# c - fastest - sorting algorithms

## Fastest sort of fixed length 6 int array (16)

An XOR swap may be useful in your swapping functions.

```
void xorSwap (int *x, int *y) {
if (*x != *y) {
*x ^= *y;
*y ^= *x;
*x ^= *y;
}
}
```

The if may cause too much divergence in your code, but if you have a guarantee that all your ints are unique this could be handy.

Answering to another Stack Overflow question (this one) I stumbled upon an interesting sub-problem. What is the fastest way to sort an array of 6 ints?

As the question is very low level:

- we can't assume libraries are available (and the call itself has its cost), only plain C
- to avoid emptying instruction pipeline (that has a
*very*high cost) we should probably minimize branches, jumps, and every other kind of control flow breaking (like those hidden behind sequence points in`&&`

or`||`

). - room is constrained and minimizing registers and memory use is an issue, ideally in place sort is probably best.

Really this question is a kind of Golf where the goal is not to minimize source length but execution time. I call it 'Zening' code as used in the title of the book Zen of Code optimization by Michael Abrash and its sequels.

As for why it is interesting, there is several layers:

- the example is simple and easy to understand and measure, not much C skill involved
- it shows effects of choice of a good algorithm for the problem, but also effects of the compiler and underlying hardware.

Here is my reference (naive, not optimized) implementation and my test set.

```
#include <stdio.h>
static __inline__ int sort6(int * d){
char j, i, imin;
int tmp;
for (j = 0 ; j < 5 ; j++){
imin = j;
for (i = j + 1; i < 6 ; i++){
if (d[i] < d[imin]){
imin = i;
}
}
tmp = d[j];
d[j] = d[imin];
d[imin] = tmp;
}
}
static __inline__ unsigned long long rdtsc(void)
{
unsigned long long int x;
__asm__ volatile (".byte 0x0f, 0x31" : "=A" (x));
return x;
}
int main(int argc, char ** argv){
int i;
int d[6][5] = {
{1, 2, 3, 4, 5, 6},
{6, 5, 4, 3, 2, 1},
{100, 2, 300, 4, 500, 6},
{100, 2, 3, 4, 500, 6},
{1, 200, 3, 4, 5, 600},
{1, 1, 2, 1, 2, 1}
};
unsigned long long cycles = rdtsc();
for (i = 0; i < 6 ; i++){
sort6(d[i]);
/*
* printf("d%d : %d %d %d %d %d %d\n", i,
* d[i][0], d[i][6], d[i][7],
* d[i][8], d[i][9], d[i][10]);
*/
}
cycles = rdtsc() - cycles;
printf("Time is %d\n", (unsigned)cycles);
}
```

**Raw results**

As number of variants is becoming large, I gathered them all in a test suite that can be found here. The actual tests used are a bit less naive than those showed above, thanks to Kevin Stock. You can compile and execute it in your own environment. I'm quite interested by behavior on different target architecture/compilers. (OK guys, put it in answers, I will +1 every contributor of a new resultset).

I gave the answer to Daniel Stutzbach (for golfing) one year ago as he was at the source of the fastest solution at that time (sorting networks).

**Linux 64 bits, gcc 4.6.1 64 bits, Intel Core 2 Duo E8400, -O2**

- Direct call to qsort library function : 689.38
- Naive implementation (insertion sort) : 285.70
- Insertion Sort (Daniel Stutzbach) : 142.12
- Insertion Sort Unrolled : 125.47
- Rank Order : 102.26
- Rank Order with registers : 58.03
- Sorting Networks (Daniel Stutzbach) : 111.68
- Sorting Networks (Paul R) : 66.36
- Sorting Networks 12 with Fast Swap : 58.86
- Sorting Networks 12 reordered Swap : 53.74
- Sorting Networks 12 reordered Simple Swap : 31.54
- Reordered Sorting Network w/ fast swap : 31.54
- Reordered Sorting Network w/ fast swap V2 : 33.63
- Inlined Bubble Sort (Paolo Bonzini) : 48.85
- Unrolled Insertion Sort (Paolo Bonzini) : 75.30

**Linux 64 bits, gcc 4.6.1 64 bits, Intel Core 2 Duo E8400, -O1**

- Direct call to qsort library function : 705.93
- Naive implementation (insertion sort) : 135.60
- Insertion Sort (Daniel Stutzbach) : 142.11
- Insertion Sort Unrolled : 126.75
- Rank Order : 46.42
- Rank Order with registers : 43.58
- Sorting Networks (Daniel Stutzbach) : 115.57
- Sorting Networks (Paul R) : 64.44
- Sorting Networks 12 with Fast Swap : 61.98
- Sorting Networks 12 reordered Swap : 54.67
- Sorting Networks 12 reordered Simple Swap : 31.54
- Reordered Sorting Network w/ fast swap : 31.24
- Reordered Sorting Network w/ fast swap V2 : 33.07
- Inlined Bubble Sort (Paolo Bonzini) : 45.79
- Unrolled Insertion Sort (Paolo Bonzini) : 80.15

I included both -O1 and -O2 results because surprisingly for several programs O2 is **less** efficient than O1. I wonder what specific optimization has this effect ?

**Comments on proposed solutions**

**Insertion Sort (Daniel Stutzbach)**

As expected minimizing branches is indeed a good idea.

**Sorting Networks (Daniel Stutzbach)**

Better than insertion sort. I wondered if the main effect was not get from avoiding the external loop. I gave it a try by unrolled insertion sort to check and indeed we get roughly the same figures (code is here).

**Sorting Networks (Paul R)**

The best so far. The actual code I used to test is here. Don't know yet why it is nearly two times as fast as the other sorting network implementation. Parameter passing ? Fast max ?

**Sorting Networks 12 SWAP with Fast Swap**

As suggested by Daniel Stutzbach, I combined his 12 swap sorting network with branchless fast swap (code is here). It is indeed faster, the best so far with a small margin (roughly 5%) as could be expected using 1 less swap.

