blqsort - fast branchless quicksort

Performance results naturally depend on the underlying hardware. The following benchmarks show the execution times for sorting 50 million doubles using different sorting implementations. The measurements were taken on an Apple M1 system using Clang and on an AMD Ryzen 3 (Linux) system using GCC, both compiled with the -O3 option. test_double.cpp

Implementation	Apple M1	AMD Ryzen
std::sort	1.33s	5.56s
pdqsort	1.33s	2.81s
blqsort (single threaded)	0.93s	1.98s

For a fair comparison, the single-threaded version of blqs was used here. On an M1, the threaded versions are another factor of 3 to 4 faster. In terms of runtime, the C++ versions differ only very little from the C version.

Branchless programming

On modern CPUs, avoiding branch misprediction is a key technique to speed up programs. This is much slower:

for (int i = 0; i < 10000; i++) {
	if (numbers[i] < 500) {
		small_numbers[smlen] = numbers[i];
		smlen += 1;
	}
}

than the branchless version:

for (int i = 0; i < 10000; i++) {
	small_numbers[smlen] = numbers[i];
	smlen += (numbers[i] < 500);
}

This paper by Edelkamp and Weiß shows how partitioning performance in Quicksort can be improved by avoiding conditional branches.

The strategy of using an auxiliary buffer for branchless partitioning is inspired by fluxsort. The “auxiliary buffer” here means a 1024‑element stack array, not heap memory.

First, 1024 elements are copied from one side into an auxiliary buffer to make room for subsequent operations. Then, we alternately copy a 1024-element block to the left and right in a branchless manner. The left pointer is only incremented if the element is smaller than the pivot, otherwise, the right pointer is incremented - branchless, of course.

This involves more more than double the necessary copy operations. For data types that are cheap to copy, however, this is much less expensive than the branch mispredictions that would otherwise occur.

Pivot strategy, bad input and sorting network

To avoid the O(n²) runtime caused by bad input data, the program can group identical elements together and switch to heapsort for that specific part if it detects a big imbalance during partitioning. The program also checks if a partition is already sorted.

For larger parts, it uses a median-of-medians strategy to find a good pivot. In addition, critical partitioning loops are explicitly unrolled.

For 2 to 16 elements, the algorithm uses custom sorting networks. This approach requires a separate code path for each size but sorts small subsets with very few swaps using a branchless sort-2 primitive. [Source for sorting networks](https://bertdobbelaere.github.io/ sorting_networks.html)

C++

For types with higher copy costs that are not is_trivially_copyable (such as strings), the buffer-based branchless approach becomes less efficient. In such cases, a BlockQuicksort variant is used instead. This processes only the element indices in a branchless manner and then moves the actual data with fewer swaps. Some ideas are from pdqsort.

Usage

You only need to include blqs.h, and it can be used just as easily as std::sort.

#include "blqs.h"

double data[SIZE];

blqs::sort(data, data + SIZE);

For the C++ multithreaded variant, which uses C++ threads, include blqs_thr.h instead of blqs.h. The function call remains the same.

C

In C, the code specialized for the data type is generated using #define.

Usage

#define BLQS_CMP(a, b) ((a) < (b))
#define BLQS_TYPE double
#include "blqsort.h"

double data[SIZE];

blqsort(data, SIZE);

For the C multithreaded variant, which uses POSIX threads, include blqsort_thr.h instead of blqsort.h. The function call remains the same here as well.

Sorting Custom Data Structures

In practice, we often need to sort custom data structures. This is where SIMD libraries like Google Highway - while very fast for simple numbers - become difficult to use.

Using std::sort or blqsort gives you much more flexibility.

C++

#include "blqs.h"

struct entry {
    int id;
    int value;

    bool operator<(const entry& other) const {
        return id < other.id;
    }
};

struct entry data[SIZE];

blqs::sort(data, data + SIZE);

C

struct entry {
	int id;
	int value;
};
#define BLQS_CMP(a, b) (((a).id) < ((b).id))
#define BLQS_TYPE struct entry
#include "blqsort.h"

struct entry data[SIZE];

blqsort(data, SIZE);

Execution times for sorting 50 million of these structs.

Implementation	Apple M1	AMD Ryzen
std::sort	3.46s	4.75s
pdqsort	3.46s	4.72s
blqsort	0.90s	2.10s

Links

When 'if' slows you down, avoid it.

Interactive sorting demo