Intro

This document describes a hybrid mergesort / quicksort named fluxsort. The sort is stable, adaptive, branchless, and has exceptional performance. A visualisation and benchmarks are available at the bottom.

Analyzer

Fluxsort starts out with an analyzer that detects fully sorted arrays and sorts reverse order arrays using n comparisons. It also obtains a measure of presortedness and switches to quadsort if the array is less than 50% random.

Partitioning

Partitioning is performed in a top-down manner similar to quicksort. Fluxsort obtains the pseudomedian of 9 for partitions smaller than 1024 elements, and the pseudomedian of 25 otherwise. The median element obtained will be referred to as the pivot. Partitions that grow smaller than 24 elements are sorted with quadsort.

After obtaining a pivot the array is parsed from start to end. Elements smaller than the pivot are copied in-place to the start of the array, elements greater than the pivot are copied to swap memory. The partitioning routine is called recursively on the two partitions in main and swap memory.

Recursively partitioning through both swap and main memory is accomplished by passing along a pointer (ptx) to either swap or main memory, so swap memory does not need to be copied back to main memory before it can be partitioned again.

Worst case handling

To avoid run-away recursion fluxsort switches to quadsort for both partitions if one partition is less than 1/16th the size of the other partition. On a distribution of random unique values the observed chance of a false positive is 1 in 1,336 for the pseudomedian of 9 and approximately 1 in 4 million for the pseudomedian of 25.

Combined with the analyzer fluxsort starts out with this makes the existence of killer patterns unlikely, other than a 33% performance slowdown by prematurely triggering the use of quadsort.

Branchless optimizations

Fluxsort uses a branchless comparison optimization. The ability of quicksort to partition branchless was first described in "BlockQuicksort: How Branch Mispredictions don't affect Quicksort" by Stefan Edelkamp and Armin Weiss. Since Fluxsort uses auxiliary memory the partitioning scheme is simpler and faster than the one used by BlockQuicksort.

Median selection uses a branchless comparison technique that selects the pseudomedian of 9 using 12 comparisons, and the pseudomedian of 25 using 42 comparisons.

These optimizations do not work as well when the comparisons themselves are branched and the largest performance increase is on 32 and 64 bit integers.

Generic data optimizations

Fluxsort uses a method popularized by pdqsort to improve generic data handling. If the same pivot is chosen twice in a row it performs a reverse partition, filtering out all elements equal to the pivot, next it carries on as usual. In addition, if after a partition all elements were smaller or equal to the pivot, a reverse partition is performed. This typically only occurs when sorting tables with many repeating values, like gender, education level, birthyear, zipcode, etc.

┌──────────────────────────────────┬───┬──────────────┐
│             E <= P               │ P │    E > P     | default partition
└──────────────────────────────────┴───┴──────────────┘

┌──────────────┬───┬───────────────────┐
│    P > E     │ P │    P <= E         |                reverse partition
└──────────────┴───┴───────────────────┘

┌──────────────┬───┬───────────────┬───┬──────────────┐
│    E < P     │ P │    E == P     │ P │     E > P    | 
└──────────────┴───┴───────────────┴───┴──────────────┘

Adaptive partitioning

Fluxsort performs low cost run detection while it partitions and switches to quadsort if a long run is detected. While the run detection is not fully robust it can result in significant performance gains at a neglible cost.

Large array optimizations

For partitions larger than 65536 elements fluxsort obtains the median of 128 or 256. It does so by copying 128 or 256 random elements to swap memory, sorting them with quadsort, and taking the center element. Using pseudomedian instead of median selection on large arrays is slower, likely due to cache pollution.

Small array optimizations

For paritions smaller than 24 elements fluxsort uses quadsort's small array sorting routine. This routine uses branchless parity merges for the first 8 or 16 elements, and twice-unguarded insertion sort to sort the remainder. This gives a significant performance gain compared to the unguarded insertion sort used by most quicksorts.

