mattsta / fluxsort

(my fixes / improvements to: A branchless stable quicksort / mergesort hybrid)

building / testing

mkdir build
cd build
cmake ..
make
./src/bench

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 splits the array in 4 segments and obtains a measure of presortedness for each segment, switching to quadsort if the segment is more than 50% ordered.

While arguably not as adaptive as the bottom-up analyzer used by quadsort, a top-down analyzer works well because quicksort significantly benefits from sorting longer ranges. This approach results in more robust overall adaptivity as fluxsort cannot be tricked into performing less efficient partitions by small sorted runs. In addition, the analyzer will execute what could be considered a true top-down merge on the 4 segments.

Increasing the segments from 4 to 16 is challenging due to register pressure.

Partitioning

Partitioning is performed in a top-down manner similar to quicksort. Fluxsort obtains the pseudomedian of 9 for partitions smaller than 2024 elements, the pseudomedian of 25 if the array is smaller than 65536, and the median of 128, 256, or 512 otherwise, making the pivot selection an approximation of the cubic root of the partition size. 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 at most a 33% performance slowdown by prematurely triggering the use of quadsort. However, for more complex comparisons, like strings, quadsort matches fluxsort, in which case it wouldn't make a difference.

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.

When sorting, branchless comparisons are primarily useful to take advantage of memory-level parallelism. After writing data, the process can continue without having to wait for the write operation to have actually finished, and the process will primarily stall when a cache line is fetched. Since quicksort partitions to two memory regions, part of the loop can continue, reducing the wait time for cache line fetches. This gives an overall performance gain, even though the branchless operation is more expensive.

When the comparison becomes more expensive (like string comparisons), the size of the type is increased, the size of the partition is increased, or the comparison accesses uncached memory regions, the benefit of memory-level parallelism is reduced, and can even result in slower overall execution. While it's possible to write to four memory regions at once, the cost-benefit is dubious for a general purpose sort, and likely hardware dependant.

Quadsort, as of September 2021, uses a branchless optimization as well, and writes to two distinct memory regions by merging both ends of an array simultaneously. For sorting strings and objects quadsort's overall branchless performance is better than fluxsort's with the exception that fluxsort is faster on random data with low cardinality.

Generic data optimizations

Fluxsort uses a method that mimicks dual-pivot quicksort to improve generic data handling. If after a partition all elements were smaller or equal to the pivot, a reverse partition is performed, filtering out all elements equal to the pivot, next it carries on as usual. This typically only occurs when sorting tables with many identical values, like gender, age, etc. Fluxsort has a small bias in its pivot selection to increase the odds of this happening. In addition, generic data performance is improved slightly by checking if the same pivot is chosen twice in a row, in which case it performs a reverse partition as well. Pivot retention was first introduced by pdqsort.

┌──────────────────────────────────┬───┬──────────────┐
│             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.

Small array optimizations

For partitions smaller than 24 elements fluxsort uses quadsort's small array sorting routine. This routine uses branchless parity merges for the first 4 or 8 elements, and twice-unguarded insertion sort to sort the remainder. If the array exceeds 15 elements it is split in 4 segments and parity merged. 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.

Fluxsort comes with the fluxsort_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. 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 additional primitive as well as custom types can be added to fluxsort.h and quadsort.h.

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.

If in-place stable sorting is desired the best option is to use blitsort, which is a properly in-place alternative to fluxsort. For in-place unstable sorting crumsort is an option as well.

Performance

Fluxsort is one of the fastest stable comparison sorts written to date.

To take full advantage of branchless operations the cmp macro can be uncommented in bench.c, which will double the performance on primitive types. Fluxsort, after crumsort, is faster than a radix sort for sorting 64 bit integers. An adaptive radix sort, like wolfsort, has better performance on 8, 16, and 32 bit types.

Fluxsort needs to be compiled using gcc -O3 for optimal performance.

Porting

People wanting to port fluxsort might want to have a look at piposort, which is a simplified implementation of quadsort. Fluxsort itself is relatively simple. Earlier versions of fluxsort have a less bulky analyzer.

