如何在现代 C++ 中实现经典排序算法?

Algorithmic building blocks

We begin by assembling the algorithmic building blocks from the Standard Library:

#include <algorithm>    // min_element, iter_swap, 
                        // upper_bound, rotate, 
                        // partition, 
                        // inplace_merge,
                        // make_heap, sort_heap, push_heap, pop_heap,
                        // is_heap, is_sorted
#include <cassert>      // assert 
#include <functional>   // less
#include <iterator>     // distance, begin, end, next

• 非成员std::begin()/std::end()等迭代器工具以及std::next() 仅适用于 C++11 及更高版本.对于 C++98，需要自己编写这些.boost::begin()/boost::end() 中的 Boost.Range 和 boost::next().
• std::is_sorted 算法仅适用于 C++11 及更高版本.对于 C++98，这可以通过 std::adjacent_find 和一个手写的函数对象来实现.Boost.Algorithm 还提供了一个 boost::algorithm::is_sorted 作为替代.
• std::is_heap 算法仅适用于 C++11 及更高版本.
• the iterator tools such as non-member std::begin() / std::end() as well as with std::next() are only available as of C++11 and beyond. For C++98, one needs to write these himself. There are substitutes from Boost.Range in boost::begin() / boost::end(), and from Boost.Utility in boost::next().
• the std::is_sorted algorithm is only available for C++11 and beyond. For C++98, this can be implemented in terms of std::adjacent_find and a hand-written function object. Boost.Algorithm also provides a boost::algorithm::is_sorted as a substitute.
• the std::is_heap algorithm is only available for C++11 and beyond.

C++14 提供了透明比较器的形式std::less<> 多态地作用于它们的参数.这避免了必须提供迭代器的类型.这可以与 C++11 的 默认函数模板参数结合使用来创建 单个重载，用于将 < 作为比较的排序算法以及具有用户定义的比较函数对象的排序算法.

C++14 provides transparent comparators of the form std::less<> that act polymorphically on their arguments. This avoids having to provide an iterator's type. This can be used in combination with C++11's default function template arguments to create a single overload for sorting algorithms that take < as comparison and those that have a user-defined comparison function object.

template<class It, class Compare = std::less<>>
void xxx_sort(It first, It last, Compare cmp = Compare{});

在 C++11 中，可以定义一个可重用的模板别名来提取一个迭代器的值类型，它为排序算法的签名添加了轻微的混乱:

In C++11, one can define a reusable template alias to extract an iterator's value type which adds minor clutter to the sort algorithms' signatures:

template<class It>
using value_type_t = typename std::iterator_traits<It>::value_type;

template<class It, class Compare = std::less<value_type_t<It>>>
void xxx_sort(It first, It last, Compare cmp = Compare{});

在 C++98 中，需要编写两个重载并使用详细的 typename xxx::type 语法

In C++98, one needs to write two overloads and use the verbose typename xxx<yyy>::type syntax

template<class It, class Compare>
void xxx_sort(It first, It last, Compare cmp); // general implementation

template<class It>
void xxx_sort(It first, It last)
{
    xxx_sort(first, last, std::less<typename std::iterator_traits<It>::value_type>());
}

• 另一个语法上的好处是，C++14 有助于通过 多态 lambdas 包装用户定义的比较器(使用 auto 参数，这些参数像函数模板参数一样推导出来).
• C++11 只有单态 lambda，需要使用上述模板别名 value_type_t.
• 在 C++98 中，要么需要编写一个独立的函数对象，要么求助于冗长的 std::bind1st/std::bind2nd/std::not1 类型的语法.
• Boost.Bind 使用 boost::bind 和 _1/_2 占位符语法对此进行了改进.
• C++11 及更高版本也有 std::find_if_not，而 C++98 需要 std::find_if 和 std::not1 围绕一个函数对象.
• Another syntactical nicety is that C++14 facilitates wrapping user-defined comparators through polymorphic lambdas (with auto parameters that are deduced like function template arguments).
• C++11 only has monomorphic lambdas, that require the use of the above template alias value_type_t.
• In C++98, one either needs to write a standalone function object or resort to the verbose std::bind1st / std::bind2nd / std::not1 type of syntax.
• Boost.Bind improves this with boost::bind and _1 / _2 placeholder syntax.
• C++11 and beyond also have std::find_if_not, whereas C++98 needs std::find_if with a std::not1 around a function object.