Big O

                 ┌───────────────────────┐┌───────────────────────┐
                 │comparisons            ││swap memory            │
┌───────────────┐├───────┬───────┬───────┤├───────┬───────┬───────┤┌──────┐┌─────────┐┌─────────┐
│name           ││min    │avg    │max    ││min    │avg    │max    ││stable││partition││adaptive │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│fluxsort       ││n      │n log n│n log n││1      │n      │n      ││yes   ││yes      ││yes      │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│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       │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│pdqsort        ││n      │n log n│n log n││1      │1      │1      ││no    ││yes      ││semi     │
└───────────────┘└───────┴───────┴───────┘└───────┴───────┴───────┘└──────┘└─────────┘└─────────┘

Data Types

The C implementation of fluxsort 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

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

Memory

Fluxsort allocates n elements of swap memory, which is shared with quadsort. Recursion requires log n stack memory.

If memory allocation fails fluxsort defaults to quadsort, which requires n / 4 elements of swap memory. If allocation fails again quadsort will sort in-place through rotations.

Performance

Fluxsort is the fastest stable comparison sort written to date.

To take full advantage of branchless operations the cmp macro needs to be uncommented in bench.c, which will double the performance on primitive types.

Porting

People wanting to port fluxsort might want to have a look at twinsort, which is a simplified implementation of quadsort. Fluxsort itself is relatively simple.

Visualization

In the visualization below four tests are performed on 512 elements: Random, Generic, Random Half, and Ascending Tiles. Partitions greater than 48 elements use the pseudomedian of 15 to select the pivot.

Benchmarks

The following benchmark was on WSL 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. The bar graph shows the best run out of 100 on 100,000 32 bit integers. Comparisons for timsort, fluxsort and std::stable_sort are inlined.

data table

Name	Items	Type	Best	Average	Compares	Samples	Distribution
stablesort	100000	64	0.006106	0.006165	1	100	random order
fluxsort	100000	64	0.001916	0.001938	1	100	random order
timsort	100000	64	0.007624	0.007680	1	100	random order

Name	Items	Type	Best	Average	Loops	Samples	Distribution
stablesort	100000	32	0.006154	0.006187	1	100	random order
fluxsort	100000	32	0.001830	0.001852	1	100	random order
timsort	100000	32	0.007637	0.007683	1	100	random order
stablesort	100000	32	0.004002	0.004041	1	100	random % 100
fluxsort	100000	32	0.000677	0.000699	1	100	random % 100
timsort	100000	32	0.005627	0.005701	1	100	random % 100
stablesort	100000	32	0.000799	0.000832	1	100	ascending order
fluxsort	100000	32	0.000044	0.000045	1	100	ascending order
timsort	100000	32	0.000045	0.000045	1	100	ascending order
stablesort	100000	32	0.001497	0.001524	1	100	ascending saw
fluxsort	100000	32	0.000812	0.000820	1	100	ascending saw
timsort	100000	32	0.000834	0.000840	1	100	ascending saw
stablesort	100000	32	0.000889	0.000913	1	100	pipe organ
fluxsort	100000	32	0.000377	0.000379	1	100	pipe organ
timsort	100000	32	0.000168	0.000171	1	100	pipe organ
stablesort	100000	32	0.000927	0.000953	1	100	descending order
fluxsort	100000	32	0.000055	0.000056	1	100	descending order
timsort	100000	32	0.000088	0.000092	1	100	descending order
stablesort	100000	32	0.001498	0.001526	1	100	descending saw
fluxsort	100000	32	0.000814	0.000823	1	100	descending saw
timsort	100000	32	0.000832	0.000841	1	100	descending saw
stablesort	100000	32	0.002158	0.002184	1	100	random tail
fluxsort	100000	32	0.000845	0.000854	1	100	random tail
timsort	100000	32	0.002010	0.002033	1	100	random tail
stablesort	100000	32	0.003652	0.003676	1	100	random half
fluxsort	100000	32	0.001160	0.001169	1	100	random half
timsort	100000	32	0.004041	0.004078	1	100	random half
stablesort	100000	32	0.001092	0.001124	1	100	ascending tiles
fluxsort	100000	32	0.000333	0.000336	1	100	ascending tiles
timsort	100000	32	0.000964	0.001025	1	100	ascending tiles
stablesort	100000	32	0.001762	0.001899	1	100	bit reversal
fluxsort	100000	32	0.001664	0.001687	1	100	bit reversal
timsort	100000	32	0.002254	0.002745	1	100	bit reversal

