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miniselect: Generic selection and partial ordering algorithms

miniselect is a C++ header-only library that contains various generic selection and partial sorting algorithms with the ease of use, testing, advice on usage and benchmarking.

Sorting is everywhere and there are many outstanding sorting algorithms that compete in speed, comparison count and cache friendliness. However, selection algorithms are always a bit outside of the competition scope, they are pretty important, for example, in databases ORDER BY LIMIT N is used extremely often which can benefit from more optimal selection and partial sorting algorithms. This library tries to solve this problem with Modern C++.

Easy: First-class, easy to use dependency and carefully documented APIs and algorithm properties.
Fast: We do care about speed of the algorithms and provide reasonable implementations.
Standard compliant: We provide C++11 compatible APIs that are compliant to the standard std::nth_element and std::partial_sort functions including custom comparators and order guarantees. Just replace the names of the functions in your project and it should work!
Well tested: We test all algorithms with a unified framework, under sanitizers and fuzzing.
Benchmarked: We gather benchmarks for all implementations to better understand good and bad spots.

Quick Start
Testing
Documentation
Performance results
Real-world usage
Contributing
Motivation
License

Quick Start

You can either include this project as a cmake dependency and then use the headers that are provided in the include folder or just pass the include folder to your compiler.

#include <iostream>
#include <vector>

#include "miniselect/median_of_ninthers.h"

int main() {
  std::vector<int> v = {1, 8, 4, 3, 2, 9, 0, 7, 6, 5};
  miniselect::median_of_ninthers_select(v.begin(), v.begin() + 5, v.end());
  for (const int i : v) {
    std::cout << i << ' ';
  }
  return 0;
}
// Compile it `clang++/g++ -I$DIRECTORY/miniselect/include/ example.cpp -std=c++11 -O3 -o example
// Possible output: 0 1 4 3 2 5 8 7 6 9
//                            ^ on the right place

Examples can be found in examples.

We support all compilers starting from GCC 7 and Clang 6. We are also planning to support Windows, for now it is best effort but no issues are known so far.

More on which algorithms are available, see documentation. For overview of this work you can read the article in the author's blog.

Testing

To test and benchmark, we use Google benchmark library. Simply do in the root directory:

# Check out the libraries.
$ git clone https://github.com/google/benchmark.git
$ git clone https://github.com/google/googletest.git
$ mkdir build && cd build
$ cmake -DMINISELECT_TESTING=on -DBENCHMARK_ENABLE_GTEST_TESTS=off -DBENCHMARK_ENABLE_TESTING=off ..
$ make -j
$ ctest -j4 --output-on-failure

It will create two tests and two benchmarks test_sort, test_select, benchmark_sort, benchmark_select. Use them to validate or contribute. You can also use ctest.

Documentation

There are several selection algorithms available, further $\large n$ is the number of elements in the array, $\large k$ is the selection element that is needed to be found (all algorithms are deterministic and not stable unless otherwise is specified):

Name	Average	Best Case	Worst Case	Comparisons	Memory
pdqselect	$\large O(n)$	$\large O(n)$	$\large O(n\log n)$	At least $\large 2n$ . Random data $\large 2.5n$	$\large O(1)$
Floyd-Rivest	$\large O(n)$	$\large O(n)$	$\displaystyle \large O(n^2 )$	Avg: $\displaystyle \large n + \min(k, n - k) + O(\sqrt{n \log n})$	$\large O(\log \log n)$
Median Of Medians	$\large O(n)$	$\large O(n)$	$\large O(n)$	Between $\large 2n$ and $\large 22n$ . Random data $\large 2.5n$	$\large O(\log n)$
Median Of Ninthers	$\large O(n)$	$\large O(n)$	$\large O(n)$	Between $\large 2n$ and $\large 21n$ . Random data $\large 2n$	$\large O(\log n)$
Median Of 3 Random	$\large O(n)$	$\large O(n)$	$\displaystyle \large O(n^2 )$	At least $\large 2n$ . Random data $\large 3n$	$\large O(\log n)$
HeapSelect	$\large O(n\log k)$	$\large O(n)$	$\large O(n\log k)$	$\large n\log k$ on average, for some data patterns might be better	$\large O(1)$
libstdc++ (introselect)	$\large O(n)$	$\large O(n)$	$\large O(n\log n)$	At least $\large 2n$ . Random data $\large 3n$	$\large O(1)$
libc++ (median of 3)	$\large O(n)$	$\large O(n)$	$\displaystyle \large O(n^2 )$	At least $\large 2n$ . Random data $\large 3n$	$\large O(1)$

For sorting the situation is similar except every line adds $\large O(k\log k)$ comparisons and pdqselect is using $\large O(\log n)$ memory.

API

All functions end either in select, either in partial_sort and their behavior is exactly the same as for std::nth_element and std::partial_sort respectively, i.e. they accept 3 arguments as first, middle, end iterators and an optional comparator. Several notes:

You should not throw exceptions from Compare function. Standard library also does not specify the behavior in that matter.
We don't support ParallelSTL for now.
C++20 constexpr specifiers might be added but currently we don't have them because of some floating point math in several algorithms.
All functions are in the miniselect namespace. See the example for that.
pdqselect
- This algorithm is based on pdqsort which is acknowledged as one of the fastest generic sort algorithms.
- Location: miniselect/pdqselect.h.
- Functions: pdqselect, pdqselect_branchless, pdqpartial_sort, pdqpartial_sort_branchless. Branchless version uses branchless partition algorithm provided by pdqsort. Use it if your comparison function is branchless, it might give performance for very big ranges.
- Performance advice: Use it when you need to sort a big chunk so that $\large k$ is close to $\large n$ .

Floyd-Rivest
- This algorithm is based on Floyd-Rivest algorithm.
- Location: miniselect/floyd_rivest_select.h.
- Functions: floyd_rivest_select, floyd_rivest_partial_sort.
- Performance advice: Given that this algorithm performs as one of the best on average case in terms of comparisons and speed, we highly advise to at least try this in your project. Especially it is good for small $\large k$ or types that are expensive to compare (for example, strings). But even for median the benchmarks show it outperforms others. It is not easy for this algorithm to build a reasonable worst case but one of examples when this algorithm does not perform well is when there are lots of similar values of linear size (random01 dataset showed some moderate penalties).

We present here two gifs, for median and for $\displaystyle \large k = n / 10$ order statistic.

Median Of Medians
- This algorithm is based on Median of Medians algorithm, one of the first deterministic linear time worst case median algorithm.
- Location: miniselect/median_of_medians.h.
- Functions: median_of_medians_select, median_of_medians_partial_sort.
- Performance advice: This algorithm does not show advantages over others, implemented for historical reasons and for bechmarking.

Median Of Ninthers
- This algorithm is based on Fast Deterministic Selection paper by Andrei Alexandrescu, one of the latest and fastest deterministic linear time worst case median algorithms.
- Location: miniselect/median_of_ninthers.h.
- Functions: median_of_ninthers_select, median_of_ninthers_partial_sort.
- Performance advice: Use this algorithm if you absolutely need linear time worst case scenario for selection algorithm. This algorithm shows some strengths over other deterministic PICK algorithms and has lower constanst than MedianOfMedians.

Median Of 3 Random
- This algorithm is based on QuickSelect with the random median of 3 pivot choice algorithm (it chooses random 3 elements in the range and takes the middle value). It is a randomized algorithm.
- Location: miniselect/median_of_3_random.h.
- Functions: median_of_3_random_select, median_of_3_random_partial_sort.
- Performance advice: This is a randomized algorithm and also it did not show any strengths against Median Of Ninthers.

Introselect
- This algorithm is based on Introselect algorithm, it is used in libstdc++ in std::nth_element, however instead of falling back to MedianOfMedians it is using HeapSelect which adds logarithm to its worst complexity.
- Location: <algorithm>.
- Functions: std::nth_element.
- Performance advice: This algorithm is used in standard library and is not recommended to use if you are looking for performance.

Median Of 3
- This algorithm is based on QuickSelect with median of 3 pivot choice algorithm (the middle value between begin, mid and end values), it is used in libc++ in std::nth_element.
- Location: <algorithm>.
- Functions: std::nth_element.
- Performance advice: This algorithm is used in standard library and is not recommended to use if you are looking for performance.

std::partial_sort or HeapSelect
- This algorithm has heap-based solutions both in libc++ and libstdc++, from the first $\large k$ elements the max heap is built, then one by one the elements are trying to be pushed to that heap with HeapSort in the end.
- Location: <algorithm>, miniselect/heap_select.h.
- Functions: std::partial_sort, heap_select, heap_partial_sort.
- Performance advice: This algorithm is very good for random data and small $\large k$ and might outperform all selection+sort algorithms. However, for descending data it starts to significantly degrade and is not recommended for use if you have such patterns in real data.

Other algorithms to come

Kiwiel modification of FloydRivest algorithm which is described in On Floyd and Rivest’s SELECT algorithm with ternary and quintary pivots.
Combination of FloydRivest and pdqsort pivot strategies, currently all experiments did not show any boost.

Performance results

We use 10 datasets and 8 algorithms with 10000000 elements to find median and other $\large k$ on Intel(R) Core(TM) i5-4200H CPU @ 2.80GHz for std::vector<int>, for median the benchmarks are the following:

For smaller $\large k$ , for example, 1000, the results are the following

Other benchmarks can be found here.

Real-world usage

Yandex ClickHouse

If you are planning to use miniselect in your product, please work from one of our releases and if you wish, you can write the acknowledgment in this section for visibility.

Contributing

Patches are welcome with new algorithms! You should add the selection algorithm together with the partial sorting algorithm in include, add tests in testing and ideally run benchmarks to see how it performs. If you also have some data cases to test against, we would be more than happy to merge them.

Motivation

Firstly the author was interested if any research had been done for small $\large k$ in selection algorithms and was struggling to find working implementations to compare different approaches from standard library and quickselect algorithms. After that it turned out that the problem is much more interesting than it looks like and after reading The Art of Computer Programming from Donald Knuth about minimum comparison sorting and selection algorithms the author decided to look through all non-popular algorithms and try them out.

The author have not found any decent library for selection algorithms and little research is published in open source, so that they decided to merge all that implementations and compare them with possible merging of different ideas into a decent one algorithm for most needs. For a big story of adventures see the author's blog post.

License

The code is made available under the Boost License 1.0.

Third-Party Libraries Used and Adjusted

Library	License
pdqsort	MIT
MedianOfNinthers	Boost License 1.0

README.md