目前还没有普遍接受的 C++14 风格.无论好坏，我都密切关注 Scott Meyers 的 draft Effective Modern C++ 和 Herb Sutter 的 改进的 GotW.我使用以下样式建议:

There is no generally acceptable C++14 style yet. For better or for worse, I closely follow Scott Meyers's draft Effective Modern C++ and Herb Sutter's revamped GotW. I use the following style recommendations:

• Herb Sutter 的 几乎总是自动" 和 Scott Meyers 的 优先使用 auto 而非特定类型声明" 建议，虽然其简洁性有时是无与伦比的，但 有争议.
• Scott Meyers 的 在创建对象时区分()和{}"并一致选择braced-initialization {} 而不是旧的带括号的初始化 ()(为了回避通用代码中所有最棘手的解析问题).
• Scott Meyers 的 "首选别名声明而不是 typedefs".对于模板来说，这是必须的，并且在任何地方都使用它而不是 typedef 可以节省时间并增加一致性.
• 我在某些地方使用 for (auto it = first; it != last; ++it) 模式，以便允许对已排序的子范围进行循环不变检查.在生产代码中，在循环中的某处使用 while (first != last) 和 ++first 可能会稍微好一些.
• Herb Sutter's "Almost Always Auto" and Scott Meyers's "Prefer auto to specific type declarations" recommendation, for which the brevity is unsurpassed, although its clarity is sometimes disputed.
• Scott Meyers's "Distinguish () and {} when creating objects" and consistently choose braced-initialization {} instead of the good old parenthesized initialization () (in order to side-step all most-vexing-parse issues in generic code).
• Scott Meyers's "Prefer alias declarations to typedefs". For templates this is a must anyway, and using it everywhere instead of typedef saves time and adds consistency.
• I use a for (auto it = first; it != last; ++it) pattern in some places, in order to allow for loop invariant checking for already sorted sub-ranges. In production code, the use of while (first != last) and a ++first somewhere inside the loop might be slightly better.

选择排序不会以任何方式适应数据，所以它的运行时间总是O(N2).然而，选择排序具有最小化交换次数的特性.在交换项目的成本很高的应用程序中，选择排序很可能是首选算法.

Selection sort does not adapt to the data in any way, so its runtime is always O(N2). However, selection sort has the property of minimizing the number of swaps. In applications where the cost of swapping items is high, selection sort very well may be the algorithm of choice.

要使用标准库实现它，重复使用 std::min_element 找到剩余的最小元素，并使用 iter_swap 将其交换到位:

To implement it using the Standard Library, repeatedly use std::min_element to find the remaining minimum element, and iter_swap to swap it into place:

template<class FwdIt, class Compare = std::less<>>
void selection_sort(FwdIt first, FwdIt last, Compare cmp = Compare{})
{
    for (auto it = first; it != last; ++it) {
        auto const selection = std::min_element(it, last, cmp);
        std::iter_swap(selection, it); 
        assert(std::is_sorted(first, std::next(it), cmp));
    }
}

请注意，selection_sort 已将已处理的范围 [first, it) 排序为其循环不变量.与 std::sort 的随机访问迭代器相比，最低要求是 前向迭代器.

Note that selection_sort has the already processed range [first, it) sorted as its loop invariant. The minimal requirements are forward iterators, compared to std::sort's random access iterators.

细节省略:

• 选择排序可以通过早期测试进行优化 if (std::distance(first, last) <= 1) return; (或者对于前向/双向迭代器:if (first == last || std::next(first) == last) return;).
• 对于双向迭代器，上述测试可以与区间[first, std::prev(last))上的循环结合，因为最后一个元素是保证是最小的剩余元素，不需要交换.
• selection sort can be optimized with an early test if (std::distance(first, last) <= 1) return; (or for forward / bidirectional iterators: if (first == last || std::next(first) == last) return;).
• for bidirectional iterators, the above test can be combined with a loop over the interval [first, std::prev(last)), because the last element is guaranteed to be the minimal remaining element and doesn't require a swap.

