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120 lines
6.2 KiB
Markdown
120 lines
6.2 KiB
Markdown
pdqsort
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-------
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Pattern-defeating quicksort (pdqsort) is a novel sorting algorithm that combines the fast average
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case of randomized quicksort with the fast worst case of heapsort, while achieving linear time on
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inputs with certain patterns. pdqsort is an extension and improvement of David Mussers introsort.
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All code is available for free under the zlib license.
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Best Average Worst Memory Stable Deterministic
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n n log n n log n log n No Yes
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### Usage
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`pdqsort` is a drop-in replacement for [`std::sort`](http://en.cppreference.com/w/cpp/algorithm/sort).
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Just replace a call to `std::sort` with `pdqsort` to start using pattern-defeating quicksort. If your
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comparison function is branchless, you can call `pdqsort_branchless` for a potential big speedup. If
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you are using C++11, the type you're sorting is arithmetic and your comparison function is not given
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or is `std::less`/`std::greater`, `pdqsort` automatically delegates to `pdqsort_branchless`.
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### Benchmark
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A comparison of pdqsort and GCC's `std::sort` and `std::stable_sort` with various input
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distributions:
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Compiled with `-std=c++11 -O2 -m64 -march=native`.
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### Visualization
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A visualization of pattern-defeating quicksort sorting a ~200 element array with some duplicates.
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Generated using Timo Bingmann's [The Sound of Sorting](http://panthema.net/2013/sound-of-sorting/)
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program, a tool that has been invaluable during the development of pdqsort. For the purposes of
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this visualization the cutoff point for insertion sort was lowered to 8 elements.
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### The best case
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pdqsort is designed to run in linear time for a couple of best-case patterns. Linear time is
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achieved for inputs that are in strictly ascending or descending order, only contain equal elements,
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or are strictly in ascending order followed by one out-of-place element. There are two separate
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mechanisms at play to achieve this.
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For equal elements a smart partitioning scheme is used that always puts equal elements in the
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partition containing elements greater than the pivot. When a new pivot is chosen it's compared to
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the greatest element in the partition before it. If they compare equal we can derive that there are
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no elements smaller than the chosen pivot. When this happens we switch strategy for this partition,
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and filter out all elements equal to the pivot.
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To get linear time for the other patterns we check after every partition if any swaps were made. If
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no swaps were made and the partition was decently balanced we will optimistically attempt to use
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insertion sort. This insertion sort aborts if more than a constant amount of moves are required to
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sort.
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### The average case
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On average case data where no patterns are detected pdqsort is effectively a quicksort that uses
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median-of-3 pivot selection, switching to insertion sort if the number of elements to be
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(recursively) sorted is small. The overhead associated with detecting the patterns for the best case
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is so small it lies within the error of measurement.
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pdqsort gets a great speedup over the traditional way of implementing quicksort when sorting large
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arrays (1000+ elements). This is due to a new technique described in "BlockQuicksort: How Branch
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Mispredictions don't affect Quicksort" by Stefan Edelkamp and Armin Weiss. In short, we bypass the
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branch predictor by using small buffers (entirely in L1 cache) of the indices of elements that need
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to be swapped. We fill these buffers in a branch-free way that's quite elegant (in pseudocode):
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```cpp
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buffer_num = 0; buffer_max_size = 64;
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for (int i = 0; i < buffer_max_size; ++i) {
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// With branch:
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if (elements[i] < pivot) { buffer[buffer_num] = i; buffer_num++; }
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// Without:
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buffer[buffer_num] = i; buffer_num += (elements[i] < pivot);
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}
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```
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This is only a speedup if the comparison function itself is branchless, however. By default pdqsort
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will detect this if you're using C++11 or higher, the type you're sorting is arithmetic (e.g.
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`int`), and you're using either `std::less` or `std::greater`. You can explicitly request branchless
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partitioning by calling `pdqsort_branchless` instead of `pdqsort`.
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### The worst case
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Quicksort naturally performs bad on inputs that form patterns, due to it being a partition-based
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sort. Choosing a bad pivot will result in many comparisons that give little to no progress in the
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sorting process. If the pattern does not get broken up, this can happen many times in a row. Worse,
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real world data is filled with these patterns.
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Traditionally the solution to this is to randomize the pivot selection of quicksort. While this
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technically still allows for a quadratic worst case, the chances of it happening are astronomically
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small. Later, in introsort, pivot selection is kept deterministic, instead switching to the
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guaranteed O(n log n) heapsort if the recursion depth becomes too big. In pdqsort we adopt a hybrid
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approach, (deterministically) shuffling some elements to break up patterns when we encounter a "bad"
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partition. If we encounter too many "bad" partitions we switch to heapsort.
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### Bad partitions
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A bad partition occurs when the position of the pivot after partitioning is under 12.5% (1/8th)
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percentile or over 87,5% percentile - the partition is highly unbalanced. When this happens we will
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shuffle four elements at fixed locations for both partitions. This effectively breaks up many
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patterns. If we encounter more than log(n) bad partitions we will switch to heapsort.
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The 1/8th percentile is not chosen arbitrarily. An upper bound of quicksorts worst case runtime can
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be approximated within a constant factor by the following recurrence:
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T(n, p) = n + T(p(n-1), p) + T((1-p)(n-1), p)
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Where n is the number of elements, and p is the percentile of the pivot after partitioning.
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`T(n, 1/2)` is the best case for quicksort. On modern systems heapsort is profiled to be
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approximately 1.8 to 2 times as slow as quicksort. Choosing p such that `T(n, 1/2) / T(n, p) ~= 1.9`
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as n gets big will ensure that we will only switch to heapsort if it would speed up the sorting.
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p = 1/8 is a reasonably close value and is cheap to compute on every platform using a bitshift.
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