It is also interesting to notice that the branchless swap seems to be much (4 times) less efficient than the simple one using if on PPC architecture.

**Calling Library qsort**

To give another reference point I also tried as suggested to just call library qsort (code is here). As expected it is much slower : 10 to 30 times slower... as it became obvious with the new test suite, the main problem seems to be the initial load of the library after the first call, and it compares not so poorly with other version. It is just between 3 and 20 times slower on my Linux. On some architecture used for tests by others it seems even to be faster (I'm really surprised by that one, as library qsort use a more complex API).

**Rank order**

Rex Kerr proposed another completely different method : for each item of the array compute directly its final position. This is efficient because computing rank order do not need branch. The drawback of this method is that it takes three times the amount of memory of the array (one copy of array and variables to store rank orders). The performance results are very surprising (and interesting). On my reference architecture with 32 bits OS and Intel Core2 Quad E8300, cycle count was slightly below 1000 (like sorting networks with branching swap). But when compiled and executed on my 64 bits box (Intel Core2 Duo) it performed much better : it became the fastest so far. I finally found out the true reason. My 32bits box use gcc 4.4.1 and my 64bits box gcc 4.4.3 and the last one seems much better at optimising this particular code (there was very little difference for other proposals).

*update*:

As published figures above shows this effect was still enhanced by later versions of gcc and Rank Order became consistently twice as fast as any other alternative.

**Sorting Networks 12 with reordered Swap**

The amazing efficiency of the Rex Kerr proposal with gcc 4.4.3 made me wonder : how could a program with 3 times as much memory usage be faster than branchless sorting networks? My hypothesis was that it had less dependencies of the kind read after write, allowing for better use of the superscalar instruction scheduler of the x86. That gave me an idea: reorder swaps to minimize read after write dependencies. More simply put: when you do `SWAP(1, 2); SWAP(0, 2);`

you have to wait for the first swap to be finished before performing the second one because both access to a common memory cell. When you do `SWAP(1, 2); SWAP(4, 5);`

the processor can execute both in parallel. I tried it and it works as expected, the sorting networks is running about 10% faster.

**Sorting Networks 12 with Simple Swap**

One year after the original post Steinar H. Gunderson suggested, that we should not try to outsmart the compiler and keep the swap code simple. It's indeed a good idea as the resulting code is about 40% faster! He also proposed a swap optimized by hand using x86 inline assembly code that can still spare some more cycles. The most surprising (it says volumes on programmer's psychology) is that one year ago none of used tried that version of swap. Code I used to test is here. Others suggested other ways to write a C fast swap, but it yields the same performances as the simple one with a decent compiler.

The "best" code is now as follow:

```
static inline void sort6_sorting_network_simple_swap(int * d){
#define min(x, y) (x<y?x:y)
#define max(x, y) (x<y?y:x)
#define SWAP(x,y) { const int a = min(d[x], d[y]); \
const int b = max(d[x], d[y]); \
d[x] = a; d[y] = b; }
SWAP(1, 2);
SWAP(4, 5);
SWAP(0, 2);
SWAP(3, 5);
SWAP(0, 1);
SWAP(3, 4);
SWAP(1, 4);
SWAP(0, 3);
SWAP(2, 5);
SWAP(1, 3);
SWAP(2, 4);
SWAP(2, 3);
#undef SWAP
#undef min
#undef max
}
```

If we believe our test set (and, yes it is quite poor, it's mere benefit is being short, simple and easy to understand what we are measuring), the average number of cycles of the resulting code for one sort is below 40 cycles (6 tests are executed). That put each swap at an average of 4 cycles. I call that amazingly fast. Any other improvements possible ?

For any optimization, it's always best to test, test, test. I would try at least sorting networks and insertion sort. If I were betting, I'd put my money on insertion sort based on past experience.

Do you know anything about the input data? Some algorithms will perform better with certain kinds of data. For example, insertion sort performs better on sorted or almost-sorted dat, so it will be the better choice if there's an above-average chance of almost-sorted data.

The algorithm you posted is similar to an insertion sort, but it looks like you've minimized the number of swaps at the cost of more comparisons. Comparisons are far more expensive than swaps, though, because branches can cause the instruction pipeline to stall.

Here's an insertion sort implementation:

```
static __inline__ int sort6(int *d){
int i, j;
for (i = 1; i < 6; i++) {
int tmp = d[i];
for (j = i; j >= 1 && tmp < d[j-1]; j--)
d[j] = d[j-1];
d[j] = tmp;
}
}
```

Here's how I'd build a sorting network. First, use this site to generate a minimal set of SWAP macros for a network of the appropriate length. Wrapping that up in a function gives me:

```
static __inline__ int sort6(int * d){
#define SWAP(x,y) if (d[y] < d[x]) { int tmp = d[x]; d[x] = d[y]; d[y] = tmp; }
SWAP(1, 2);
SWAP(0, 2);
SWAP(0, 1);
SWAP(4, 5);
SWAP(3, 5);
SWAP(3, 4);
SWAP(0, 3);
SWAP(1, 4);
SWAP(2, 5);
SWAP(2, 4);
SWAP(1, 3);
SWAP(2, 3);
#undef SWAP
}
```

Here is my contribution to this thread: an optimized 1, 4 gap shellsort for a 6-member int vector (valp) containing unique values.

```
void shellsort (int *valp)
{
int c,a,*cp,*ip=valp,*ep=valp+5;
c=*valp; a=*(valp+4);if (c>a) {*valp= a;*(valp+4)=c;}
c=*(valp+1);a=*(valp+5);if (c>a) {*(valp+1)=a;*(valp+5)=c;}
cp=ip;
do
{
c=*cp;
a=*(cp+1);
do
{
if (c<a) break;
*cp=a;
*(cp+1)=c;
cp-=1;
c=*cp;
} while (cp>=valp);
ip+=1;
cp=ip;
} while (ip<ep);
}
```

On my HP dv7-3010so laptop with a dual-core Athlon M300 @ 2 Ghz (DDR2 memory) it executes in 165 clock cycles. This is an average calculated from timing every unique sequence (6!/720 in all). Compiled to Win32 using OpenWatcom 1.8. The loop is essentially an insertion sort and is 16 instructions/37 bytes long.

