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Intro

This document describes a stable bottom-up adaptive branchless merge sort named quadsort. A visualisation and benchmarks are available at the bottom.

The quad swap analyzer

Quadsort starts out with an analyzer that has the following tasks:

1. Detect ordered data with minimal comparisons.
2. Detect reverse order data with minimal comparisons.
3. Do the above without impacting performance on random data.
4. Exit the quad swap analyzer with sorted blocks of 8 elements.

Ordered data handling

Quadsort's analyzer examines the array 8 elements at a time. It performs 4 comparisons on elements (1,2), (3,4), (5,6), and (7,8) of which the result is stored and a bitmask is created with a value between 0 and 15 for all the possible combinations. If the result is 0 it means the 4 comparisons were all in order.

What remains is 3 more comparisons on elements (2,3), (4,5), and (6,7) to determine if all 8 elements are in order. Traditional sorts would do this with 7 branched individual comparisons, which should result in 3.2 branch mispredictions on random data on average. Using quadsort's method should result in 0.2 branch mispredictions on random data on average.

If the data is in order quadsort moves on to the next 8 elements. If the data turns out to be neither in order or in reverse order, 4 branchless swaps are performed using the stored comparison results, followed by a branchless parity merge. More on that later.

Reverse order handling

Reverse order data is typically moved using a simple reversal function, as following.

int reverse(int array[], int start, int end, int swap)
{
    while (start < end)
    {
        swap = array[start];
        array[start++] = array[end];
        array[end--] = swap;
    }
}

While random data can only be sorted using n log n comparisons and n log n moves, reverse-order data can be sorted using n comparisons and n moves through run detection and a reversal routine. Without run detection the best you can do is sort it in n comparisons and n log n moves.

Run detection, as the name implies, comes with a detection cost. Thanks to the laws of probability a quad swap can cheat however. The chance of 4 separate pairs of elements being in reverse order is 1 in 16. So there's a 6.25% chance quadsort makes a wasteful run check.

What about run detection for in-order data? While we're turning n log n moves into n moves with reverse order run detection, we'd be turning 0 moves into 0 moves with forward run detection. So there's no point in doing so.

The next optimization is to write the quad swap analyzer in such a way that we can perform a simple check to see if the entire array was in reverse order, if so, the sort is finished.

At the end of the loop the array has been turned into a series of ordered blocks of 8 elements.

Ping-Pong Quad Merge

Most textbook mergesort examples merge two blocks to swap memory, then copy them back to main memory.

main memory ┌────────┐┌────────┐
            └────────┘└────────┘
                  ↓ merge ↓
swap memory ┌──────────────────┐
            └──────────────────┘
                  ↓ copy ↓
main memory ┌──────────────────┐
            └──────────────────┘

This doubles the amount of moves and we can fix this by merging 4 blocks at once using a quad merge / ping-pong merge like so:

main memory ┌────────┐┌────────┐┌────────┐┌────────┐
            └────────┘└────────┘└────────┘└────────┘
                  ↓ merge ↓           ↓ merge ↓
swap memory ┌──────────────────┐┌──────────────────┐
            └──────────────────┘└──────────────────┘
                            ↓ merge ↓
main memory ┌──────────────────────────────────────┐
            └──────────────────────────────────────┘

It is possible to interleave the two merges to swap memory for increased memory-level parallelism, but this can both increase and decrease performance.

Skipping

Just like with the quad swap it is beneficial to check whether the 4 blocks are in-order.

In the case of the 4 blocks being in-order the merge operation is skipped, as this would be pointless. Because reverse order data is handled in the quad swap there is no need to check for reverse order blocks.

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

Parity merge

A parity merge takes advantage of the fact that if you have two n length arrays, you can fully merge the two arrays by performing n merge operations on the start of each array, and n merge operations on the end of each array. The arrays must be of equal length. Another way to describe the parity merge would be as a bidirectional unguarded merge.

The main advantage of a parity merge over a traditional merge is that the loop of a parity merge can be fully unrolled.

