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README.md

Intro

This document describes a stable non-recursive adaptive 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 swap variables. During the first stage the four variables are partially sorted in the four swap variables, in the second stage they are fully sorted back to the original four variables.

            ╭─╮             ╭─╮                  ╭─╮          ╭─╮
            │A├─╮         ╭─┤S├────────┬─────────┤?├─╮    ╭───┤F│
            ╰─╯    ╭─╮    ╰─╯                 ╰┬╯    ╭┴╮  ╰─╯
                ├───┤?├───┤                   ╭──╯  ╰───┤?
            ╭─╮    ╰─╯    ╭─╮                        ╰┬╯  ╭─╮
            │A├─╯         ╰─┤S├────────│────────╮         ╰───┤F│
            ╰─╯             ╰┬╯               ││             ╰─╯
                            ╭┴╮ ╭─╮   ╭┴╮ ╭─╮  ││
                            ?├─┤F│   ?├─┤F│  ││
                            ╰┬╯ ╰─╯   ╰┬╯ ╰─╯  ││
            ╭─╮             ╭┴╮               ││             ╭─╮
            │A├─╮         ╭─┤S├────────│───────╯│         ╭───┤F│
            ╰─╯    ╭─╮    ╰─╯                ╰─╮      ╭┴╮  ╰─╯
                ├───┤?├───┤                        ╭───┤?
            ╭─╮    ╰─╯    ╭─╮                 ╭┴╮    ╰┬╯  ╭─╮
            │A├─╯         ╰─┤S├────────┴─────────┤?├─╯    ╰───┤F│
            ╰─╯             ╰─╯                  ╰─╯          ╰─╯

This process is visualized in the diagram above.

After the first round of sorting a single if check determines if the four swap variables are sorted in order, if that's the case the swap finishes up immediately. Next it checks if the swap variables are sorted in reverse-order, if that's the case the sort finishes up immediately. If both checks fail the final arrangement is known and two checks remain to determine the final order.

This eliminates 1 wasteful comparison for in-order sequences while creating 1 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.

There is also an overall performance increase due to the elimination of wasteful swapping. In C the basic quad swap 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.

The quad swap above is implemented in-place in quadsort.

quad merge

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 uses an approach similar to the quad swap to detect in-order and reverse-order arrangements, but as it's sorting blocks of 4, 16, 64, or more elements, the final step needs to be handled like the traditional merge sort.

This can be visualized as following:

main memory: AAAA BBBB CCCC DDDD

swap memory: ABABABAB  CDCDCDCD

main memory: ABCDABCDABCDABCD

In the first row quad swap has been used to create 4 blocks of 4 sorted elements each. In the second row quad merge has been used to merge the blocks into 2 blocks of 8 sorted elements each in swap memory. In the last row the blocks are merged back to main memory and we're left with 1 block of 16 sorted elements. The following is a visualization.

quadsort visualization

These operations do require doubling the memory overhead for the swap space. More on this later.

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 the 4 blocks being in reverse order an in-place stable swap is performed.

In the case only 2 out of 4 blocks are in order or in reverse-order the comparisons in the merge itself are unnecessary and subsequently omitted. The data still needs to be copied to swap memory, but this is a less computational intensive procedure.

This allows quadsort to sort in order and reverse-order sequences using 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:

    while (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. This looks as following:

    if (val[a_max] <= val[b_max])
        while (a <= a_max)
        {
            while (a > b)
                [insert b++]
            [insert a++]
        }
    else
        while (b <= b_max)
        {
            while (a <= b)
                [insert a++]
            [insert b++]
        }

tail merge

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 (64, 256, or 1024) to do so.

Another problem is that a sub-optimal array results in wasteful swapping. To work around these two problems the quad merge routine is aborted when the block size reaches 1/8th of the array size, and the remainder of the array is sorted using a tail merge.

The main advantage of the tail merge is that it allows reducing the swap space of quadsort to n / 2 without notably impacting performance.

Big O

Name Best Average Worst Stable Memory
quadsort n n log n n log n yes n

Quadsort makes n comparisons when the data is already sorted or reverse sorted.

Cache

Because quadsort uses n / 2 swap memory its cache utilization is not as ideal as in-place sorts. However, in-place sorting of random data results in suboptimal swapping. Based on my benchmarks it appears that quadsort is always faster than in-place sorts for array sizes that do not exhaust the L1 cache, which can be up to 64KB on modern processors.

Wolfsort and flowsort

wolfsort and flowsort are a hybrid radixsort / quadsort with improved performance on random data. They're mostly a proof of concept that only work on unsigned 32 and 64 bit integers.

Visualization

In the visualization below four tests are performed. The first test is on a random distribution, the second on an ascending distribution, the third on a descending distribution, and the fourth on an ascending distribution with a random tail.

The upper half shows the swap memory and the bottom half shows the main memory. Colors are used to differentiate between skip, swap, merge, and copy operations.

quadsort benchmark

Benchmark: quadsort vs std::stable_sort vs timsort vs pdqsort vs wolfsort

The following benchmark was on WSL gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04.1). The source code was compiled using g++ -O3 quadsort.cpp. Each test was ran 100 times and only the best run is reported.

It should be noted that pdqsort is not a stable sort which is the reason it's much faster on generic order data.

The X axis of the bar graph below is the execution time.

Graph

quadsort vs std::stable_sort vs timsort vs pdqsort vs wolfsort data table | Name | Items | Type | Best | Average | Comparisons | Distribution | | --------- | -------- | ---- | -------- | -------- | ----------- | ---------------- | | quadsort | 1000000 | i32 | 0.070469 | 0.070635 | | random order | |stablesort | 1000000 | i32 | 0.073865 | 0.074078 | | random order | | timsort | 1000000 | i32 | 0.089192 | 0.089301 | | random order | | pdqsort | 1000000 | i32 | 0.029911 | 0.029948 | | random order | | wolfsort | 1000000 | i32 | 0.017359 | 0.017744 | | random order | | | | | | | | | | quadsort | 1000000 | i32 | 0.000485 | 0.000511 | | ascending | |stablesort | 1000000 | i32 | 0.010188 | 0.010261 | | ascending | | timsort | 1000000 | i32 | 0.000331 | 0.000342 | | ascending | | pdqsort | 1000000 | i32 | 0.000863 | 0.000875 | | ascending | | wolfsort | 1000000 | i32 | 0.000539 | 0.000579 | | ascending | | | | | | | | | | quadsort | 1000000 | i32 | 0.008901 | 0.008934 | | ascending saw | |stablesort | 1000000 | i32 | 0.017093 | 0.017275 | | ascending saw | | timsort | 1000000 | i32 | 0.008615 | 0.008674 | | ascending saw | | pdqsort | 1000000 | i32 | 0.047785 | 0.047921 | | ascending saw | | wolfsort | 1000000 | i32 | 0.012490 | 0.012554 | | ascending saw | | | | | | | | | | quadsort | 1000000 | i32 | 0.038260 | 0.038343 | | generic order | |stablesort | 1000000 | i32 | 0.042292 | 0.042388 | | generic order | | timsort | 1000000 | i32 | 0.055855 | 0.055967 | | generic order | | pdqsort | 1000000 | i32 | 0.008093 | 0.008168 | | generic order | | wolfsort | 1000000 | i32 | 0.038320 | 0.038417 | | generic order | | | | | | | | | | quadsort | 1000000 | i32 | 0.000559 | 0.000576 | | descending order | |stablesort | 1000000 | i32 | 0.010343 | 0.010438 | | descending order | | timsort | 1000000 | i32 | 0.000891 | 0.000900 | | descending order | | pdqsort | 1000000 | i32 | 0.001882 | 0.001897 | | descending order | | wolfsort | 1000000 | i32 | 0.000604 | 0.000625 | | descending order | | | | | | | | | | quadsort | 1000000 | i32 | 0.006174 | 0.006245 | | descending saw | |stablesort | 1000000 | i32 | 0.014679 | 0.014767 | | descending saw | | timsort | 1000000 | i32 | 0.006419 | 0.006468 | | descending saw | | pdqsort | 1000000 | i32 | 0.016603 | 0.016629 | | descending saw | | wolfsort | 1000000 | i32 | 0.006264 | 0.006329 | | descending saw | | | | | | | | | | quadsort | 1000000 | i32 | 0.018675 | 0.018729 | | random tail | |stablesort | 1000000 | i32 | 0.026384 | 0.026508 | | random tail | | timsort | 1000000 | i32 | 0.023226 | 0.023395 | | random tail | | pdqsort | 1000000 | i32 | 0.028599 | 0.028674 | | random tail | | wolfsort | 1000000 | i32 | 0.009517 | 0.009631 | | random tail | | | | | | | | | | quadsort | 1000000 | i32 | 0.037593 | 0.037679 | | random half | |stablesort | 1000000 | i32 | 0.043755 | 0.043899 | | random half | | timsort | 1000000 | i32 | 0.047008 | 0.047112 | | random half | | pdqsort | 1000000 | i32 | 0.029800 | 0.029847 | | random half | | wolfsort | 1000000 | i32 | 0.013238 | 0.013355 | | random half | | | | | | | | | | quadsort | 1000000 | i32 | 0.009605 | 0.009673 | | wave order | |stablesort | 1000000 | i32 | 0.013667 | 0.013785 | | wave order | | timsort | 1000000 | i32 | 0.007994 | 0.008138 | | wave order | | pdqsort | 1000000 | i32 | 0.024683 | 0.024727 | | wave order | | wolfsort | 1000000 | i32 | 0.009642 | 0.009709 | | wave order |

The following benchmark was on WSL gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04.1). The source code was compiled using g++ -O3 quadsort.cpp. Each test was ran 100 times and only the best run is reported. It measures the performance on random data with array sizes ranging from 256 to 1,048,576.

The X axis of the graph below is the number of elements, the Y axis is the execution time.

Graph

quadsort vs std::stable_sort vs timsort vs pdqsort vs wolfsort vs flowsort data table
Name Items Type Best Average Comparisons Distribution
quadsort 256 i32 0.008306 0.009226 random order
stablesort 256 i32 0.009325 0.022037 random order
timsort 256 i32 0.015605 0.026554 random order
pdqsort 256 i32 0.010840 0.015047 random order
wolfsort 256 i32 0.008287 0.008338 random order
flowsort 256 i32 0.008332 0.009783 random order
quadsort 512 i32 0.007497 0.012202 random order
stablesort 512 i32 0.014719 0.023305 random order
timsort 512 i32 0.014791 0.026926 random order
pdqsort 512 i32 0.010218 0.015399 random order
wolfsort 512 i32 0.005434 0.005711 random order
flowsort 512 i32 0.008525 0.008842 random order
quadsort 1024 i32 0.012224 0.013265 random order
stablesort 1024 i32 0.027686 0.033828 random order
timsort 1024 i32 0.033677 0.043642 random order
pdqsort 1024 i32 0.011030 0.015985 random order
wolfsort 1024 i32 0.005497 0.007087 random order
flowsort 1024 i32 0.008267 0.009476 random order
quadsort 4096 i32 0.040918 0.041427 random order
stablesort 4096 i32 0.043393 0.044039 random order
timsort 4096 i32 0.059069 0.059289 random order
pdqsort 4096 i32 0.019076 0.020721 random order
wolfsort 4096 i32 0.007710 0.009736 random order
flowsort 4096 i32 0.010430 0.012855 random order
quadsort 16384 i32 0.051287 0.051480 random order
stablesort 16384 i32 0.052293 0.052393 random order
timsort 16384 i32 0.068145 0.068304 random order
pdqsort 16384 i32 0.024337 0.024414 random order
wolfsort 16384 i32 0.012334 0.012433 random order
flowsort 16384 i32 0.015112 0.015173 random order
quadsort 65536 i32 0.058787 0.058863 random order
stablesort 65536 i32 0.060166 0.060262 random order
timsort 65536 i32 0.076500 0.076612 random order
pdqsort 65536 i32 0.026368 0.026425 random order
wolfsort 65536 i32 0.013164 0.013208 random order
flowsort 65536 i32 0.015327 0.015362 random order
quadsort 262144 i32 0.066391 0.066484 random order
stablesort 262144 i32 0.068144 0.068255 random order
timsort 262144 i32 0.084703 0.084835 random order
pdqsort 262144 i32 0.028397 0.028457 random order
wolfsort 262144 i32 0.013937 0.014095 random order
flowsort 262144 i32 0.016058 0.016107 random order
quadsort 1048576 i32 0.074302 0.074495 random order
stablesort 1048576 i32 0.076274 0.076419 random order
timsort 1048576 i32 0.093351 0.093517 random order
pdqsort 1048576 i32 0.030378 0.030446 random order
wolfsort 1048576 i32 0.034210 0.034403 random order
flowsort 1048576 i32 0.017668 0.017795 random order

Benchmark: quadsort vs qsort (mergesort)

The following benchmark was on WSL gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04.1). The source code was compiled using gcc -O3 quadsort.c. Each test was ran 100 times and only the best run is reported.

MO: lists the number of comparisons that are performed for 1 million items.

         quadsort: sorted 1000000 i32s in 0.092399 seconds. MO:   19305366 (random order)
            qsort: sorted 1000000 i32s in 0.103581 seconds. MO:   18673007 (random order)

         quadsort: sorted 1000000 i32s in 0.002191 seconds. MO:     999999 (ascending)
            qsort: sorted 1000000 i32s in 0.026788 seconds. MO:    9884992 (ascending)

         quadsort: sorted 1000000 i32s in 0.013560 seconds. MO:    4008160 (ascending saw)
            qsort: sorted 1000000 i32s in 0.034882 seconds. MO:   10884985 (ascending saw)

         quadsort: sorted 1000000 i32s in 0.057610 seconds. MO:   19241914 (generic order)
            qsort: sorted 1000000 i32s in 0.070901 seconds. MO:   18617580 (generic order)

         quadsort: sorted 1000000 i32s in 0.001780 seconds. MO:     999999 (descending order)
            qsort: sorted 1000000 i32s in 0.026404 seconds. MO:   10066432 (descending order)