Variants

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

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

• 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.

• glidesort is a hybrid stable quicksort / timsort written in Rust. The timsort is enhanced with quadsort's branchless merge logic and powersort's optimization to run lengths. Partitioning is similar to fluxsort, except that it is bidirectional like a parity merge, writing to 4 instead of 2 memory regions. Similarly, the memory regions of the merge routine are increase from 2 to 4 through partitioning and conjoining quad merges. Reportedly, doing this gives a performance benefit on the most recent hardware, while decreasing performance on older hardware. Auxiliary memory use is reduced by up to n / 8 for large arrays.

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	128	0.010946	0.011070	0	100	random order
fluxsort	100000	128	0.008525	0.008616	0	100	random order
timsort	100000	128	0.012787	0.012885	0	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
stablesort	100000	64	0.006119	0.006166	0	100	random order
fluxsort	100000	64	0.001947	0.001974	0	100	random order
timsort	100000	64	0.007788	0.007839	0	100	random order

Name	Items	Type	Best	Average	Loops	Samples	Distribution
stablesort	100000	32	0.005996	0.006033	0	100	random order
fluxsort	100000	32	0.001852	0.001872	0	100	random order
timsort	100000	32	0.007584	0.007616	0	100	random order
stablesort	100000	32	0.003818	0.003845	0	100	random % 100
fluxsort	100000	32	0.000657	0.000668	0	100	random % 100
timsort	100000	32	0.005606	0.005644	0	100	random % 100
stablesort	100000	32	0.000670	0.000688	0	100	ascending order
fluxsort	100000	32	0.000044	0.000045	0	100	ascending order
timsort	100000	32	0.000045	0.000046	0	100	ascending order
stablesort	100000	32	0.001359	0.001389	0	100	ascending saw
fluxsort	100000	32	0.000328	0.000331	0	100	ascending saw
timsort	100000	32	0.000842	0.000853	0	100	ascending saw
stablesort	100000	32	0.001124	0.001152	0	100	pipe organ
fluxsort	100000	32	0.000206	0.000207	0	100	pipe organ
timsort	100000	32	0.000465	0.000472	0	100	pipe organ
stablesort	100000	32	0.000905	0.000919	0	100	descending order
fluxsort	100000	32	0.000055	0.000055	0	100	descending order
timsort	100000	32	0.000092	0.000093	0	100	descending order
stablesort	100000	32	0.001602	0.001618	0	100	descending saw
fluxsort	100000	32	0.000418	0.000421	0	100	descending saw
timsort	100000	32	0.000785	0.000797	0	100	descending saw
stablesort	100000	32	0.002030	0.002063	0	100	random tail
fluxsort	100000	32	0.000627	0.000630	0	100	random tail
timsort	100000	32	0.001992	0.002012	0	100	random tail
stablesort	100000	32	0.003491	0.003531	0	100	random half
fluxsort	100000	32	0.001080	0.001095	0	100	random half
timsort	100000	32	0.004022	0.004040	0	100	random half
stablesort	100000	32	0.000919	0.000942	0	100	ascending tiles
fluxsort	100000	32	0.000294	0.000297	0	100	ascending tiles
timsort	100000	32	0.000836	0.000878	0	100	ascending tiles
stablesort	100000	32	0.001172	0.001201	0	100	bit reversal
fluxsort	100000	32	0.001717	0.001743	0	100	bit reversal
timsort	100000	32	0.002240	0.002352	0	100	bit reversal

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 10 to 10,000,000. 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.

