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Move contrib/pdqsort from submodule to source

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proller 2019-02-11 18:30:51 +03:00
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.gitmodules vendored
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[submodule "contrib/cppkafka"]
path = contrib/cppkafka
url = https://github.com/ClickHouse-Extras/cppkafka.git
[submodule "contrib/pdqsort"]
path = contrib/pdqsort
url = https://github.com/orlp/pdqsort

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contrib/pdqsort vendored

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Subproject commit 08879029ab8dcb80a70142acb709e3df02de5d37

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Source from https://github.com/orlp/pdqsort
Mandatory for Clickhouse, not available in OS packages, we can't use it as submodule.

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Copyright (c) 2015 Orson Peters <orsonpeters@gmail.com>
This software is provided 'as-is', without any express or implied warranty. In no event will the
authors be held liable for any damages arising from the use of this software.
Permission is granted to anyone to use this software for any purpose, including commercial
applications, and to alter it and redistribute it freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not claim that you wrote the
original software. If you use this software in a product, an acknowledgment in the product
documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be misrepresented as
being the original software.
3. This notice may not be removed or altered from any source distribution.

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/*
pdqsort.h - Pattern-defeating quicksort.
Copyright (c) 2015 Orson Peters
This software is provided 'as-is', without any express or implied warranty. In no event will the
authors be held liable for any damages arising from the use of this software.
Permission is granted to anyone to use this software for any purpose, including commercial
applications, and to alter it and redistribute it freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not claim that you wrote the
original software. If you use this software in a product, an acknowledgment in the product
documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be misrepresented as
being the original software.
3. This notice may not be removed or altered from any source distribution.
*/
#ifndef PDQSORT_H
#define PDQSORT_H
#include <algorithm>
#include <cstddef>
#include <functional>
#include <utility>
#include <iterator>
#if __cplusplus >= 201103L
#include <cstdint>
#include <type_traits>
#define PDQSORT_PREFER_MOVE(x) std::move(x)
#else
#define PDQSORT_PREFER_MOVE(x) (x)
#endif
namespace pdqsort_detail {
enum {
// Partitions below this size are sorted using insertion sort.
insertion_sort_threshold = 24,
// Partitions above this size use Tukey's ninther to select the pivot.
ninther_threshold = 128,
// When we detect an already sorted partition, attempt an insertion sort that allows this
// amount of element moves before giving up.
partial_insertion_sort_limit = 8,
// Must be multiple of 8 due to loop unrolling, and < 256 to fit in unsigned char.
block_size = 64,
// Cacheline size, assumes power of two.
cacheline_size = 64
};
#if __cplusplus >= 201103L
template<class T> struct is_default_compare : std::false_type { };
template<class T> struct is_default_compare<std::less<T>> : std::true_type { };
template<class T> struct is_default_compare<std::greater<T>> : std::true_type { };
#endif
// Returns floor(log2(n)), assumes n > 0.
template<class T>
inline int log2(T n) {
int log = 0;
while (n >>= 1) ++log;
return log;
}
// Sorts [begin, end) using insertion sort with the given comparison function.
template<class Iter, class Compare>
inline void insertion_sort(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
if (begin == end) return;
for (Iter cur = begin + 1; cur != end; ++cur) {
Iter sift = cur;
Iter sift_1 = cur - 1;
// Compare first so we can avoid 2 moves for an element already positioned correctly.
if (comp(*sift, *sift_1)) {
T tmp = PDQSORT_PREFER_MOVE(*sift);
do { *sift-- = PDQSORT_PREFER_MOVE(*sift_1); }
while (sift != begin && comp(tmp, *--sift_1));
*sift = PDQSORT_PREFER_MOVE(tmp);
}
}
}
// Sorts [begin, end) using insertion sort with the given comparison function. Assumes
// *(begin - 1) is an element smaller than or equal to any element in [begin, end).
template<class Iter, class Compare>
inline void unguarded_insertion_sort(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
if (begin == end) return;
for (Iter cur = begin + 1; cur != end; ++cur) {
Iter sift = cur;
Iter sift_1 = cur - 1;
// Compare first so we can avoid 2 moves for an element already positioned correctly.