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 bench.c. The bar graph shows the best run out of 100 on 100,000 32 bit integers. Comparisons for fluxsort and qsort are not inlined. The stdlib qsort() in the benchmark is a mergesort variant.

data table

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	64	0.016893	0.017751	1536381	100	random string
fluxsort	100000	64	0.010192	0.010726	1884907	100	random string

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	128	0.018242	0.019490	1536491	100	random order
fluxsort	100000	128	0.011403	0.011871	1884438	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	64	0.009503	0.009730	1536491	100	random order
fluxsort	100000	64	0.004176	0.004258	1884438	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	32	0.008824	0.009017	1536634	100	random order
fluxsort	100000	32	0.003597	0.003648	1892483	100	random order
qsort	100000	32	0.006820	0.006973	1532465	100	random % 100
fluxsort	100000	32	0.001460	0.001500	972114	100	random % 100
qsort	100000	32	0.002606	0.002765	815024	100	ascending order
fluxsort	100000	32	0.000173	0.000177	99999	100	ascending order
qsort	100000	32	0.003379	0.003525	915020	100	ascending saw
fluxsort	100000	32	0.001257	0.001278	553308	100	ascending saw
qsort	100000	32	0.002672	0.002800	884462	100	pipe organ
fluxsort	100000	32	0.000712	0.000716	428065	100	pipe organ
qsort	100000	32	0.002472	0.002632	853904	100	descending order
fluxsort	100000	32	0.000168	0.000168	99999	100	descending order
qsort	100000	32	0.003293	0.003425	953892	100	descending saw
fluxsort	100000	32	0.001260	0.001276	565476	100	descending saw
qsort	100000	32	0.004208	0.004596	1012073	100	random tail
fluxsort	100000	32	0.001433	0.001501	670123	100	random tail
qsort	100000	32	0.005965	0.006254	1200713	100	random half
fluxsort	100000	32	0.002344	0.002421	1076830	100	random half
qsort	100000	32	0.003899	0.004497	1209200	100	ascending tiles
fluxsort	100000	32	0.002171	0.002221	789580	100	ascending tiles
qsort	100000	32	0.005194	0.005992	1553378	100	bit reversal
fluxsort	100000	32	0.003480	0.003622	1898395	100	bit reversal

The following benchmark was on WSL 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. The bar graph shows the best run out of 100 on 100,000 32 bit integers. Comparisons for fluxsort and pdqsort are inlined.

data table

Name	Items	Type	Best	Average	Compares	Samples	Distribution
fluxsort	100000	64	0.001907	0.001953	1	100	random order
pdqsort	100000	64	0.002681	0.002707	1	100	random order

Name	Items	Type	Best	Average	Loops	Samples	Distribution
fluxsort	100000	32	0.001835	0.001855	1	100	random order
pdqsort	100000	32	0.002685	0.002699	1	100	random order
fluxsort	100000	32	0.000677	0.000692	1	100	random % 100
pdqsort	100000	32	0.000777	0.000785	1	100	random % 100
fluxsort	100000	32	0.000044	0.000044	1	100	ascending order
pdqsort	100000	32	0.000084	0.000086	1	100	ascending order
fluxsort	100000	32	0.000810	0.000831	1	100	ascending saw
pdqsort	100000	32	0.003462	0.003481	1	100	ascending saw
fluxsort	100000	32	0.000377	0.000383	1	100	pipe organ
pdqsort	100000	32	0.002832	0.002903	1	100	pipe organ
fluxsort	100000	32	0.000056	0.000057	1	100	descending order
pdqsort	100000	32	0.000188	0.000192	1	100	descending order
fluxsort	100000	32	0.000809	0.000819	1	100	descending saw
pdqsort	100000	32	0.003232	0.003258	1	100	descending saw
fluxsort	100000	32	0.000843	0.000854	1	100	random tail
pdqsort	100000	32	0.002564	0.002589	1	100	random tail
fluxsort	100000	32	0.001161	0.001166	1	100	random half
pdqsort	100000	32	0.002666	0.002682	1	100	random half
fluxsort	100000	32	0.000334	0.000354	1	100	ascending tiles
pdqsort	100000	32	0.002316	0.002346	1	100	ascending tiles
fluxsort	100000	32	0.001661	0.001686	1	100	bit reversal
pdqsort	100000	32	0.002664	0.002679	1	100	bit reversal