From 0539dbf1c8ed11e32f2a111c5d6da928c0b61f9f Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Wed, 9 Mar 2022 15:23:23 -0500 Subject: Candidate selection and sampling-based heuristics --- docs/implementation/primitive/sort.html | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) (limited to 'docs/implementation/primitive/sort.html') diff --git a/docs/implementation/primitive/sort.html b/docs/implementation/primitive/sort.html index 06a23fce..0e5d590d 100644 --- a/docs/implementation/primitive/sort.html +++ b/docs/implementation/primitive/sort.html @@ -26,23 +26,26 @@
  • Branchless quicksorts are the solid choice for larger types, particularly since they can track ranges and call counting and other distribution sorts when appropriate.
  • But for 2- and 4-byte data, radix sort can be a lot faster? For 2-byte sort, I think it's a better bridge than fluxsort between insertion and counting sort (but scan for sortedness first); for 4-byte, hard to say.
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    Grade is basically the same (now that fluxsort gives us a good stable quicksort), excepth moves get more expensive relative to comparisons. Counting sort needs to be switch to the much slower bucket sort.

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    Grade is basically the same (now that fluxsort gives us a good stable quicksort), except moves get more expensive relative to comparisons. Counting sort needs to be switch to the much slower bucket sort.

    A branchless binary search is adequate for Bins but in many cases—very small or large 𝕨, and small range—there are better methods.

    Distribution sorts

    Both counting and bucket sort are small-range algorithms that begin by counting the number of each possible value. Bucket sort, as used here, means that the counts are then used to place values in the appropriate position in the result in another pass. Counting sort does not read from the initial values again and instead reconstructs them from the counts. It might be written (//⁼)⌾(-⟜min) in BQN, relying on the extension of /⁼ to unsorted arguments.

    Bucket sort can be used for Grade or sort-by (⍋⊸⊏), but counting sort only works for sorting itself. It's not-even-unstable: there's no connection between result values and the input values except that they are constructed to be equal. But with fast Indices, counting sort is vastly more powerful, and is effective with a range four to eight times the argument length. This is large enough that it might pose a memory usage problem, but the memory use can be made arbitrarily low by partitioning.

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    I developed Robin Hood Sort as an algorithm with similar properties to bucket sort that relies on uniformly-distributed data rather than a small range. It uses a buffer a few times larger than the input array, and inserts values in a manner similar to a hash table with linear probing, shifting large clumps out if they appear—they're merge-sorted back in at the end. Like counting sort, the substantial memory use can be cut down by partitioning. And it should be cheap to detect probable uniformity during median selection, making this a good fit for quicksorts. Emphasis on probable: it's still very important that RHsort has decent worst-case performance.

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    I developed Robin Hood Sort as an algorithm with similar properties to bucket sort that relies on uniformly-distributed data rather than a small range. It uses a buffer a few times larger than the input array, and inserts values in a manner similar to a hash table with linear probing, shifting large clumps out if they appear—they're merge-sorted back in at the end. Like counting sort, the substantial memory use can be cut down by partitioning. And a random selection of √n samples is enough to make a good decision about whether to use it (see candidate selection), which is a good fit for quicksorts. Of course this can only be probabilistic, so it's still important that RHsort has decent worst-case performance. When quadsort is used for merging, the worst case appears to be about half as fast as fluxsort, very solid.

    Radix sort

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    LSD radix sort is really fast, like three times faster than fluxsort on random 4-byte data. The idea is: bucket sort according to the last byte, then the second-to-last, on up to the first byte. Array is now sorted, after most likely having been scrambled substantially (but stably!). It's tricky to implement right though. The sort_inline functions from ska_sort_copy here are good. They count buckets for every step in one pass, and move back and forth from the array to a buffer instead of adding more memory. Radix sort uses memory proportional to the input array length, plus a constant. But that constant is a liability on short arrays, so it's only really useful for sizes above a few hundred.

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    LSD radix sort is really fast, like three times faster than fluxsort on random 4-byte data. The idea is: bucket sort according to the last byte, then the second-to-last, on up to the first byte. Array is now sorted, after most likely having been scrambled substantially (but stably!). It's tricky to implement right though. The sort_inline functions from ska_sort_copy here are good. They count buckets for every step in one pass, and move back and forth from the array to a buffer instead of adding more memory. Radix sort uses memory proportional to the input array length, plus a constant. But that constant is a liability on short arrays, so it's only really useful for sizes above a hundred or so (to get down to this limit, use 1-byte counts and sum with SIMD or at least SWAR).