if (comp(*sift, *sift_1)) {
T tmp = PDQSORT_PREFER_MOVE(*sift);
do { *sift-- = PDQSORT_PREFER_MOVE(*sift_1); }
while (comp(tmp, *--sift_1));
*sift = PDQSORT_PREFER_MOVE(tmp);
}
}
}
// Attempts to use insertion sort on [begin, end). Will return false if more than
// partial_insertion_sort_limit elements were moved, and abort sorting. Otherwise it will
// successfully sort and return true.
template<class Iter, class Compare>
inline bool partial_insertion_sort(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
if (begin == end) return true;
int limit = 0;
for (Iter cur = begin + 1; cur != end; ++cur) {
if (limit > partial_insertion_sort_limit) return false;
Iter sift = cur;
Iter sift_1 = cur - 1;
// Compare first so we can avoid 2 moves for an element already positioned correctly.
if (comp(*sift, *sift_1)) {
T tmp = PDQSORT_PREFER_MOVE(*sift);
do { *sift-- = PDQSORT_PREFER_MOVE(*sift_1); }
while (sift != begin && comp(tmp, *--sift_1));
*sift = PDQSORT_PREFER_MOVE(tmp);
limit += cur - sift;
}
}
return true;
}
template<class Iter, class Compare>
inline void sort2(Iter a, Iter b, Compare comp) {
if (comp(*b, *a)) std::iter_swap(a, b);
}
// Sorts the elements *a, *b and *c using comparison function comp.
template<class Iter, class Compare>
inline void sort3(Iter a, Iter b, Iter c, Compare comp) {
sort2(a, b, comp);
sort2(b, c, comp);
sort2(a, b, comp);
}
template<class T>
inline T* align_cacheline(T* p) {
#if defined(UINTPTR_MAX) && __cplusplus >= 201103L
std::uintptr_t ip = reinterpret_cast<std::uintptr_t>(p);
#else
std::size_t ip = reinterpret_cast<std::size_t>(p);
#endif
ip = (ip + cacheline_size - 1) & -cacheline_size;
return reinterpret_cast<T*>(ip);
}
template<class Iter>
inline void swap_offsets(Iter first, Iter last,
unsigned char* offsets_l, unsigned char* offsets_r,
int num, bool use_swaps) {
typedef typename std::iterator_traits<Iter>::value_type T;
if (use_swaps) {
// This case is needed for the descending distribution, where we need
// to have proper swapping for pdqsort to remain O(n).
for (int i = 0; i < num; ++i) {
std::iter_swap(first + offsets_l[i], last - offsets_r[i]);
}
} else if (num > 0) {
Iter l = first + offsets_l[0]; Iter r = last - offsets_r[0];
T tmp(PDQSORT_PREFER_MOVE(*l)); *l = PDQSORT_PREFER_MOVE(*r);
for (int i = 1; i < num; ++i) {
l = first + offsets_l[i]; *r = PDQSORT_PREFER_MOVE(*l);
r = last - offsets_r[i]; *l = PDQSORT_PREFER_MOVE(*r);
}
*r = PDQSORT_PREFER_MOVE(tmp);
}
}
// Partitions [begin, end) around pivot *begin using comparison function comp. Elements equal
// to the pivot are put in the right-hand partition. Returns the position of the pivot after
// partitioning and whether the passed sequence already was correctly partitioned. Assumes the
// pivot is a median of at least 3 elements and that [begin, end) is at least
// insertion_sort_threshold long. Uses branchless partitioning.
template<class Iter, class Compare>
inline std::pair<Iter, bool> partition_right_branchless(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
// Move pivot into local for speed.
T pivot(PDQSORT_PREFER_MOVE(*begin));
Iter first = begin;
Iter last = end;
// Find the first element greater than or equal than the pivot (the median of 3 guarantees
// this exists).
while (comp(*++first, pivot));
// Find the first element strictly smaller than the pivot. We have to guard this search if
// there was no element before *first.
if (first - 1 == begin) while (first < last && !comp(*--last, pivot));
else while ( !comp(*--last, pivot));
// If the first pair of elements that should be swapped to partition are the same element,
// the passed in sequence already was correctly partitioned.