I do not have a 64-bit environment to compile on.

Here's an implementation using sorting networks:

```
inline void Sort2(int *p0, int *p1)
{
const int temp = min(*p0, *p1);
*p1 = max(*p0, *p1);
*p0 = temp;
}
inline void Sort3(int *p0, int *p1, int *p2)
{
Sort2(p0, p1);
Sort2(p1, p2);
Sort2(p0, p1);
}
inline void Sort4(int *p0, int *p1, int *p2, int *p3)
{
Sort2(p0, p1);
Sort2(p2, p3);
Sort2(p0, p2);
Sort2(p1, p3);
Sort2(p1, p2);
}
inline void Sort6(int *p0, int *p1, int *p2, int *p3, int *p4, int *p5)
{
Sort3(p0, p1, p2);
Sort3(p3, p4, p5);
Sort2(p0, p3);
Sort2(p2, p5);
Sort4(p1, p2, p3, p4);
}
```

You really need very efficient branchless `min`

and `max`

implementations for this, since that is effectively what this code boils down to - a sequence of `min`

and `max`

operations (13 of each, in total). I leave this as an exercise for the reader.

Note that this implementation lends itself easily to vectorization (e.g. SIMD - most SIMD ISAs have vector min/max instructions) and also to GPU implementations (e.g. CUDA - being branchless there are no problems with warp divergence etc).

See also: Fast algorithm implementation to sort very small list

I found that at least on my system, the functions `sort6_iterator()`

and `sort6_iterator_local()`

defined below both ran at least as fast, and frequently noticeably faster, than the above current record holder:

```
#define MIN(x, y) (x<y?x:y)
#define MAX(x, y) (x<y?y:x)
template<class IterType>
inline void sort6_iterator(IterType it)
{
#define SWAP(x,y) { const auto a = MIN(*(it + x), *(it + y)); \
const auto b = MAX(*(it + x), *(it + y)); \
*(it + x) = a; *(it + y) = b; }
SWAP(1, 2) SWAP(4, 5)
SWAP(0, 2) SWAP(3, 5)
SWAP(0, 1) SWAP(3, 4)
SWAP(1, 4) SWAP(0, 3)
SWAP(2, 5) SWAP(1, 3)
SWAP(2, 4)
SWAP(2, 3)
#undef SWAP
}
```

I passed this function a `std::vector`

's iterator in my timing code. I suspect that using iterators gives g++ certain assurances about what can and can't happen to the memory that the iterator refers to, which it otherwise wouldn't have and it is these assurances that allow g++ to better optimize the sorting code (which if I remember correctly, is also *part* of the reason why so many `std`

algorithms, such as `std::sort()`

, generally have such obscenely good performance). However, while timing I noticed that the context (i.e. surrounding code) in which the call to the sorting function was made had a significant impact on performance, which is likely due to the function being inlined and then optimized. For instance, if the program was sufficiently simple then there usually wasn't much of a difference in performance between passing the sorting function a pointer versus passing it an iterator; otherwise using iterators usually resulted in noticeably better performance and never (in my experience so far at least) any noticeably worse performance. I suspect that this may be because g++ can globally optimize sufficiently simple code.

Moreover, `sort6_iterator()`

is *some*times (again, depending on the context in which the function is called) consistently outperformed by the following sorting function:

```
template<class IterType>
inline void sort6_iterator_local(IterType it)
{
#define SWAP(x,y) { const auto a = MIN(data##x, data##y); \
const auto b = MAX(data##x, data##y); \
data##x = a; data##y = b; }
//DD = Define Data
#define DD1(a) auto data##a = *(it + a);
#define DD2(a,b) auto data##a = *(it + a), data##b = *(it + b);
//CB = Copy Back
#define CB(a) *(it + a) = data##a;
DD2(1,2) SWAP(1, 2)
DD2(4,5) SWAP(4, 5)
DD1(0) SWAP(0, 2)
DD1(3) SWAP(3, 5)
SWAP(0, 1) SWAP(3, 4)
SWAP(1, 4) SWAP(0, 3) CB(0)
SWAP(2, 5) CB(5)
SWAP(1, 3) CB(1)
SWAP(2, 4) CB(4)
SWAP(2, 3) CB(2) CB(3)
#undef CB
#undef DD2
#undef DD1
#undef SWAP
}
```

Note that defining `SWAP()`

as follows *some*times results in slightly better performance although most of the time it results in slightly worse performance or a negligible difference in performance.

```
#define SWAP(x,y) { const auto a = MIN(data##x, data##y); \
data##y = MAX(data##x, data##y); \
data##x = a; }
```

If you just want a sorting algorithm that gcc -O3 is consistently good at optimizing then, depending on how you pass the input, try one of the following two algorithms:

```
template<class T> inline void sort6(T it) {
#define SORT2(x,y) {if(data##x>data##y){auto a=std::move(data##y);data##y=std::move(data##x);data##x=std::move(a);}}
#define DD1(a) register auto data##a=*(it+a);
#define DD2(a,b) register auto data##a=*(it+a);register auto data##b=*(it+b);
#define CB1(a) *(it+a)=data##a;
#define CB2(a,b) *(it+a)=data##a;*(it+b)=data##b;
DD2(1,2) SORT2(1,2)
DD2(4,5) SORT2(4,5)
DD1(0) SORT2(0,2)
DD1(3) SORT2(3,5)
SORT2(0,1) SORT2(3,4) SORT2(2,5) CB1(5)
SORT2(1,4) SORT2(0,3) CB1(0)
SORT2(2,4) CB1(4)
SORT2(1,3) CB1(1)
SORT2(2,3) CB2(2,3)
#undef CB1
#undef CB2
#undef DD1
#undef DD2
#undef SORT2
}
```