    LSD radix sort suffers from problems of cache associativity. Now, usually (for, say, non-blocked transpose) such problems strike only at power of 2 lengths. But by picking out input bytes, radix sort tends to create its own powers of 2. Consider an input consisting of ascending natural numbers ↕n. Lowest byte is fine: the lengths are around n÷256. Next byte up, problems: this byte only changes once every 256 inputs, so every bucket but one has a multiple of 256 length! And writes will cycle around these buckets, so they stay roughly in sync. This is enough to overwhelm any set-associative cache. I measured a degradation of about 5 times on that pass and 3 times overall. The case with bucket lengths near multiples of 256—they need to be separated by an entire cache line not to conflict—is detectable after the cheap counting pass, but it's not the only way this pattern can arise. For example, put a bunch of zeros at the beginning of the array. The first bucket now has some arbitrary length, but once the zeros are processed the gap between it and the next is back to being a multiple of 256. The good news is that it still requires a lot of space to start kicking out a bunch of cache lines: below 10,000 4-byte elements I could never measure significant degradation. So if the lack of adaptivity (and O(n) memory of course) doesn't bother you, radix sort is kind of the best thing going for 4-byte values in the 500 to 20,000 range.

    Quicksort

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    Fluxsort attains high performance with a branchless stable partition that places one half on top of existing data and the other half somewhere else. One half ends up in the appropriate place in the sorted array. The other is in swap memory, and will be shifted back by subsequent partitions and base-case sorting. Aside from the partitioning strategy, Fluxsort makes a number of other decisions differently from pdqsort, including a fairly complicated merge sort (Quadsort) as the base case. I haven't looked into Quadsort, but did discuss other features with the author in this issue. Pivot selection is an important one—it seems pdqsort uses far fewer pivots than it should. Picking out a larger sample of pivots also opens up the opportunity of performing statistics on them, or checking for a run while the cache line's hot.

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    Fluxsort attains high performance with a branchless stable partition that places one half on top of existing data and the other half somewhere else. One half ends up in the appropriate place in the sorted array. The other is in swap memory, and will be shifted back by subsequent partitions and base-case sorting. Aside from the partitioning strategy, fluxsort makes a number of other decisions differently from pdqsort, including a fairly complicated merge sort (Quadsort) as the base case.

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    Crumsort is an in-place adaptation of fluxsort, which uses a (constant) small amount of external memory to take an approach to partitioning that's slightly more ordered than Hoare partitioning. It's more complicated than fluxsort but performs similarly at small sizes and faster at large ones.

    This paper gives a good description of pdqsort. I'd start with the Rust version, which has some advantages but can still be improved further. The subsections below describe improved partitioning and an initial pass with several benefits. I also found that the pivot randomization methods currently used are less effective because they swap elements that won't become pivots soon; the pivot candidates and randomization targets need to be chosen to overlap. The optimistic insertion sort can also be improved: when a pair of elements is swapped the smaller one should be inserted as usual but the larger one can also be pushed forward at little cost, potentially saving many swaps and handling too-large elements as gracefully as too-small ones.

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    While the stable partitioning for Fluxsort seems to be an overall better choice, pdqsort's unstable partitioning is what I've worked with in the past. The following sections are written from the perspective of pdqsort and will be rewritten for Fluxsort as the methods are adapted.

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    Most likely fluxsort/crumsort partitioning make pdqsort's partitioning obselete for sorting. Because it's easily invertible if comparison results are saved (it's a self-inverse) it's still useful for the partitioning approach to binary search mentioned later.

    Partitioning

    In-place quicksort relies on a partitioning algorithm that exchanges elements in order to split them into two contiguous groups. The Hoare partition scheme does this, and BlockQuicksort showed that it can be performed quickly with branchless index generation; this method was then adopted by pdqsort. But the bit booleans to indices method is faster and fits well with vectorized comparisons.

    It's simplest to define an operation P that partitions a list 𝕩 according to a boolean list 𝕨. Partitioning permutes 𝕩 so that all elements corresponding to 0 in 𝕨 come before those corresponding to 1. The quicksort partition step, with pivot t, is (t≤𝕩)P𝕩, and the comparison can be vectorized. Interleaving comparison and partitioning in chunks would save memory (a fraction of the size of 𝕩, which should have 32- or 64-bit elements because plain counting sort is best for smaller ones) but hardly speeds things up: only a few percent, and only for huge lists with hundreds of millions of elements. The single-step P is also good for Bins, where the boolean 𝕨 will have to be saved.

    For binary search 𝕨⍋𝕩, partitioning allows one pivot element t from 𝕨 to be compared to all of 𝕩 at once, instead of the normal strategy of working with one element from 𝕩 at a time. 𝕩 is partitioned according to t≤𝕩, then result values are found by searching the first half of 𝕨 for the smaller elements and the second half for the larger ones, and then they are put back in the correct positions by reversing the partitioning. Because Hoare partitioning works by swapping independent pairs of elements, P is a self inverse, identical to P⁼. So the last step is simple, provided the partitioning information t≤𝕩 is saved.

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    Scans and heuristics

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    Because quicksort does its work before recursing, it's well suited to statistical techniques that allow the algorithm to be changed.

    Initial pass

    An initial pass for pdqsort (or another in-place quicksort) provides a few advantages: