Let’s do this

Let’s get in the ring and show what AVX/AVX2 intrinsics can really do for a non-trivial problem, and even discuss potential improvements that future CoreCLR versions could bring to the table.

Everyone needs to sort arrays, once in a while, and many algorithms we take for granted rely on being able to do so. We think of it as a solved problem and that nothing can be further done about it in 2020, except for waiting for newer, marginally faster machines to pop-up1. However, that is not the case, and while I’m not the first to have thoughts about it; or the best at implementing it, if you join me in this rather long journey, we’ll end up with a replacement function for Array.Sort, written in pure C# that can outperform CoreCLR’s own C++2 based code by a factor north of 10x on most modern Intel machines, and even north of 11x on mine.
Sounds interesting? If so, down the rabbit hole we go…

Note that in the final days before posting this series, Intel started seeding a CPU microcode update that is/was affecting the performance of the released version of CoreCLR 3.0/3.1 quite considerably. I managed to stir up a small commotion as this was unraveling in my benchmarks. As it happened, my code was (not coincidentally) less affected by this change, while CoreCLRs Array.Sort() took a 20% nosedive. Let it never be said I’m nothing less than chivalrous, for I rolled back the microcode update and for this entire series I’m going to run against a much faster version of Array.Sort() than what you, the reader, are probably using, Assuming you update your machine from time to time. For the technically inclined, here’s a whole footnote3 on how to double-check what your machine is actually running. I also opened two issues in the CoreCLR repo about attempting to mitigate this both in CoreCLRs C++ code and separately in the JIT. If/when there is movement on those fronts, the microcode you’re running will become less of an issue, to begin with, but for now, this just adds another level of unwarranted complexity to our lives.

A while back now, I was reading the post by Stephen Toub about Improvements in CoreCLR 3.0, and it became apparent that hardware intrinsics were common to many of these, and that so many parts of CoreCLR could still be sped up with these techniques, that one thing led to another, and I decided an attempt to apply hardware intrinsics to a larger problem than I had previously done myself was in order. To see if I could rise to the challenge, I decided to take on array sorting and see how far I can go.

What I came up with eventually would become a re-write of QuickSort/Array.Sort() with AVX2 hardware intrinsics. Fortunately, choosing sorting, and specifically, QuickSort, makes for a great blog post series, since:

I started with searching various keywords and found an interesting paper titled: Fast Quicksort Implementation Using AVX Instructions by Shay Gueron and Vlad Krasnov. That title alone made me think this is about to be a walk in the park. While initially promising, it wasn’t good enough as a drop-in replacement for Array.Sort for reasons I’ll shortly go into. I ended up having a lot of fun expanding on their idea. I will submit a proper pull-request to start a discussion with CoreCLR devs about integrating this code into the main dotnet/runtime repository, but for now, let’s talk about sorting.

Since there’s a lot to go over here, I’ve split it up into no less than 6 parts:

  1. In this part, we start with a refresher on QuickSort and how it compares to Array.Sort(). If you don’t need a refresher, skip it and get right down to part 2 and onwards. I recommend skimming through, mostly because I’ve got excellent visualizations which should be in the back of everyone’s mind as we deal with vectorization & optimization later.
  2. In part 2, we go over the basics of vectorized hardware intrinsics, vector types, and go over a handful of vectorized instructions we’ll use in part 3. We still won’t be sorting anything.
  3. In part 3, we go through the initial code for the vectorized sorting, and we’ll start seeing some payoff. We finish agonizing courtesy of the CPU’s Branch Predictor, throwing a wrench into our attempts.
  4. In part 4, we go over a handful of optimization approaches that I attempted trying to get the vectorized partitioning to run faster. We’ll see what worked and what didn’t.
  5. In part 5, we’ll see how we can almost get rid of all the remaining scalar code- by implementing small-constant size array sorting. We’ll use, drum roll…, yet more AVX2 vectorization.
  6. Finally, in part 6, I’ll list the outstanding stuff/ideas I have for getting more juice and functionality out of my vectorized code.

QuickSort Crash Course

QuickSort is deceivingly simple.
No, it really is.
In 20 lines of C# or whatever language you can sort numbers. Lots of them, and incredibly fast. However, try and change something about it; nudge it in the wrong way, and it will quickly turn around and teach you a lesson in humility. It is hard to improve on it without breaking any of the tenants it is built upon.

In words

Before we discuss any of that, let’s describe QuickSort in words, code, pictures, and statistics:

That last point, referring to in-place sorting, sounds simple and neat, and it sure is from the perspective of the user: no additional memory allocation needs to occur regardless of how much data they’re sorting. While that’s great, I’ve spent days trying to overcome the correctness and performance challenges that arise from it, specifically in the context of vectorization. It is also essential to remain in-place since I intend for this to become a drop-in replacement for Array.Sort.

More concretely, QuickSort works like this:

  1. Pick a pivot value.
  2. Partition the array around the pivot value.
  3. Recurse on the left side of the pivot.
  4. Recurse on the right side of the pivot.

Picking a pivot could be a mini-post in itself, but again, in the context of competing with Array.Sort we don’t need to dive into it, we’ll copy whatever CoreCLR does, and get on with our lives.
CoreCLR uses a pretty standard scheme of median-of-three for pivot selection, which can be summed up as: “Let’s sort these 3 elements: In the first, middle and last positions, then pick the middle one of those three as the pivot”.

Partitioning the array is where we spend most of the execution time: we take our selected pivot value and rearrange the array segment that was handed to us such that all numbers smaller-than the pivot are in the beginning or left, in no particular order amongst themselves. Then comes the pivot, in its final resting position, and following it are all elements greater-than the pivot, again in no particular order amongst themselves.

After partitioning is complete, we recurse to the left and right of the pivot, as previously described.

That’s all there is: this gets millions, billions of numbers sorted, in-place, efficiently as we know how to do 60+ years after its invention.

Bonus trivia points for those who are still here with me: Tony Hoare, who invented QuickSort back in the early 60s also took responsibility for inventing the null pointer concept. So I guess there really is no good without evil in this world.

In code

void QuickSort(int[] items) => QuickSort(items, 0, items.Length - 1);

void QuickSort(int[] items, int left, int right)
    if (left == right) return;
    int pivot = PickPivot(items, left, right);
    int pivotPos = Partition(items, pivot, left, right);
    QuickSort(items, left, pivotPos);
    QuickSort(items, pivotPos + 1, right);

int PickPivot(int[] items, int left, int right)
    var mid = left + ((right - left) / 2);
    SwapIfGreater(ref items[left],  ref items[mid]);
    SwapIfGreater(ref items[left],  ref items[right]);
    SwapIfGreater(ref items[mid],   ref items[right]);
    var pivot = items[mid];

int Partition(int[] array, int pivot, int left, int right)
    while (left <= right) {
        while (array[left]  < pivot) left++;
        while (array[right] > pivot) right--;

        if (left <= right) {
            var t = array[left];
            array[left++]  = array[right];
            array[right--] = t;
    return left;

I did say it is deceptively simple, and grasping how QuickSort really works sometimes feels like trying to lift sand through your fingers; To that end I’ve included two more visualizations of QuickSort, which are derivatives of the amazing work done by Michael Bostock (@mbostock) with d3.js.

Visualizing QuickSort’s recursion

One thing that we have to keep in mind is that the same data is partitioned over-and-over again, many times, with ever-shrinking partition sizes until we end up having a partition size of 2 or 3, in which case we can trivially sort the partition as-is and return.

To help see this better, we’ll use this way of visualizing arrays and their intermediate states in QuickSort:

QuickSort Legend

Here, we see an unsorted array of 200 elements (in the process of getting sorted).
The different sticks represent numbers in the [-45°..+45°] range, and the angle of each individual stick represents its value, as I hope it is easy to discern.
We represent the pivots with two colors:

Our ultimate goal is to go from the image above to the image below:

QuickSort Sorted

What follows is a static (e.g., non-animated) visualization that shows how pivots are randomly selected at each level of recursion and how, by the next step, the unsorted segments around them become partitioned until we finally have a completely sorted array. Here is how the whole thing looks:

These visuals are auto-generated in Javascript + d3.js, so feel free to hit that “Reload” button and/or change the number of elements in the array if you feel you want to see a new set of random sticks sorted.

I encourage you to look at this and try to explain to yourself what QuickSort “does” here, at every level. What you can witness here is the interaction between pivot selection, where it “lands” in the next recursion level (or row), and future pivots to its left and right and in the next levels of recursion. We also see how, with every level of recursion, the partition sizes decrease in until finally, every element is a pivot, which means sorting is complete.

Visualizing QuickSort’s Comparisons/Swaps

While the above visualization really does a lot to help understand how QuickSort works, I also wanted to leave you with an impression of the total amount of work done by QuickSort:

Above is an animation of the whole process as it goes over the same array, slowly and recursively going from an unsorted mess to a completely sorted array.

We can witness just how many comparisons and swap operations need to happen for a 200 element QuickSort to complete successfully. There’s genuinely a lot of work that needs to happen per element (when considering how we re-partition virtually all elements again and again) for the whole thing to finish.

Array.Sort vs. QuickSort

It’s important to note that Array.Sort uses a couple of more tricks to get better performance. I would be irresponsible if I didn’t mention those since in the later posts, I borrow at least one idea from its play-book, and improve upon it with intrinsics.