data table

Name	Items	Type	Best	Average	Compares	Samples	Distribution
stablesort	10	32	0.127779	0.128149	0.0	10	random 10
fluxsort	10	32	0.049679	0.049883	0.0	10	random 10
timsort	10	32	0.141079	0.144350	0.0	10	random 10
stablesort	100	32	0.242744	0.243288	0.0	10	random 100
fluxsort	100	32	0.112359	0.113083	0.0	10	random 100
timsort	100	32	0.341105	0.341710	0.0	10	random 100
stablesort	1000	32	0.362902	0.363586	0.0	10	random 1000
fluxsort	1000	32	0.137569	0.138066	0.0	10	random 1000
timsort	1000	32	0.493047	0.493622	0.0	10	random 1000
stablesort	10000	32	0.476859	0.477168	0.0	10	random 10000
fluxsort	10000	32	0.160565	0.160774	0.0	10	random 10000
timsort	10000	32	0.637224	0.641950	0.0	10	random 10000
stablesort	100000	32	0.602276	0.602838	0.0	10	random 100000
fluxsort	100000	32	0.187420	0.187915	0.0	10	random 100000
timsort	100000	32	0.762078	0.762648	0.0	10	random 100000
stablesort	1000000	32	0.731715	0.734021	0.0	10	random 1000000
fluxsort	1000000	32	0.217196	0.219207	0.0	10	random 1000000
timsort	1000000	32	0.895532	0.896547	0.0	10	random 1000000
stablesort	10000000	32	0.891028	0.895325	0	10	random 10000000
fluxsort	10000000	32	0.266910	0.269159	0	10	random 10000000
timsort	10000000	32	1.081709	1.089263	0	10	random 10000000

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 qsort, fluxsort and quadsort 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	0.017033	0.017227	1536381	100	random string
fluxsort	100000	64	0.009928	0.010212	1782460	100	random string
quadsort	100000	64	0.010777	0.010909	1684673	100	random string
qsort	100000	64	0.015398	0.015551	1536491	100	random double
fluxsort	100000	64	0.007646	0.007811	1781640	100	random double
quadsort	100000	64	0.008702	0.008818	1684633	100	random double
qsort	100000	64	0.011014	0.011124	1536491	100	random long
fluxsort	100000	64	0.004859	0.004965	1781640	100	random long
quadsort	100000	64	0.006043	0.006120	1684633	100	random long
qsort	100000	64	0.010834	0.010965	1536634	100	random int
fluxsort	100000	64	0.004596	0.004662	1790032	100	random int
quadsort	100000	64	0.005365	0.005422	1684734	100	random int

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	128	0.018219	0.019014	1536491	100	random order
fluxsort	100000	128	0.011178	0.011290	1781640	100	random order
quadsort	100000	128	0.011141	0.011258	1684633	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	64	0.009328	0.009505	1536491	100	random order
fluxsort	100000	64	0.004025	0.004095	1781640	100	random order
quadsort	100000	64	0.004103	0.004143	1684633	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
qsort	100000	32	0.008932	0.009094	1536634	100	random order
fluxsort	100000	32	0.003403	0.003453	1790032	100	random order
quadsort	100000	32	0.003373	0.003419	1684734	100	random order
qsort	100000	32	0.006692	0.006831	1532465	100	random % 100
fluxsort	100000	32	0.001322	0.001352	897246	100	random % 100
quadsort	100000	32	0.002723	0.002816	1415417	100	random % 100
qsort	100000	32	0.002233	0.002371	815024	100	ascending order
fluxsort	100000	32	0.000174	0.000175	99999	100	ascending order
quadsort	100000	32	0.000159	0.000161	99999	100	ascending order
qsort	100000	32	0.003067	0.003177	915020	100	ascending saw
fluxsort	100000	32	0.000545	0.000549	300011	100	ascending saw
quadsort	100000	32	0.000897	0.000915	379624	100	ascending saw
qsort	100000	32	0.002480	0.002523	884462	100	pipe organ
fluxsort	100000	32	0.000381	0.000382	200006	100	pipe organ
quadsort	100000	32	0.000457	0.000462	277113	100	pipe organ
qsort	100000	32	0.002459	0.002535	853904	100	descending order
fluxsort	100000	32	0.000186	0.000187	99999	100	descending order
quadsort	100000	32	0.000164	0.000166	99999	100	descending order
qsort	100000	32	0.003289	0.003361	953892	100	descending saw
fluxsort	100000	32	0.000555	0.000561	300013	100	descending saw
quadsort	100000	32	0.000922	0.000930	391547	100	descending saw
qsort	100000	32	0.003943	0.004005	1012073	100	random tail
fluxsort	100000	32	0.001181	0.001207	623604	100	random tail
quadsort	100000	32	0.001203	0.001221	592061	100	random tail
qsort	100000	32	0.005791	0.005986	1200713	100	random half
fluxsort	100000	32	0.001974	0.002004	1028725	100	random half
quadsort	100000	32	0.002055	0.002092	1006728	100	random half
qsort	100000	32	0.004085	0.004223	1209200	100	ascending tiles
fluxsort	100000	32	0.001523	0.001557	528889	100	ascending tiles
quadsort	100000	32	0.002073	0.002130	671244	100	ascending tiles
qsort	100000	32	0.005143	0.005429	1553378	100	bit reversal
fluxsort	100000	32	0.003263	0.003385	1798806	100	bit reversal
quadsort	100000	32	0.003179	0.003218	1727134	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 pdqsort, fluxsort and crumsort are inlined.