If the arrays are not of equal length a hybrid parity merge can be performed. One way to do so is using n parity merges where n is the size of the smaller array, before switching to a traditional merge.

Branchless parity merge

Since the parity merge can be unrolled it's very suitable for branchless optimizations to speed up the sorting of random data. Another advantage is that two separate memory regions are accessed in the same loop, allowing memory-level parallelism. This makes the routine up to 2.5 times faster for random data on most hardware.

Increasing the memory regions from two to four can result in both performance gains and performance losses.

The following is a visualization of an array with 256 random elements getting turned into sorted blocks of 32 elements using ping-pong parity merges.

Cross merge

While a branchless parity merge sorts random data faster, it sorts ordered data slower. One way to solve this problem is by using a method with a resemblance to the galloping merge concept first introduced by timsort.

The cross merge works in a similar way to the quad swap. Instead of merging two arrays two items at a time, it merges four items at a time.

┌───┐┌───┐┌───┐    ┌───┐┌───┐┌───┐            ╭───╮  ┌───┐┌───┐┌───┐
│ A ││ B ││ C │    │ X ││ Y ││ Z │        ┌───│B<X├──┤ A ││ B ││C/X│
└─┬─┘└─┬─┘└───┘    └─┬─┘└─┬─┘└───┘        │   ╰─┬─╯  └───┘└───┘└───┘
  └────┴─────────────┴────┴───────────────┘     │  ╭───╮  ┌───┐┌───┐┌───┐
                                                └──│A>Y├──┤ X ││ Y ││A/Z│
                                                   ╰─┬─╯  └───┘└───┘└───┘
                                                     │    ┌───┐┌───┐┌───┐
                                                     └────│A/X││X/A││B/Y│
                                                          └───┘└───┘└───┘

When merging ABC and XYZ it first checks if B is smaller or equal to X. If that's the case A and B are copied to swap. If not, it checks if A is greater than Y. If that's the case X and Y are copied to swap.

If either check is false it's known that the two remaining distributions are A X and X A. This allows performing an optimal branchless merge. Since it's known each block still has at least 1 item remaining (B and Y) and there is a high probability of the data to be random, another (sub-optimal) branchless merge can be performed.

In C this looks as following:

while (l < l_size - 1 && r < r_size - 1)
{
    if (left[l + 1] <= right[r])
    {
        swap[s++] = left[l++];
        swap[s++] = left[l++];
    }
    else if (left[l] > right[r + 1])
    {
        swap[s++] = right[r++];
        swap[s++] = right[r++];
    }
    else
    {
        a = left[l] > right[r];
        x = !a;
        swap[s + a] = left[l++];
        swap[s + x] = right[r++];
        s += 2;
    }
}

Overall the cross merge gives a decent performance gain for both ordered and random data, particularly when the two arrays are of unequal length. When two arrays are of near equal length quadsort looks 8 elements ahead, and performs an 8 element parity merge if it can't skip ahead.

Merge strategy

Quadsort will merge blocks of 8 into blocks of 32, which it will merge into blocks of 128, 512, 2048, 8192, etc.

For each ping-pong merge quadsort will perform two comparisons to see if it will be faster to use a parity merge or a cross merge, and pick the best option.

Tail merge

When sorting an array of 33 elements you end up with a sorted array of 32 elements and a sorted array of 1 element in the end. If a program sorts in intervals it should pick an optimal array size (32, 128, 512, etc) to do so.

To minimalize the impact the remainder of the array is sorted using a tail merge.

Big O

                 ┌───────────────────────┐┌───────────────────────┐
                 │comparisons            ││swap memory            │
┌───────────────┐├───────┬───────┬───────┤├───────┬───────┬───────┤┌──────┐┌─────────┐┌─────────┐
│name           ││min    │avg    │max    ││min    │avg    │max    ││stable││partition││adaptive │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│mergesort      ││n log n│n log n│n log n││n      │n      │n      ││yes   ││no       ││no       │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│quadsort       ││n      │n log n│n log n││1      │n      │n      ││yes   ││no       ││yes      │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│quicksort      ││n log n│n log n│n²     ││1      │1      │1      ││no    ││yes      ││no       │
└───────────────┘└───────┴───────┴───────┘└───────┴───────┴───────┘└──────┘└─────────┘└─────────┘

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

Data Types

The C implementation of quadsort supports long doubles and 8, 16, 32, and 64 bit data types. By using pointers it's possible to sort any other data type, like strings.

