Intro

This document describes a partitioning stable adaptive comparison-based sort named fluxsort. Benchmarks and a visualization 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 25% 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 15 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 through the ptx pointer, which despite being simple in implementation is likely a novel technique since the logic behind it is a bit of a brain-twister.

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 chance of a false positive is 1 in 65,536 for the pseudomedian of 9 and 1 in 16,777,216 for the pseudomedian of 15.

Combined with the analyzer fluxsort starts out with this makes the existence of killer patterns unlikely, other than a 2x performance slowdown by triggering the use of quadsort prematurely.

Branchless optimizations

Fluxsort uses a branchless comparison optimization similar to the method described in "BlockQuicksort: How Branch Mispredictions don't affect Quicksort" by Stefan Edelkamp and Armin Weiss.

Median selection uses a novel branchless comparison technique that selects the pseudomedian of 9 using between 8 and 12 (10.66 average) comparisons, and the pseudomedian of 15 using between 14 and 25 (21.33 average) 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.

Memory

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

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.

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      │n log n│n²     ││1      │1      │1      ││no    ││yes      ││no       │
├───────────────┤├───────┼───────┼───────┤├───────┼───────┼───────┤├──────┤├─────────┤├─────────┤
│introsort      ││n log n│n log n│n log n││1      │1      │1      ││no    ││yes      ││no       │
└───────────────┘└───────┴───────┴───────┘└───────┴───────┴───────┘└──────┘└─────────┘└─────────┘

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 fluxsort and std::stable_sort are inlined.

data table

Name	Items	Type	Best	Average	Loops	Samples	Distribution
stablesort	100000	64	0.006123	0.006153	1	100	random order
fluxsort	100000	64	0.002477	0.002488	1	100	random order

Name	Items	Type	Best	Average	Loops	Samples	Distribution
stablesort	100000	32	0.006008	0.006033	1	100	random order
fluxsort	100000	32	0.002329	0.002345	1	100	random order
stablesort	100000	32	0.000679	0.000683	1	100	ascending order
fluxsort	100000	32	0.000037	0.000037	1	100	ascending order
stablesort	100000	32	0.001375	0.001402	1	100	ascending saw
fluxsort	100000	32	0.000842	0.000854	1	100	ascending saw
stablesort	100000	32	0.003827	0.003853	1	100	generic order
fluxsort	100000	32	0.001129	0.001140	1	100	generic order
stablesort	100000	32	0.000901	0.000912	1	100	descending order
fluxsort	100000	32	0.000048	0.000048	1	100	descending order
stablesort	100000	32	0.001021	0.001035	1	100	descending saw
fluxsort	100000	32	0.000351	0.000365	1	100	descending saw
stablesort	100000	32	0.002034	0.002060	1	100	random tail
fluxsort	100000	32	0.001485	0.001492	1	100	random tail
stablesort	100000	32	0.003520	0.003542	1	100	random half
fluxsort	100000	32	0.002077	0.002088	1	100	random half
stablesort	100000	32	0.000922	0.000956	1	100	ascending tiles
fluxsort	100000	32	0.000674	0.000692	1	100	ascending tiles

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	0.009427	0.009512	1536491	100	random order
fluxsort	100000	64	0.004853	0.004862	2001035	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	32	0.008472	0.008609	1536634	100	random order
fluxsort	100000	32	0.004117	0.004136	1990342	100	random order
qsort	100000	32	0.002017	0.002194	815024	100	ascending order
fluxsort	100000	32	0.000140	0.000140	99999	100	ascending order
qsort	100000	32	0.002817	0.002855	915019	100	ascending saw
fluxsort	100000	32	0.001514	0.001525	558847	100	ascending saw
qsort	100000	32	0.006382	0.006435	1532339	100	generic order
fluxsort	100000	32	0.002333	0.002349	1269601	100	generic order
qsort	100000	32	0.002450	0.002479	853904	100	descending order
fluxsort	100000	32	0.000150	0.000150	99999	100	descending order
qsort	100000	32	0.002826	0.002916	1063907	100	descending saw
fluxsort	100000	32	0.001171	0.001200	697343	100	descending saw
qsort	100000	32	0.003705	0.003751	1012028	100	random tail
fluxsort	100000	32	0.002251	0.002261	681125	100	random tail
qsort	100000	32	0.005447	0.005497	1200835	100	random half
fluxsort	100000	32	0.003728	0.003747	1889402	100	random half
qsort	100000	32	0.003873	0.004301	1209200	100	ascending tiles
fluxsort	100000	32	0.000999	0.001005	400063	100	ascending tiles