bool already_partitioned = first >= last;
if (!already_partitioned) {
std::iter_swap(first, last);
++first;
}
// The following branchless partitioning is derived from "BlockQuicksort: How Branch
// Mispredictions don’t affect Quicksort" by Stefan Edelkamp and Armin Weiss.
unsigned char offsets_l_storage[block_size + cacheline_size];
unsigned char offsets_r_storage[block_size + cacheline_size];
unsigned char* offsets_l = align_cacheline(offsets_l_storage);
unsigned char* offsets_r = align_cacheline(offsets_r_storage);
int num_l, num_r, start_l, start_r;
num_l = num_r = start_l = start_r = 0;
while (last - first > 2 * block_size) {
// Fill up offset blocks with elements that are on the wrong side.
if (num_l == 0) {
start_l = 0;
Iter it = first;
for (unsigned char i = 0; i < block_size;) {
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
}
}
if (num_r == 0) {
start_r = 0;
Iter it = last;
for (unsigned char i = 0; i < block_size;) {
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
}
}
// Swap elements and update block sizes and first/last boundaries.
int num = std::min(num_l, num_r);
swap_offsets(first, last, offsets_l + start_l, offsets_r + start_r,
num, num_l == num_r);
num_l -= num; num_r -= num;
start_l += num; start_r += num;
if (num_l == 0) first += block_size;
if (num_r == 0) last -= block_size;
}
int l_size = 0, r_size = 0;
int unknown_left = (last - first) - ((num_r || num_l) ? block_size : 0);
if (num_r) {
// Handle leftover block by assigning the unknown elements to the other block.
l_size = unknown_left;
r_size = block_size;
} else if (num_l) {
l_size = block_size;
r_size = unknown_left;
} else {
// No leftover block, split the unknown elements in two blocks.
l_size = unknown_left/2;
r_size = unknown_left - l_size;
}
// Fill offset buffers if needed.
if (unknown_left && !num_l) {
start_l = 0;
Iter it = first;
for (unsigned char i = 0; i < l_size;) {
offsets_l[num_l] = i++; num_l += !comp(*it, pivot); ++it;
}
}
if (unknown_left && !num_r) {
start_r = 0;
Iter it = last;
for (unsigned char i = 0; i < r_size;) {
offsets_r[num_r] = ++i; num_r += comp(*--it, pivot);
}
}
int num = std::min(num_l, num_r);
swap_offsets(first, last, offsets_l + start_l, offsets_r + start_r, num, num_l == num_r);
num_l -= num; num_r -= num;
start_l += num; start_r += num;
if (num_l == 0) first += l_size;
if (num_r == 0) last -= r_size;
// We have now fully identified [first, last)'s proper position. Swap the last elements.
if (num_l) {
offsets_l += start_l;
while (num_l--) std::iter_swap(first + offsets_l[num_l], --last);
first = last;
}
if (num_r) {
offsets_r += start_r;
while (num_r--) std::iter_swap(last - offsets_r[num_r], first), ++first;
last = first;
}
// Put the pivot in the right place.
Iter pivot_pos = first - 1;
*begin = PDQSORT_PREFER_MOVE(*pivot_pos);
*pivot_pos = PDQSORT_PREFER_MOVE(pivot);
return std::make_pair(pivot_pos, already_partitioned);
}
// Partitions [begin, end) around pivot *begin using comparison function comp. Elements equal
// to the pivot are put in the right-hand partition. Returns the position of the pivot after
// partitioning and whether the passed sequence already was correctly partitioned. Assumes the
// pivot is a median of at least 3 elements and that [begin, end) is at least
// insertion_sort_threshold long.
template<class Iter, class Compare>
inline std::pair<Iter, bool> partition_right(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
// Move pivot into local for speed.
T pivot(PDQSORT_PREFER_MOVE(*begin));
Iter first = begin;
Iter last = end;
// Find the first element greater than or equal than the pivot (the median of 3 guarantees
// this exists).
while (comp(*++first, pivot));
// Find the first element strictly smaller than the pivot. We have to guard this search if
// there was no element before *first.
if (first - 1 == begin) while (first < last && !comp(*--last, pivot));
else while ( !comp(*--last, pivot));
// If the first pair of elements that should be swapped to partition are the same element,
// the passed in sequence already was correctly partitioned.
bool already_partitioned = first >= last;
// Keep swapping pairs of elements that are on the wrong side of the pivot. Previously
// swapped pairs guard the searches, which is why the first iteration is special-cased
// above.
while (first < last) {
std::iter_swap(first, last);
while (comp(*++first, pivot));
while (!comp(*--last, pivot));
}
// Put the pivot in the right place.