Interface

Quadsort uses the same interface as qsort, which is described in man qsort.

In addition to supporting (l - r) and ((l > r) - (l < r)) for the comparison function, (l > r) is valid as well. Special note should be taken that C++ sorts use (l < r) for the comparison function, which is incompatible with the C standard. When porting quadsort to C++ or Rust, switch (l, r) to (r, l) for every comparison.

Quadsort comes with the quadsort_prim(void *array, size_t nmemb, size_t size) function to perform primitive comparisons on arrays of 32 and 64 bit integers. Nmemb is the number of elements, while size should be either sizeof(int) or sizeof(long long) for signed integers, and sizeof(int) + 1 or sizeof(long long) + 1 for unsigned integers. Support for the char, short, float, double, and long double types can be easily added in quadsort.h.

Quadsort comes with the quadsort_size(void *array, size_t nmemb, size_t size, CMPFUNC *cmp) function to sort elements of any given size. The comparison function needs to be by reference, instead of by value, as if you are sorting an array of pointers.

Memory

By default quadsort uses n swap memory. If memory allocation fails quadsort will switch to sorting in-place through rotations. The minimum memory requirement is 32 elements of stack memory.

Rotations can be performed with minimal performance loss by using branchless binary searches and trinity / bridge rotations.

Sorting in-place through rotations will increase the number of moves from n log n to n log² n. The overall impact on performance is minor on array sizes below 1M elements.

Performance

Quadsort is one of the fastest merge sorts written to date. It is faster than quicksort for most data distributions, with the notable exception of generic data. Data type is important as well, and overall quadsort is faster for sorting referenced objects.

Compared to Timsort, Quadsort has similar overall adaptivity while being much faster on random data, even without branchless optimizations.

Quicksort has a slight advantage on random data as the array size increases beyond the L1 cache. For small arrays quadsort has a significant advantage due to quicksort's inability to cost effectively pick a reliable pivot. Subsequently, the only way for quicksort to rival quadsort is to cheat and become a hybrid sort, by using branchless merges to sort small partitions.

When using the clang compiler quadsort can use branchless ternary comparisons. Since most programming languages only support ternary merges ? : and not ternary partitions : ? this gives branchless mergesorts an additional advantage over branchless quicksorts. However, since the gcc compiler doesn't support branchless ternary merges, and the hack to perform branchless merges is less efficient than the hack to perform branchless partitions, branchless quicksorts have an advantage for gcc.

To take full advantage of branchless operations the cmp macro needs to be uncommented in bench.c, which will increase the performance by 30% on primitive types. The quadsort_prim function can be used to access primitive comparisons directly.

Variants

• blitsort is a hybrid stable in-place rotate quicksort / quadsort.

• crumsort is a hybrid unstable in-place quicksort / quadsort.

• fluxsort is a hybrid stable quicksort / quadsort.

• gridsort is a hybrid stable cubesort / quadsort. Gridsort is an online sort and might be of interest to those interested in data structures and sorting very large arrays.

• twinsort is a simplified quadsort with a much smaller code size. Twinsort might be of use to people who want to port or understand quadsort; it does not use pointers or gotos. It is a bit dated and isn't branchless.

• piposort is a simplified branchless quadsort with a much smaller code size and complexity while still being very fast. Piposort might be of use to people who want to port quadsort. This is a lot easier when you start out small.

• wolfsort is a hybrid stable radixsort / fluxsort with improved performance on random data. It's mostly a proof of concept that only works on unsigned 32 bit integers.