Or

```
template<class T> inline void sort6(T& e0, T& e1, T& e2, T& e3, T& e4, T& e5) {
#define SORT2(x,y) {if(data##x>data##y)std::swap(data##x,data##y);}
#define DD1(a) register auto data##a=e##a;
#define DD2(a,b) register auto data##a=e##a;register auto data##b=e##b;
#define CB1(a) e##a=data##a;
#define CB2(a,b) e##a=data##a;e##b=data##b;
DD2(1,2) SORT2(1,2)
DD2(4,5) SORT2(4,5)
DD1(0) SORT2(0,2)
DD1(3) SORT2(3,5)
SORT2(0,1) SORT2(3,4) SORT2(2,5) CB1(5)
SORT2(1,4) SORT2(0,3) CB1(0)
SORT2(2,4) CB1(4)
SORT2(1,3) CB1(1)
SORT2(2,3) CB2(2,3)
#undef CB1
#undef CB2
#undef DD1
#undef DD2
#undef SORT2
}
```

The reason for using the `register`

keyword is because this is one of the few times that you know that you want these values in registers. Without `register`

, the compiler will figure this out most of the time but sometimes it doesn't. Using the `register`

keyword helps solve this issue. Normally, however, don't use the `register`

keyword since it's more likely to slow your code than speed it up.

Also, note the use of templates. This is done on purpose since, even with the `inline`

keyword, template functions are generally much more aggressively optimized by gcc than vanilla C functions (this has to do with gcc needing to deal with function pointers for vanilla C functions but not with template functions).

I know I'm super-late, but I was interested in experimenting with some different solutions. First, I cleaned up that paste, made it compile, and put it into a repository. I kept some undesirable solutions as dead-ends so that others wouldn't try it. Among this was my first solution, which attempted to ensure that x1>x2 was calculated once. After optimization, it is no faster than the other, simple versions.

I added a looping version of rank order sort, since my own application of this study is for sorting 2-8 items, so since there are a variable number of arguments, a loop is necessary. This is also why I ignored the sorting network solutions.

The test code didn't test that duplicates were handled correctly, so while the existing solutions were all correct, I added a special case to the test code to ensure that duplicates were handled correctly.

Then, I wrote an insertion sort that is entirely in AVX registers. On my machine it is 25% faster than the other insertion sorts, but 100% slower than rank order. I did this purely for experiment and had no expectation of this being better due to the branching in insertion sort.

```
static inline void sort6_insertion_sort_avx(int* d) {
__m256i src = _mm256_setr_epi32(d[0], d[1], d[2], d[3], d[4], d[5], 0, 0);
__m256i index = _mm256_setr_epi32(0, 1, 2, 3, 4, 5, 6, 7);
__m256i shlpermute = _mm256_setr_epi32(7, 0, 1, 2, 3, 4, 5, 6);
__m256i sorted = _mm256_setr_epi32(d[0], INT_MAX, INT_MAX, INT_MAX,
INT_MAX, INT_MAX, INT_MAX, INT_MAX);
__m256i val, gt, permute;
unsigned j;
// 8 / 32 = 2^-2
#define ITER(I) \
val = _mm256_permutevar8x32_epi32(src, _mm256_set1_epi32(I));\
gt = _mm256_cmpgt_epi32(sorted, val);\
permute = _mm256_blendv_epi8(index, shlpermute, gt);\
j = ffs( _mm256_movemask_epi8(gt)) >> 2;\
sorted = _mm256_blendv_epi8(_mm256_permutevar8x32_epi32(sorted, permute),\
val, _mm256_cmpeq_epi32(index, _mm256_set1_epi32(j)))
ITER(1);
ITER(2);
ITER(3);
ITER(4);
ITER(5);
int x[8];
_mm256_storeu_si256((__m256i*)x, sorted);
d[0] = x[0]; d[1] = x[1]; d[2] = x[2]; d[3] = x[3]; d[4] = x[4]; d[5] = x[5];
#undef ITER
}
```

Then, I wrote a rank order sort using AVX. This matches the speed of the other rank-order solutions, but is no faster. The issue here is that I can only calculate the indices with AVX, and then I have to make a table of indices. This is because the calculation is destination-based rather than source-based. See Converting from Source-based Indices to Destination-based Indices

```
static inline void sort6_rank_order_avx(int* d) {
__m256i ror = _mm256_setr_epi32(5, 0, 1, 2, 3, 4, 6, 7);
__m256i one = _mm256_set1_epi32(1);
__m256i src = _mm256_setr_epi32(d[0], d[1], d[2], d[3], d[4], d[5], INT_MAX, INT_MAX);
__m256i rot = src;
__m256i index = _mm256_setzero_si256();
__m256i gt, permute;
__m256i shl = _mm256_setr_epi32(1, 2, 3, 4, 5, 6, 6, 6);
__m256i dstIx = _mm256_setr_epi32(0,1,2,3,4,5,6,7);
__m256i srcIx = dstIx;
__m256i eq = one;
__m256i rotIx = _mm256_setzero_si256();
#define INC(I)\
rot = _mm256_permutevar8x32_epi32(rot, ror);\
gt = _mm256_cmpgt_epi32(src, rot);\
index = _mm256_add_epi32(index, _mm256_and_si256(gt, one));\
index = _mm256_add_epi32(index, _mm256_and_si256(eq,\
_mm256_cmpeq_epi32(src, rot)));\
eq = _mm256_insert_epi32(eq, 0, I)
INC(0);
INC(1);
INC(2);
INC(3);
INC(4);
int e[6];
e[0] = d[0]; e[1] = d[1]; e[2] = d[2]; e[3] = d[3]; e[4] = d[4]; e[5] = d[5];
int i[8];
_mm256_storeu_si256((__m256i*)i, index);
d[i[0]] = e[0]; d[i[1]] = e[1]; d[i[2]] = e[2]; d[i[3]] = e[3]; d[i[4]] = e[4]; d[i[5]] = e[5];
}
```

The repo can be found here: https://github.com/eyepatchParrot/sort6/

I ported the test suite to a PPC architecture machine I can not identify (didn't have to touch code, just increase the iterations of the test, use 8 test cases to avoid polluting results with mods and replace the x86 specific rdtsc):

**Direct call to qsort library function** : 101

**Naive implementation (insertion sort)** : 299

**Insertion Sort (Daniel Stutzbach)** : 108

**Insertion Sort Unrolled** : 51

**Sorting Networks (Daniel Stutzbach)** : 26

**Sorting Networks (Paul R)** : 85

**Sorting Networks 12 with Fast Swap** : 117

**Sorting Networks 12 reordered Swap** : 116

**Rank Order** : 56

I stumbled onto this question from Google a few days ago because I also had a need to quickly sort a fixed length array of 6 integers. In my case however, my integers are only 8 bits (instead of 32) and I do not have a strict requirement of only using C. I thought I would share my findings anyways, in case they might be helpful to someone...