Iter pivot_pos = first - 1;
*begin = PDQSORT_PREFER_MOVE(*pivot_pos);
*pivot_pos = PDQSORT_PREFER_MOVE(pivot);
return std::make_pair(pivot_pos, already_partitioned);
}
// Similar function to the one above, except elements equal to the pivot are put to the left of
// the pivot and it doesn't check or return if the passed sequence already was partitioned.
// Since this is rarely used (the many equal case), and in that case pdqsort already has O(n)
// performance, no block quicksort is applied here for simplicity.
template<class Iter, class Compare>
inline Iter partition_left(Iter begin, Iter end, Compare comp) {
typedef typename std::iterator_traits<Iter>::value_type T;
T pivot(PDQSORT_PREFER_MOVE(*begin));
Iter first = begin;
Iter last = end;
while (comp(pivot, *--last));
if (last + 1 == end) while (first < last && !comp(pivot, *++first));
else while ( !comp(pivot, *++first));
while (first < last) {
std::iter_swap(first, last);
while (comp(pivot, *--last));
while (!comp(pivot, *++first));
}
Iter pivot_pos = last;
*begin = PDQSORT_PREFER_MOVE(*pivot_pos);
*pivot_pos = PDQSORT_PREFER_MOVE(pivot);
return pivot_pos;
}
template<class Iter, class Compare, bool Branchless>
inline void pdqsort_loop(Iter begin, Iter end, Compare comp, int bad_allowed, bool leftmost = true) {
typedef typename std::iterator_traits<Iter>::difference_type diff_t;
// Use a while loop for tail recursion elimination.
while (true) {
diff_t size = end - begin;
// Insertion sort is faster for small arrays.
if (size < insertion_sort_threshold) {
if (leftmost) insertion_sort(begin, end, comp);
else unguarded_insertion_sort(begin, end, comp);
return;
}
// Choose pivot as median of 3 or pseudomedian of 9.
diff_t s2 = size / 2;
if (size > ninther_threshold) {
sort3(begin, begin + s2, end - 1, comp);
sort3(begin + 1, begin + (s2 - 1), end - 2, comp);
sort3(begin + 2, begin + (s2 + 1), end - 3, comp);
sort3(begin + (s2 - 1), begin + s2, begin + (s2 + 1), comp);
std::iter_swap(begin, begin + s2);
} else sort3(begin + s2, begin, end - 1, comp);
// If *(begin - 1) is the end of the right partition of a previous partition operation
// there is no element in [begin, end) that is smaller than *(begin - 1). Then if our
// pivot compares equal to *(begin - 1) we change strategy, putting equal elements in
// the left partition, greater elements in the right partition. We do not have to
// recurse on the left partition, since it's sorted (all equal).
if (!leftmost && !comp(*(begin - 1), *begin)) {
begin = partition_left(begin, end, comp) + 1;
continue;
}
// Partition and get results.
std::pair<Iter, bool> part_result =
Branchless ? partition_right_branchless(begin, end, comp)
: partition_right(begin, end, comp);
Iter pivot_pos = part_result.first;
bool already_partitioned = part_result.second;
// Check for a highly unbalanced partition.
diff_t l_size = pivot_pos - begin;
diff_t r_size = end - (pivot_pos + 1);
bool highly_unbalanced = l_size < size / 8 || r_size < size / 8;
// If we got a highly unbalanced partition we shuffle elements to break many patterns.
if (highly_unbalanced) {
// If we had too many bad partitions, switch to heapsort to guarantee O(n log n).