• Robin Hood Sort is a hybrid stable radixsort / dropsort with improved performance on random and generic data. It has a compilation option to use quadsort for its merging.

Credits

I personally invented the quad swap analyzer, cross merge, parity merge, branchless parity merge, monobound binary search, bridge rotation, and trinity rotation.

The ping-pong quad merge had been independently implemented in wikisort prior to quadsort, and likely by others as well.

The monobound binary search has been independently implemented, often referred to as a branchless binary search. I published a working concept in 2014, which appears to pre-date others.

Special kudos to Musiccombo and Co for getting me interested in rotations and branchless logic.

Visualization

In the visualization below nine tests are performed on 256 elements.

1. Random order
2. Ascending order
3. Ascending Saw
4. Generic random order
5. Descending order
6. Descending Saw
7. Random tail
8. Random half
9. Ascending tiles.

The upper half shows the swap memory and the bottom half shows the main memory. Colors are used to differentiate various operations. Quad swaps are in cyan, reversals in magenta, skips in green, parity merges in orange, bridge rotations in yellow, and trinity rotations are in violet.

The visualization is available on YouTube and there's also a YouTube video of a java port of quadsort in ArrayV on a wide variety of data distributions.

Benchmark: quadsort vs std::stable_sort vs timsort

The following benchmark was on WSL 2 gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04) using the wolfsort benchmark. The source code was compiled using g++ -O3 -w -fpermissive bench.c. Stablesort is g++'s std:stablesort function. Each test was ran 100 times on 100,000 elements. A table with the best and average time in seconds can be uncollapsed below the bar graph.

The following benchmark was on WSL 2 gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04) using the wolfsort benchmark. The source code was compiled using g++ -O3 -w -fpermissive bench.c. It measures the performance on random data with array sizes ranging from 1 to 1024. It's generated by running the benchmark using 1024 0 0 as the argument. The benchmark is weighted, meaning the number of repetitions halves each time the number of items doubles. A table with the best and average time in seconds can be uncollapsed below the bar graph.

Benchmark: quadsort vs qsort (mergesort)

The following benchmark was on WSL 2 gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04). The source code was compiled using gcc -O3 bench.c. Each test was ran 100 times. It's generated by running the benchmark using 100000 100 1 as the argument. In the benchmark quadsort is compared against glibc qsort() using the same general purpose interface and without any known unfair advantage, like inlining. A table with the best and average time in seconds can be uncollapsed below the bar graph.

The following benchmark was on WSL 2 gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04). The source code was compiled using gcc -O3 bench.c. Each test was ran 100 times. It's generated by running the benchmark using 10000000 0 0 as the argument. The benchmark is weighted, meaning the number of repetitions halves each time the number of items doubles. A table with the best and average time in seconds can be uncollapsed below the bar graph.

Benchmark: quadsort vs pdqsort vs fluxsort

The following benchmark was on WSL 2 gcc version 7.5.0 (Ubuntu 7.5.0-3ubuntu1~18.04) using the wolfsort benchmark. The source code was compiled using g++ -O3 -w -fpermissive bench.c. Pdqsort is a branchless quicksort/heapsort/insertionsort hybrid. Fluxsort is a branchless quicksort/mergesort hybrid. Each test was ran 100 times on 100,000 elements. Comparisons are fully inlined. A table with the best and average time in seconds can be uncollapsed below the bar graph.

The following benchmark was on WSL clang version 10 (10.0.0-4ubuntu1~18.04.2) using rhsort's wolfsort benchmark. The source code was compiled using clang -O3. The bar graph shows the best run out of 100 on 131,072 32 bit integers. Comparisons for quadsort, fluxsort and glidesort are inlined.

Some additional context is required for this benchmark. Glidesort is written and compiled in Rust which supports branchless ternary operations, subsequently fluxsort and quadsort are compiled using clang with branchless ternary operations in place for the merge and small-sort routines. Since fluxsort and quadsort are optimized for gcc there is a performance penalty, with some of the routines running 2-3x slower than they do in gcc.