         quadsort: sorted 1000000 i32s in 0.015482 seconds. MO:    9519209 (descending saw)
            qsort: sorted 1000000 i32s in 0.034839 seconds. MO:   13906008 (descending saw)

         quadsort: sorted 1000000 i32s in 0.026516 seconds. MO:    6786305 (random tail)
            qsort: sorted 1000000 i32s in 0.046344 seconds. MO:   12248243 (random tail)

         quadsort: sorted 1000000 i32s in 0.050595 seconds. MO:   11381790 (random half)
            qsort: sorted 1000000 i32s in 0.067199 seconds. MO:   14528949 (random half)

         quadsort: sorted 1000000 i32s in 0.024795 seconds. MO:   15328606 (wave order)
            qsort: sorted 1000000 i32s in 0.035221 seconds. MO:   14656080 (wave order)

         quadsort: sorted 1000000 i32s in 0.024867 seconds. MO:   15328606 (stable)
            qsort: sorted 1000000 i32s in 0.035251 seconds. MO:   14656080 (stable)

         quadsort: sorted    1023 i32s in 0.013662 seconds.                (random 1-1023)
            qsort: sorted    1023 i32s in 0.025581 seconds.                (random 1-1023)

In the benchmark above quadsort is compared against glibc qsort() using the same general purpose interface and without any known unfair advantage, like inlining.

     random order: 635, 202,  47, 229, etc
  ascending order: 1, 2, 3, 4, etc
    uniform order: 1, 1, 1, 1, etc
 descending order: 999, 998, 997, 996, etc
       wave order: 100, 1, 102, 2, 103, 3, etc
  stable/unstable: 100, 1, 102, 1, 103, 1, etc
     random range: time to sort 1000 arrays ranging from size 0 to 999 containing random data

Benchmark: quadsort vs qsort (quicksort)

This particular test was performed using the qsort() implementation from Cygwin which uses quicksort under the hood. The source code was compiled using gcc -O3 quadsort.c. Each test was ran 100 times and only the best run is reported.

         quadsort: sorted 1000000 i32s in 0.119437 seconds. MO:   19308657 (random order)
            qsort: sorted 1000000 i32s in 0.133077 seconds. MO:   21083741 (random order)

         quadsort: sorted 1000000 i32s in 0.002071 seconds. MO:     999999 (ascending)
            qsort: sorted 1000000 i32s in 0.007265 seconds. MO:    3000004 (ascending)

         quadsort: sorted 1000000 i32s in 0.019239 seconds. MO:    4007580 (ascending saw)
            qsort: sorted 1000000 i32s in 0.071322 seconds. MO:   20665677 (ascending saw)

         quadsort: sorted 1000000 i32s in 0.076605 seconds. MO:   19242642 (generic order)
            qsort: sorted 1000000 i32s in 0.038389 seconds. MO:    6221917 (generic order)

         quadsort: sorted 1000000 i32s in 0.002305 seconds. MO:     999999 (descending order)
            qsort: sorted 1000000 i32s in 0.009659 seconds. MO:    4000015 (descending order)

         quadsort: sorted 1000000 i32s in 0.022940 seconds. MO:    9519209 (descending saw)
            qsort: sorted 1000000 i32s in 0.042135 seconds. MO:   13152042 (descending saw)

         quadsort: sorted 1000000 i32s in 0.034456 seconds. MO:    6787656 (random tail)
            qsort: sorted 1000000 i32s in 0.098691 seconds. MO:   20584424 (random tail)

         quadsort: sorted 1000000 i32s in 0.066240 seconds. MO:   11383441 (random half)
            qsort: sorted 1000000 i32s in 0.118086 seconds. MO:   20572142 (random half)

         quadsort: sorted 1000000 i32s in 0.038116 seconds. MO:   15328606 (wave order)
            qsort: sorted 1000000 i32s in 4.471858 seconds. MO: 1974047339 (wave order)

         quadsort: sorted 1000000 i32s in 0.060989 seconds. MO:   15328606 (stable)
            qsort: sorted 1000000 i32s in 0.043175 seconds. MO:   10333679 (unstable)

         quadsort: sorted    1023 i32s in 0.016126 seconds.       (random 1-1023)
            qsort: sorted    1023 i32s in 0.033132 seconds.       (random 1-1023)

In this benchmark it becomes clear why quicksort is often preferred above a traditional mergesort, it has fewer comparisons for ascending, uniform, and descending order data. However, it performs worse than quadsort on most tests. Quicksort also has an extremely poor sorting speed for wave order data. The random range test shows quadsort to be more than twice as fast when sorting small arrays.

Quicksort is faster on the generic and stable tests, but only because it is unstable.

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Quadsort is a stable adaptive merge sort which is faster than quicksort.

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