Array.Sort isn’t strictly QuickSort; it is a variation on it called Introspective Sort invented by David Musser in 1997. What it roughly does is combine QuickSort, HeapSort, and Insertion Sort, and switch dynamically between them according to the recursion depth and the partition size. This last trick, where it switches to using “Insertion Sort” on small partitions, is critical both for the general case and also for intrinsics/vectorization. It is a sound optimization because it replaces (up to) the last 4 levels or so of recursion (for partition sizes <= 16) with a single call to an insertion sort. This reduces the overhead associated with recursion with simpler loop-based code, which ultimately runs faster for such small partitions.

As mentioned, I ended up borrowing this idea for my code as the issues around smaller partition sizes are exacerbated when using vectorized intrinsics in the following posts.

Comparing Scalar Variants

With all this new information, this is a good time to measure how a couple of different scalar (e.g. non-vectorized) versions compare to Array.Sort. I’ll show some results generated using BenchmarkDotNet (BDN) with:

I’ve prepared this last version to show that with unsafe code + InsertionSort, we can remove most of the performance gap between C# and C++ for this type of code, which mainly stems from bounds-checking, that the JIT cannot elide for these sort of random-access patterns as well as the jump-to InsertionSort optimization.

Note that for this series, We’ll benchmark each sorting method with various array sizes (BDN parameter: N): . I’ve added a custom column to the BDN column to the report: Time / N. This represents the time spent sorting per element in the array, and as such, very useful to compare the results on a more uniform scale.

Here are the results in the form of charts and tables. I’ve included a handy large button you can press to get a quick tour of what each tab contains, what we have here is:

  1. A chart scaling the performance of various implementations being compared to Array.Sort as a ratio.
  2. A chart showing time spent sorting a single element in an array of N elements (Time / N).
  3. BDN results in a friendly table form.
  4. Statistics/Counters that teach us about what is actually going on under the hood.

Surprisingly4, the unmanaged C# version is running slightly faster than Array.Sort, but with one caveat: it only outperforms the C++ version for large inputs. Otherwise, everything is as expected: The purely Managed variant is just slow, and the Unamanged one mostly is on par with Array.Sort.
These C# implementations were written to verify that we can get to Array.Sort like performance in C#, and they do just that. Running 5% faster for some input sizes will not cut it for me; I want it much faster. An equally important reason for re-implementing these basic versions is that we can now sprinkle statistics-collecting-code magic fairy dust5 on them so that we have even more numbers to dig into in the “Statistics” tab: These counters will assist us in deciphering and comparing future results and implementations. In this post they serve us by establishing a baseline. We can see, per each N value (with some commentary):

Some of these statistics will remain pretty much the same for the rest of this series, regardless of what we do next in future versions, while others radically change; We’ll observe and make use of these as key inputs in helping us to figure out how/why something worked, or not!

All Warmed Up?

We’ve spent quite some time polishing our foundations concerning QuickSort and Array.Sort. I know lengthy introductions are somewhat dull, but I think time spent on this post will pay off with dividend when we next encounter our actual implementation in the 3rd post and later on. This might be also a time to confess that just doing the leg-work to provide this refresher helped me come up with at least one, super non-trivial optimization, which I think I’ll keep the lid on all the way until the 6th and final post. So never underestimate the importance of “just” covering the basics.

Before we write vectorized code, we need to pick up some knowhow specific to vectorized intrinsics and introduce a few select intrinsics we’ll be using, so, this is an excellent time to break off this post, grab a fresh cup of coffee and head to the next post.

  1. Which is increasingly taking more and more time to happen, due to Dennard scaling and the slow-down of Moore’s law… 

  2. Since CoreCLR 3.0 was release, a PR to provide a span based version of this has been recently merged into the 5.0 master branch, but I’ll ignore this for the time being as it doesn’t seem to matter in this context. 

  3. You can grab your microcode signature in one of the following methods: On Windows, the easiest way is to install and run the excellent HWiNFO64 application, it will show you the microcode signature. On line a grep -i microcode /proc/cpuinfo does the tricks, and macOs: sysctl -a | grep -i microcode will get the job done. Unfortunately you’ll have to consult your specific CPU model to figure out the before/after signature, and I can’t help you there, except to point out that the microcode update in question came out in November 13th and is about mitigating the JCC errata. 

  4. Believe it or not, I pretty much wrote every other version features in this series before I wrote the Unmanaged one, so I really was quite surprised that it ended up being slightly faster that Array.Sort 

  5. I have a special build configuration called Stats which compiles in a bunch of calls into various conditionally compiled functions that bump various counters, and finally, dump it all to json and it eventually makes it all the way into these posts (if you dig deep you can get the actual json files :)