if (--bad_allowed == 0) {
std::make_heap(begin, end, comp);
std::sort_heap(begin, end, comp);
return;
}
if (l_size >= insertion_sort_threshold) {
std::iter_swap(begin, begin + l_size / 4);
std::iter_swap(pivot_pos - 1, pivot_pos - l_size / 4);
if (l_size > ninther_threshold) {
std::iter_swap(begin + 1, begin + (l_size / 4 + 1));
std::iter_swap(begin + 2, begin + (l_size / 4 + 2));
std::iter_swap(pivot_pos - 2, pivot_pos - (l_size / 4 + 1));
std::iter_swap(pivot_pos - 3, pivot_pos - (l_size / 4 + 2));
}
}
if (r_size >= insertion_sort_threshold) {
std::iter_swap(pivot_pos + 1, pivot_pos + (1 + r_size / 4));
std::iter_swap(end - 1, end - r_size / 4);
if (r_size > ninther_threshold) {
std::iter_swap(pivot_pos + 2, pivot_pos + (2 + r_size / 4));
std::iter_swap(pivot_pos + 3, pivot_pos + (3 + r_size / 4));
std::iter_swap(end - 2, end - (1 + r_size / 4));
std::iter_swap(end - 3, end - (2 + r_size / 4));
}
}
} else {
// If we were decently balanced and we tried to sort an already partitioned
// sequence try to use insertion sort.
if (already_partitioned && partial_insertion_sort(begin, pivot_pos, comp)
&& partial_insertion_sort(pivot_pos + 1, end, comp)) return;
}
// Sort the left partition first using recursion and do tail recursion elimination for
// the right-hand partition.
pdqsort_loop<Iter, Compare, Branchless>(begin, pivot_pos, comp, bad_allowed, leftmost);
begin = pivot_pos + 1;
leftmost = false;
}
}
}
template<class Iter, class Compare>
inline void pdqsort(Iter begin, Iter end, Compare comp) {
if (begin == end) return;
#if __cplusplus >= 201103L
pdqsort_detail::pdqsort_loop<Iter, Compare,
pdqsort_detail::is_default_compare<typename std::decay<Compare>::type>::value &&
std::is_arithmetic<typename std::iterator_traits<Iter>::value_type>::value>(
begin, end, comp, pdqsort_detail::log2(end - begin));
#else
pdqsort_detail::pdqsort_loop<Iter, Compare, false>(
begin, end, comp, pdqsort_detail::log2(end - begin));
#endif
}
template<class Iter>
inline void pdqsort(Iter begin, Iter end) {
typedef typename std::iterator_traits<Iter>::value_type T;
pdqsort(begin, end, std::less<T>());
}
template<class Iter, class Compare>
inline void pdqsort_branchless(Iter begin, Iter end, Compare comp) {
if (begin == end) return;
pdqsort_detail::pdqsort_loop<Iter, Compare, true>(
begin, end, comp, pdqsort_detail::log2(end - begin));
}
template<class Iter>
inline void pdqsort_branchless(Iter begin, Iter end) {
typedef typename std::iterator_traits<Iter>::value_type T;
pdqsort_branchless(begin, end, std::less<T>());
}
#undef PDQSORT_PREFER_MOVE
#endif

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pdqsort
-------
Pattern-defeating quicksort (pdqsort) is a novel sorting algorithm that combines the fast average
case of randomized quicksort with the fast worst case of heapsort, while achieving linear time on
inputs with certain patterns. pdqsort is an extension and improvement of David Mussers introsort.
All code is available for free under the zlib license.
Best Average Worst Memory Stable Deterministic
n n log n n log n log n No Yes
### Usage
`pdqsort` is a drop-in replacement for [`std::sort`](http://en.cppreference.com/w/cpp/algorithm/sort).
Just replace a call to `std::sort` with `pdqsort` to start using pattern-defeating quicksort. If your
comparison function is branchless, you can call `pdqsort_branchless` for a potential big speedup. If
you are using C++11, the type you're sorting is arithmetic and your comparison function is not given
or is `std::less`/`std::greater`, `pdqsort` automatically delegates to `pdqsort_branchless`.
### Benchmark
A comparison of pdqsort and GCC's `std::sort` and `std::stable_sort` with various input
distributions:
![Performance graph](http://i.imgur.com/1RnIGBO.png)
Compiled with `-std=c++11 -O2 -m64 -march=native`.
### Visualization
A visualization of pattern-defeating quicksort sorting a ~200 element array with some duplicates.
Generated using Timo Bingmann's [The Sound of Sorting](http://panthema.net/2013/sound-of-sorting/)
program, a tool that has been invaluable during the development of pdqsort. For the purposes of
this visualization the cutoff point for insertion sort was lowered to 8 elements.
![Visualization](http://i.imgur.com/QzFG09F.gif)
### The best case
pdqsort is designed to run in linear time for a couple of best-case patterns. Linear time is
achieved for inputs that are in strictly ascending or descending order, only contain equal elements,
or are strictly in ascending order followed by one out-of-place element. There are two separate
mechanisms at play to achieve this.
For equal elements a smart partitioning scheme is used that always puts equal elements in the
partition containing elements greater than the pivot. When a new pivot is chosen it's compared to
the greatest element in the partition before it. If they compare equal we can derive that there are
no elements smaller than the chosen pivot. When this happens we switch strategy for this partition,
and filter out all elements equal to the pivot.
To get linear time for the other patterns we check after every partition if any swaps were made. If
no swaps were made and the partition was decently balanced we will optimistically attempt to use
insertion sort. This insertion sort aborts if more than a constant amount of moves are required to
sort.
### The average case
On average case data where no patterns are detected pdqsort is effectively a quicksort that uses
median-of-3 pivot selection, switching to insertion sort if the number of elements to be
(recursively) sorted is small. The overhead associated with detecting the patterns for the best case
is so small it lies within the error of measurement.
pdqsort gets a great speedup over the traditional way of implementing quicksort when sorting large
arrays (1000+ elements). This is due to a new technique described in "BlockQuicksort: How Branch
Mispredictions don't affect Quicksort" by Stefan Edelkamp and Armin Weiss. In short, we bypass the
branch predictor by using small buffers (entirely in L1 cache) of the indices of elements that need
to be swapped. We fill these buffers in a branch-free way that's quite elegant (in pseudocode):
```cpp
buffer_num = 0; buffer_max_size = 64;
for (int i = 0; i < buffer_max_size; ++i) {
// With branch:
if (elements[i] < pivot) { buffer[buffer_num] = i; buffer_num++; }
// Without:
buffer[buffer_num] = i; buffer_num += (elements[i] < pivot);
}
```
This is only a speedup if the comparison function itself is branchless, however. By default pdqsort
will detect this if you're using C++11 or higher, the type you're sorting is arithmetic (e.g.
`int`), and you're using either `std::less` or `std::greater`. You can explicitly request branchless
partitioning by calling `pdqsort_branchless` instead of `pdqsort`.
### The worst case
Quicksort naturally performs bad on inputs that form patterns, due to it being a partition-based
sort. Choosing a bad pivot will result in many comparisons that give little to no progress in the
sorting process. If the pattern does not get broken up, this can happen many times in a row. Worse,
real world data is filled with these patterns.
Traditionally the solution to this is to randomize the pivot selection of quicksort. While this
technically still allows for a quadratic worst case, the chances of it happening are astronomically
small. Later, in introsort, pivot selection is kept deterministic, instead switching to the
guaranteed O(n log n) heapsort if the recursion depth becomes too big. In pdqsort we adopt a hybrid
approach, (deterministically) shuffling some elements to break up patterns when we encounter a "bad"
partition. If we encounter too many "bad" partitions we switch to heapsort.
### Bad partitions
A bad partition occurs when the position of the pivot after partitioning is under 12.5% (1/8th)
percentile or over 87,5% percentile - the partition is highly unbalanced. When this happens we will
shuffle four elements at fixed locations for both partitions. This effectively breaks up many
patterns. If we encounter more than log(n) bad partitions we will switch to heapsort.
The 1/8th percentile is not chosen arbitrarily. An upper bound of quicksorts worst case runtime can
be approximated within a constant factor by the following recurrence:
T(n, p) = n + T(p(n-1), p) + T((1-p)(n-1), p)
Where n is the number of elements, and p is the percentile of the pivot after partitioning.
`T(n, 1/2)` is the best case for quicksort. On modern systems heapsort is profiled to be
approximately 1.8 to 2 times as slow as quicksort. Choosing p such that `T(n, 1/2) / T(n, p) ~= 1.9`
as n gets big will ensure that we will only switch to heapsort if it would speed up the sorting.
p = 1/8 is a reasonably close value and is cheap to compute on every platform using a bitshift.