I implemented a variant of a network sort in assembly that uses SSE to vectorize the compare and swap operations, to the extent possible. It takes six "passes" to completely sort the array. I used a novel mechanism to directly convert the results of PCMPGTB (vectorized compare) to shuffle parameters for PSHUFB (vectorized swap), using only a PADDB (vectorized add) and in some cases also a PAND (bitwise AND) instruction.

This approach also had the side effect of yielding a *truly* branchless function. There are no jump instructions whatsoever.

It appears that this implementation **is about 38% faster** than the implementation which is currently marked as the fastest option in the question ("Sorting Networks 12 with Simple Swap"). I modified that implementation to use `char`

array elements during my testing, to make the comparison fair.

I should note that this approach can be applied to any array size up to 16 elements. I expect the relative speed advantage over the alternatives to grow larger for the bigger arrays.

The code is written in MASM for x86_64 processors with SSSE3. The function uses the "new" Windows x64 calling convention. Here it is...

```
PUBLIC simd_sort_6
.DATA
ALIGN 16
pass1_shuffle OWORD 0F0E0D0C0B0A09080706040503010200h
pass1_add OWORD 0F0E0D0C0B0A09080706050503020200h
pass2_shuffle OWORD 0F0E0D0C0B0A09080706030405000102h
pass2_and OWORD 00000000000000000000FE00FEFE00FEh
pass2_add OWORD 0F0E0D0C0B0A09080706050405020102h
pass3_shuffle OWORD 0F0E0D0C0B0A09080706020304050001h
pass3_and OWORD 00000000000000000000FDFFFFFDFFFFh
pass3_add OWORD 0F0E0D0C0B0A09080706050404050101h
pass4_shuffle OWORD 0F0E0D0C0B0A09080706050100020403h
pass4_and OWORD 0000000000000000000000FDFD00FDFDh
pass4_add OWORD 0F0E0D0C0B0A09080706050403020403h
pass5_shuffle OWORD 0F0E0D0C0B0A09080706050201040300h
pass5_and OWORD 0000000000000000000000FEFEFEFE00h
pass5_add OWORD 0F0E0D0C0B0A09080706050403040300h
pass6_shuffle OWORD 0F0E0D0C0B0A09080706050402030100h
pass6_add OWORD 0F0E0D0C0B0A09080706050403030100h
.CODE
simd_sort_6 PROC FRAME
.endprolog
; pxor xmm4, xmm4
; pinsrd xmm4, dword ptr [rcx], 0
; pinsrb xmm4, byte ptr [rcx + 4], 4
; pinsrb xmm4, byte ptr [rcx + 5], 5
; The benchmarked 38% faster mentioned in the text was with the above slower sequence that tied up the shuffle port longer. Same on extract
; avoiding pins/extrb also means we don't need SSE 4.1, but SSSE3 CPUs without SSE4.1 (e.g. Conroe/Merom) have slow pshufb.
movd xmm4, dword ptr [rcx]
pinsrw xmm4, word ptr [rcx + 4], 2 ; word 2 = bytes 4 and 5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass1_shuffle]
pcmpgtb xmm5, xmm4
paddb xmm5, oword ptr [pass1_add]
pshufb xmm4, xmm5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass2_shuffle]
pcmpgtb xmm5, xmm4
pand xmm5, oword ptr [pass2_and]
paddb xmm5, oword ptr [pass2_add]
pshufb xmm4, xmm5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass3_shuffle]
pcmpgtb xmm5, xmm4
pand xmm5, oword ptr [pass3_and]
paddb xmm5, oword ptr [pass3_add]
pshufb xmm4, xmm5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass4_shuffle]
pcmpgtb xmm5, xmm4
pand xmm5, oword ptr [pass4_and]
paddb xmm5, oword ptr [pass4_add]
pshufb xmm4, xmm5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass5_shuffle]
pcmpgtb xmm5, xmm4
pand xmm5, oword ptr [pass5_and]
paddb xmm5, oword ptr [pass5_add]
pshufb xmm4, xmm5
movdqa xmm5, xmm4
pshufb xmm5, oword ptr [pass6_shuffle]
pcmpgtb xmm5, xmm4
paddb xmm5, oword ptr [pass6_add]
pshufb xmm4, xmm5
;pextrd dword ptr [rcx], xmm4, 0 ; benchmarked with this
;pextrb byte ptr [rcx + 4], xmm4, 4 ; slower version
;pextrb byte ptr [rcx + 5], xmm4, 5
movd dword ptr [rcx], xmm4
pextrw word ptr [rcx + 4], xmm4, 2 ; x86 is little-endian, so this is the right order
ret
simd_sort_6 ENDP
END
```

You can compile this to an executable object and link it into your C project. For instructions on how to do this in Visual Studio, you can read this article. You can use the following C prototype to call the function from your C code:

```
void simd_sort_6(char *values);
```

Looking forward to trying my hand at this and learning from these examples, but first some timings from my 1.5 GHz PPC Powerbook G4 w/ 1 GB DDR RAM. (I borrowed a similar rdtsc-like timer for PPC from http://www.mcs.anl.gov/~kazutomo/rdtsc.html for the timings.) I ran the program a few times and the absolute results varied but the consistently fastest test was "Insertion Sort (Daniel Stutzbach)", with "Insertion Sort Unrolled" a close second.