data table

Name	Items	Type	Best	Average	Compares	Samples	Distribution
pdqsort	100000	128	0.005854	0.005954	0	100	random order
fluxsort	100000	128	0.008555	0.008662	0	100	random order
crumsort	100000	128	0.008253	0.008312	0	100	random order

Name	Items	Type	Best	Average	Compares	Samples	Distribution
pdqsort	100000	64	0.002660	0.002683	0	100	random order
fluxsort	100000	64	0.001950	0.001973	0	100	random order
crumsort	100000	64	0.001850	0.001864	0	100	random order

Name	Items	Type	Best	Average	Loops	Samples	Distribution
pdqsort	100000	32	0.002687	0.002711	0	100	random order
fluxsort	100000	32	0.001826	0.001857	0	100	random order
crumsort	100000	32	0.001799	0.001815	0	100	random order
pdqsort	100000	32	0.000785	0.000792	0	100	random % 100
fluxsort	100000	32	0.000656	0.000669	0	100	random % 100
crumsort	100000	32	0.000560	0.000565	0	100	random % 100
pdqsort	100000	32	0.000091	0.000091	0	100	ascending order
fluxsort	100000	32	0.000044	0.000044	0	100	ascending order
crumsort	100000	32	0.000044	0.000044	0	100	ascending order
pdqsort	100000	32	0.003465	0.003482	0	100	ascending saw
fluxsort	100000	32	0.000329	0.000337	0	100	ascending saw
crumsort	100000	32	0.000628	0.000636	0	100	ascending saw
pdqsort	100000	32	0.002840	0.002862	0	100	pipe organ
fluxsort	100000	32	0.000215	0.000218	0	100	pipe organ
crumsort	100000	32	0.000359	0.000364	0	100	pipe organ
pdqsort	100000	32	0.000195	0.000200	0	100	descending order
fluxsort	100000	32	0.000055	0.000056	0	100	descending order
crumsort	100000	32	0.000056	0.000056	0	100	descending order
pdqsort	100000	32	0.003236	0.003275	0	100	descending saw
fluxsort	100000	32	0.000329	0.000331	0	100	descending saw
crumsort	100000	32	0.000637	0.000648	0	100	descending saw
pdqsort	100000	32	0.002566	0.002587	0	100	random tail
fluxsort	100000	32	0.000624	0.000629	0	100	random tail
crumsort	100000	32	0.000879	0.000888	0	100	random tail
pdqsort	100000	32	0.002670	0.002697	0	100	random half
fluxsort	100000	32	0.001064	0.001081	0	100	random half
crumsort	100000	32	0.001199	0.001216	0	100	random half
pdqsort	100000	32	0.002316	0.002425	0	100	ascending tiles
fluxsort	100000	32	0.000293	0.000299	0	100	ascending tiles
crumsort	100000	32	0.001518	0.001545	0	100	ascending tiles
pdqsort	100000	32	0.002670	0.002694	0	100	bit reversal
fluxsort	100000	32	0.001692	0.001737	0	100	bit reversal
crumsort	100000	32	0.001785	0.001805	0	100	bit reversal