sort.

This can be visualized as following:

AAAA BBBB CCCC DDDD EEEE FFFF GGGG HHHH IIII JJJJ KKKK LLLL MMMM NNNN OOOO PPPP

ABCDABCDABCDABCD EFGHEFGHEFGHEFGH IJKLIJKLIJKLIJKL MNOPMNOPMNOPMNOP

ABCDEFGHIJKLMNOPABCDEFGHIJKLMNOPABCDEFGHIJKLMNOPABCDEFGHIJKLMNOP

In the first row quadswap has been used to create 16 blocks of 4 sorted elements each. In the second row quadsort has been used to create 4 blocks of 16 sorted elements each. In the last row we're left with 1 block of 64 sorted elements. The following is a visualization.

These operations do require doubling the memory overhead for the swap space.

Skipping

Another difference is that due to the increased cost of merge operations it is beneficial to check whether the 4 blocks are in order or in reverse-order. In the case of the 4 blocks being in order the merge operation is skipped, as this would be pointless. This does however require an extra if check, and for randomly sorted data this if check becomes increasingly unlikely to be true as the block size increases. Fortunately the frequency of this if check is quartered each loop, while the potential benefit is quadrupled each loop.

In the case of partial order or partial reverse-order the comparisons in the merge itself are unnecessary and subsequently omitted. The data still needs to be swapped but this is a less computational intensive procedure.

This allows quadsort to sort in order and reverse-order sequences using n + log n comparisons instead of n * log n comparisons.

Boundary checks

Another issue with the traditional merge sort is that it performs wasteful boundary checks. This looks as following:

if (a < a_max && b < b_max) if (a < b) [insert a] else [insert b]

To optimize this quadsort compares the last element of sequence A against the last element of sequence B. If the last element of sequence A is smaller than the last element of sequence B we know that the (b < b_max) if check will always be false because sequence A will be fully merged first.

Similarly if the last element of sequence A is greater than the last element of sequence B we know that the (a < a_max) if check will always be false.

Alignment

When sorting an array of 65 elements you end up with a sorted array of 64 elements and a sorted array of 1 element in the end. Due to the ability to skip this will result in no additional swap operation if the entire sequence is in order. Regardless, if a program sorts in intervals it should pick an optimal array size to do so.

Since a suboptimal array size isn't disastrous for quadsort I haven't bothered implementing a solution.

Cache optimization

As currently implemented quadsort is not cache efficient for large arrays. This can be alleviated by pre-sorting large blocks. This (and other hardware specific optimizations) is not within the scope of the current implementation.

Big O

The best case performance is O(n) comparisons for sorted data and reverse-sorted data. The worst case performance is O(n log n). The average case performance is O(n log n).

Benchmarks

The following benchmark was on WSL gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04.1) with minimal background interference. Each test was ran 100 times and only the best run is reported.

Quad Sort: sorted 1000000 elements in 0.127645 seconds. (random order)
   Quick Sort: sorted 1000000 elements in 0.136306 seconds. (random order)

    Quad Sort: sorted 1000000 elements in 0.002521 seconds. (forward order)
   Quick Sort: sorted 1000000 elements in 0.029430 seconds. (forward order)
   
    Quad Sort: sorted 1000000 elements in 0.014468 seconds. (reverse order)
   Quick Sort: sorted 1000000 elements in 0.030888 seconds. (reverse order)

    Quad Sort: sorted 1000000 elements in 0.032876 seconds. (random tail)
   Quick Sort: sorted 1000000 elements in 0.053318 seconds. (random tail)

For the sake completeness I ran this benchmark once again.

Quad Sort: sorted 1000000 elements in 0.127452 seconds. (random order)
   Quick Sort: sorted 1000000 elements in 0.136210 seconds. (random order)

    Quad Sort: sorted 1000000 elements in 0.002457 seconds. (forward order)
   Quick Sort: sorted 1000000 elements in 0.029419 seconds. (forward order)

    Quad Sort: sorted 1000000 elements in 0.014460 seconds. (reverse order)
   Quick Sort: sorted 1000000 elements in 0.030880 seconds. (reverse order)

    Quad Sort: sorted 1000000 elements in 0.032865 seconds. (random tail)
   Quick Sort: sorted 1000000 elements in 0.053329 seconds. (random tail)

As the numbers are similar enough I consider them valid. I would say they speak for themselves. If you want to quickly run an independent benchmark yourself you can quickly do so at this link.

https://www.onlinegdb.com/ry6JnKZXI