Here's the last set of times:

```
**Direct call to qsort library function** : 164
**Naive implementation (insertion sort)** : 138
**Insertion Sort (Daniel Stutzbach)** : 85
**Insertion Sort Unrolled** : 97
**Sorting Networks (Daniel Stutzbach)** : 457
**Sorting Networks (Paul R)** : 179
**Sorting Networks 12 with Fast Swap** : 238
**Sorting Networks 12 reordered Swap** : 236
**Rank Order** : 116
```

Looks like I got to the party a year late, but here we go...

Looking at the assembly generated by gcc 4.5.2 I observed that loads and stores are being done for every swap, which really isn't needed. It would be better to load the 6 values into registers, sort those, and store them back into memory. I ordered the loads at stores to be as close as possible to there the registers are first needed and last used. I also used Steinar H. Gunderson's SWAP macro. Update: I switched to Paolo Bonzini's SWAP macro which gcc converts into something similar to Gunderson's, but gcc is able to better order the instructions since they aren't given as explicit assembly.

I used the same swap order as the reordered swap network given as the best performing, although there may be a better ordering. If I find some more time I'll generate and test a bunch of permutations.

I changed the testing code to consider over 4000 arrays and show the average number of cycles needed to sort each one. On an i5-650 I'm getting ~34.1 cycles/sort (using -O3), compared to the original reordered sorting network getting ~65.3 cycles/sort (using -O1, beats -O2 and -O3).

```
#include <stdio.h>
static inline void sort6_fast(int * d) {
#define SWAP(x,y) { int dx = x, dy = y, tmp; tmp = x = dx < dy ? dx : dy; y ^= dx ^ tmp; }
register int x0,x1,x2,x3,x4,x5;
x1 = d[1];
x2 = d[2];
SWAP(x1, x2);
x4 = d[4];
x5 = d[5];
SWAP(x4, x5);
x0 = d[0];
SWAP(x0, x2);
x3 = d[3];
SWAP(x3, x5);
SWAP(x0, x1);
SWAP(x3, x4);
SWAP(x1, x4);
SWAP(x0, x3);
d[0] = x0;
SWAP(x2, x5);
d[5] = x5;
SWAP(x1, x3);
d[1] = x1;
SWAP(x2, x4);
d[4] = x4;
SWAP(x2, x3);
d[2] = x2;
d[3] = x3;
#undef SWAP
#undef min
#undef max
}
static __inline__ unsigned long long rdtsc(void)
{
unsigned long long int x;
__asm__ volatile ("rdtsc; shlq $32, %%rdx; orq %%rdx, %0" : "=a" (x) : : "rdx");
return x;
}
void ran_fill(int n, int *a) {
static int seed = 76521;
while (n--) *a++ = (seed = seed *1812433253 + 12345);
}
#define NTESTS 4096
int main() {
int i;
int d[6*NTESTS];
ran_fill(6*NTESTS, d);
unsigned long long cycles = rdtsc();
for (i = 0; i < 6*NTESTS ; i+=6) {
sort6_fast(d+i);
}
cycles = rdtsc() - cycles;
printf("Time is %.2lf\n", (double)cycles/(double)NTESTS);
for (i = 0; i < 6*NTESTS ; i+=6) {
if (d[i+0] > d[i+1] || d[i+1] > d[i+2] || d[i+2] > d[i+3] || d[i+3] > d[i+4] || d[i+4] > d[i+5])
printf("d%d : %d %d %d %d %d %d\n", i,
d[i+0], d[i+1], d[i+2],
d[i+3], d[i+4], d[i+5]);
}
return 0;
}
```

I changed modified the test suite to also report clocks per sort and run more tests (the cmp function was updated to handle integer overflow as well), here are the results on some different architectures. I attempted testing on an AMD cpu but rdtsc isn't reliable on the X6 1100T I have available.

```
Clarkdale (i5-650)
==================
Direct call to qsort library function 635.14 575.65 581.61 577.76 521.12
Naive implementation (insertion sort) 538.30 135.36 134.89 240.62 101.23
Insertion Sort (Daniel Stutzbach) 424.48 159.85 160.76 152.01 151.92
Insertion Sort Unrolled 339.16 125.16 125.81 129.93 123.16
Rank Order 184.34 106.58 54.74 93.24 94.09
Rank Order with registers 127.45 104.65 53.79 98.05 97.95
Sorting Networks (Daniel Stutzbach) 269.77 130.56 128.15 126.70 127.30
Sorting Networks (Paul R) 551.64 103.20 64.57 73.68 73.51
Sorting Networks 12 with Fast Swap 321.74 61.61 63.90 67.92 67.76
Sorting Networks 12 reordered Swap 318.75 60.69 65.90 70.25 70.06
Reordered Sorting Network w/ fast swap 145.91 34.17 32.66 32.22 32.18
Kentsfield (Core 2 Quad)
========================
Direct call to qsort library function 870.01 736.39 723.39 725.48 721.85
Naive implementation (insertion sort) 503.67 174.09 182.13 284.41 191.10
Insertion Sort (Daniel Stutzbach) 345.32 152.84 157.67 151.23 150.96
Insertion Sort Unrolled 316.20 133.03 129.86 118.96 105.06
Rank Order 164.37 138.32 46.29 99.87 99.81
Rank Order with registers 115.44 116.02 44.04 116.04 116.03
Sorting Networks (Daniel Stutzbach) 230.35 114.31 119.15 110.51 111.45
Sorting Networks (Paul R) 498.94 77.24 63.98 62.17 65.67
Sorting Networks 12 with Fast Swap 315.98 59.41 58.36 60.29 55.15
Sorting Networks 12 reordered Swap 307.67 55.78 51.48 51.67 50.74
Reordered Sorting Network w/ fast swap 149.68 31.46 30.91 31.54 31.58
Sandy Bridge (i7-2600k)
=======================
Direct call to qsort library function 559.97 451.88 464.84 491.35 458.11
Naive implementation (insertion sort) 341.15 160.26 160.45 154.40 106.54
Insertion Sort (Daniel Stutzbach) 284.17 136.74 132.69 123.85 121.77
Insertion Sort Unrolled 239.40 110.49 114.81 110.79 117.30
Rank Order 114.24 76.42 45.31 36.96 36.73
Rank Order with registers 105.09 32.31 48.54 32.51 33.29
Sorting Networks (Daniel Stutzbach) 210.56 115.68 116.69 107.05 124.08
Sorting Networks (Paul R) 364.03 66.02 61.64 45.70 44.19
Sorting Networks 12 with Fast Swap 246.97 41.36 59.03 41.66 38.98
Sorting Networks 12 reordered Swap 235.39 38.84 47.36 38.61 37.29
Reordered Sorting Network w/ fast swap 115.58 27.23 27.75 27.25 26.54
Nehalem (Xeon E5640)
====================
Direct call to qsort library function 911.62 890.88 681.80 876.03 872.89
Naive implementation (insertion sort) 457.69 236.87 127.68 388.74 175.28
Insertion Sort (Daniel Stutzbach) 317.89 279.74 147.78 247.97 245.09
Insertion Sort Unrolled 259.63 220.60 116.55 221.66 212.93
Rank Order 140.62 197.04 52.10 163.66 153.63
Rank Order with registers 84.83 96.78 50.93 109.96 54.73
Sorting Networks (Daniel Stutzbach) 214.59 220.94 118.68 120.60 116.09
Sorting Networks (Paul R) 459.17 163.76 56.40 61.83 58.69
Sorting Networks 12 with Fast Swap 284.58 95.01 50.66 53.19 55.47
Sorting Networks 12 reordered Swap 281.20 96.72 44.15 56.38 54.57
Reordered Sorting Network w/ fast swap 128.34 50.87 26.87 27.91 28.02
```

Never optimize min/max without benchmarking and looking at actual compiler generated assembly. If I let GCC optimize min with conditional move instructions I get a 33% speedup:

```
#define SWAP(x,y) { int dx = d[x], dy = d[y], tmp; tmp = d[x] = dx < dy ? dx : dy; d[y] ^= dx ^ tmp; }
```

(280 vs. 420 cycles in the test code). Doing max with ?: is more or less the same, almost lost in the noise, but the above is a little bit faster. This SWAP is faster with both GCC and Clang.

Compilers are also doing an exceptional job at register allocation and alias analysis, effectively moving d[x] into local variables upfront, and only copying back to memory at the end. In fact, they do so even better than if you worked entirely with local variables (like `d0 = d[0], d1 = d[1], d2 = d[2], d3 = d[3], d4 = d[4], d5 = d[5]`

). I'm writing this because you are assuming strong optimization and yet trying to outsmart the compiler on min/max. :)

By the way, I tried Clang and GCC. They do the same optimization, but due to scheduling differences the two have some variation in the results, can't say really which is faster or slower. GCC is faster on the sorting networks, Clang on the quadratic sorts.

Just for completeness, unrolled bubble sort and insertion sorts are possible too. Here is the bubble sort:

```
SWAP(0,1); SWAP(1,2); SWAP(2,3); SWAP(3,4); SWAP(4,5);
SWAP(0,1); SWAP(1,2); SWAP(2,3); SWAP(3,4);
SWAP(0,1); SWAP(1,2); SWAP(2,3);
SWAP(0,1); SWAP(1,2);
SWAP(0,1);
```

and here is the insertion sort:

```
//#define ITER(x) { if (t < d[x]) { d[x+1] = d[x]; d[x] = t; } }
//Faster on x86, probably slower on ARM or similar:
#define ITER(x) { d[x+1] ^= t < d[x] ? d[x] ^ d[x+1] : 0; d[x] = t < d[x] ? t : d[x]; }
static inline void sort6_insertion_sort_unrolled_v2(int * d){
int t;
t = d[1]; ITER(0);
t = d[2]; ITER(1); ITER(0);
t = d[3]; ITER(2); ITER(1); ITER(0);
t = d[4]; ITER(3); ITER(2); ITER(1); ITER(0);
t = d[5]; ITER(4); ITER(3); ITER(2); ITER(1); ITER(0);
```

This insertion sort is faster than Daniel Stutzbach's, and is especially good on a GPU or a computer with predication because ITER can be done with only 3 instructions (vs. 4 for SWAP). For example, here is the `t = d[2]; ITER(1); ITER(0);`

line in ARM assembly:

```
MOV r6, r2
CMP r6, r1
MOVLT r2, r1
MOVLT r1, r6
CMP r6, r0
MOVLT r1, r0
MOVLT r0, r6
```

For six elements the insertion sort is competitive with the sorting network (12 swaps vs. 15 iterations balances 4 instructions/swap vs. 3 instructions/iteration); bubble sort of course is slower. But it's not going to be true when the size grows, since insertion sort is O(n^2) while sorting networks are O(n log n).

Since these are integers and compares are fast, why not compute the rank order of each directly:

```
inline void sort6(int *d) {
int e[6];
memcpy(e,d,6*sizeof(int));
int o0 = (d[0]>d[1])+(d[0]>d[2])+(d[0]>d[3])+(d[0]>d[4])+(d[0]>d[5]);
int o1 = (d[1]>=d[0])+(d[1]>d[2])+(d[1]>d[3])+(d[1]>d[4])+(d[1]>d[5]);
int o2 = (d[2]>=d[0])+(d[2]>=d[1])+(d[2]>d[3])+(d[2]>d[4])+(d[2]>d[5]);
int o3 = (d[3]>=d[0])+(d[3]>=d[1])+(d[3]>=d[2])+(d[3]>d[4])+(d[3]>d[5]);
int o4 = (d[4]>=d[0])+(d[4]>=d[1])+(d[4]>=d[2])+(d[4]>=d[3])+(d[4]>d[5]);
int o5 = 15-(o0+o1+o2+o3+o4);
d[o0]=e[0]; d[o1]=e[1]; d[o2]=e[2]; d[o3]=e[3]; d[o4]=e[4]; d[o5]=e[5];
}
```

The test code is pretty bad; it overflows the initial array (don't people here read compiler warnings?), the printf is printing out the wrong elements, it uses .byte for rdtsc for no good reason, there's only one run (!), there's nothing checking that the end results are actually correct (so it's very easy to “optimize” into something subtly wrong), the included tests are very rudimentary (no negative numbers?) and there's nothing to stop the compiler from just discarding the entire function as dead code.

That being said, it's also pretty easy to improve on the bitonic network solution; simply change the min/max/SWAP stuff to

```
#define SWAP(x,y) { int tmp; asm("mov %0, %2 ; cmp %1, %0 ; cmovg %1, %0 ; cmovg %2, %1" : "=r" (d[x]), "=r" (d[y]), "=r" (tmp) : "0" (d[x]), "1" (d[y]) : "cc"); }
```

and it comes out about 65% faster for me (Debian gcc 4.4.5 with -O2, amd64, Core i7).

This question is becoming quite old, but I actually had to solve the same problem these days: fast agorithms to sort small arrays. I thought it would be a good idea to share my knowledge. While I first started by using sorting networks, I finally managed to find other algorithms for which the total number of comparisons performed to sort every permutation of 6 values was smaller than with sorting networks, and smaller than with insertion sort. I didn't count the number of swaps; I would expect it to be roughly equivalent (maybe a bit higher sometimes).

The algorithm `sort6`

uses the algorithm `sort4`

which uses the algorithm `sort3`

. Here is the implementation in some light C++ form (the original is template-heavy so that it can work with any random-access iterator and any suitable comparison function).

# Sorting 3 values

The following algorithm is an unrolled insertion sort. When two swaps (6 assignments) have to be performed, it uses 4 assignments instead:

```
void sort3(int* array)
{
if (array[1] < array[0]) {
if (array[2] < array[0]) {
if (array[2] < array[1]) {
std::swap(array[0], array[2]);
} else {
int tmp = array[0];
array[0] = array[1];
array[1] = array[2];
array[2] = tmp;
}
} else {
std::swap(array[0], array[1]);
}
} else {
if (array[2] < array[1]) {
if (array[2] < array[0]) {
int tmp = array[2];
array[2] = array[1];
array[1] = array[0];
array[0] = tmp;
} else {
std::swap(array[1], array[2]);
}
}
}
}
```

It looks a bit complex because the sort has more or less one branch for every possible permutation of the array, using 2~3 comparisons and at most 4 assignments to sort the three values.

# Sorting 4 values

This one calls `sort3`

then performs an unrolled insertion sort with the last element of the array:

```
void sort4(int* array)
{
// Sort the first 3 elements
sort3(array);
// Insert the 4th element with insertion sort
if (array[3] < array[2]) {
std::swap(array[2], array[3]);
if (array[2] < array[1]) {
std::swap(array[1], array[2]);
if (array[1] < array[0]) {
std::swap(array[0], array[1]);
}
}
}
}
```

This algorithm performs 3 to 6 comparisons and at most 5 swaps. It is easy to unroll an insertion sort, but we will be using another algorithm for the last sort...

# Sorting 6 values

This one uses an unrolled version of what I called a *double insertion sort*. The name isn't that great, but it's quite descriptive, here is how it works:

- Sort everything but the first and the last elements of the array.
- Swap the first and the elements of the array if the first is greater than the last.
- Insert the first element into the sorted sequence from the front then the last element from the back.

After the swap, the first element is always smaller than the last, which means that, when inserting them into the sorted sequence, there won't be more than N comparisons to insert the two elements in the worst case: for example, if the first element has been insert in the 3rd position, then the last one can't be inserted lower than the 4th position.

```
void sort6(int* array)
{
// Sort everything but first and last elements
sort4(array+1);
// Switch first and last elements if needed
if (array[5] < array[0]) {
std::swap(array[0], array[5]);
}
// Insert first element from the front
if (array[1] < array[0]) {
std::swap(array[0], array[1]);
if (array[2] < array[1]) {
std::swap(array[1], array[2]);
if (array[3] < array[2]) {
std::swap(array[2], array[3]);
if (array[4] < array[3]) {
std::swap(array[3], array[4]);
}
}
}
}
// Insert last element from the back
if (array[5] < array[4]) {
std::swap(array[4], array[5]);
if (array[4] < array[3]) {
std::swap(array[3], array[4]);
if (array[3] < array[2]) {
std::swap(array[2], array[3]);
if (array[2] < array[1]) {
std::swap(array[1], array[2]);
}
}
}
}
}
```

My tests on every permutation of 6 values ever show that this algorithms always performs between 6 and 13 comparisons. I didn't compute the number of swaps performed, but I don't expect it to be higher than 11 in the worst case.

I hope that this helps, even if this question may not represent an actual problem anymore :)

*EDIT:* after putting it in the provided benchmark, it is cleary slower than most of the interesting alternatives. It tends to perform a bit better than the unrolled insertion sort, but that's pretty much it. Basically, it isn't the best sort for integers but could be interesting for types with an expensive comparison operation.

Well, if it's only 6 elements and you can leverage parallelism, want to minimize conditional branching, etc. Why you don't generate all the combinations and test for order? I would venture that in some architectures, it can be pretty fast (as long as you have the memory preallocated)

While I really like the swap macro provided:

```
#define min(x, y) (y ^ ((x ^ y) & -(x < y)))
#define max(x, y) (x ^ ((x ^ y) & -(x < y)))
#define SWAP(x,y) { int tmp = min(d[x], d[y]); d[y] = max(d[x], d[y]); d[x] = tmp; }
```

I see an improvement (which a good compiler might make):

```
#define SWAP(x,y) { int tmp = ((x ^ y) & -(y < x)); y ^= tmp; x ^= tmp; }
```

We take note of how min and max work and pull the common sub-expression explicitly. This eliminates the min and max macros completely.