虽然它是具有 O(N2) 最坏情况时间的基本排序算法之一，但 插入排序是当数据接近排序(因为它是自适应)或问题规模较小(因为它的开销很低).由于这些原因，并且由于它也是稳定，插入排序通常用作递归基本情况(当问题规模较小时)用于更高开销的分治排序算法，例如合并排序或快速排序.

Although it is one of the elementary sorting algorithms with O(N2) worst-case time, insertion sort is the algorithm of choice either when the data is nearly sorted (because it is adaptive) or when the problem size is small (because it has low overhead). For these reasons, and because it is also stable, insertion sort is often used as the recursive base case (when the problem size is small) for higher overhead divide-and-conquer sorting algorithms, such as merge sort or quick sort.

用标准库实现insertion_sort，重复使用std::upper_bound找到当前元素需要去的位置，使用std::rotate 将输入范围内的剩余元素向上移动:

To implement insertion_sort with the Standard Library, repeatedly use std::upper_bound to find the location where the current element needs to go, and use std::rotate to shift the remaining elements upward in the input range:

template<class FwdIt, class Compare = std::less<>>
void insertion_sort(FwdIt first, FwdIt last, Compare cmp = Compare{})
{
    for (auto it = first; it != last; ++it) {
        auto const insertion = std::upper_bound(first, it, *it, cmp);
        std::rotate(insertion, it, std::next(it)); 
        assert(std::is_sorted(first, std::next(it), cmp));
    }
}

请注意，insertion_sort 已将已处理的范围 [first, it) 排序为其循环不变量.插入排序也适用于前向迭代器.

Note that insertion_sort has the already processed range [first, it) sorted as its loop invariant. Insertion sort also works with forward iterators.

细节省略:

• 插入排序可以通过早期测试进行优化 if (std::distance(first, last) <= 1) return; (或者对于前向/双向迭代器:if (first == last || std::next(first) == last) return;) 和区间 [std::next(first), last) 上的循环，因为第一个元素保证就位，不需要旋转.
• 对于双向迭代器，可以使用标准库的std::find_if_not 将查找插入点的二进制搜索替换为反向线性搜索代码>算法.
• insertion sort can be optimized with an early test if (std::distance(first, last) <= 1) return; (or for forward / bidirectional iterators: if (first == last || std::next(first) == last) return;) and a loop over the interval [std::next(first), last), because the first element is guaranteed to be in place and doesn't require a rotate.
• for bidirectional iterators, the binary search to find the insertion point can be replaced with a reverse linear search using the Standard Library's std::find_if_not algorithm.

四个实时示例(C++14、C++11、C++98 和 Boost, C++98) 用于以下片段:

Four Live Examples (C++14, C++11, C++98 and Boost, C++98) for the fragment below:

using RevIt = std::reverse_iterator<BiDirIt>;
auto const insertion = std::find_if_not(RevIt(it), RevIt(first), 
    [=](auto const& elem){ return cmp(*it, elem); }
).base();

• 对于随机输入，这给出了 O(N2) 比较，但对于几乎排序的输入，这改进为 O(N) 比较.二进制搜索总是使用 O(N log N) 比较.
• 对于较小的输入范围，线性搜索的更好的内存局部性(缓存、预取)也可能主导二分搜索(当然，应该对此进行测试).
• For random inputs this gives O(N2) comparisons, but this improves to O(N) comparisons for almost sorted inputs. The binary search always uses O(N log N) comparisons.
• For small input ranges, the better memory locality (cache, prefetching) of a linear search might also dominate a binary search (one should test this, of course).

如果仔细实施，快速排序是健壮的并且具有<代码>O(N log N) 预期复杂度，但 O(N2) 最坏情况复杂度可以通过对抗性选择的输入数据触发.当不需要稳定排序时，快速排序是一种出色的通用排序.

When carefully implemented, quick sort is robust and has O(N log N) expected complexity, but with O(N2) worst-case complexity that can be triggered with adversarially chosen input data. When a stable sort is not needed, quick sort is an excellent general-purpose sort.

即使对于最简单的版本，使用标准库实现快速排序也比其他经典排序算法要复杂得多.下面的方法使用一些迭代器实用程序将输入范围 [first, last) 的 middle element 定位为枢轴，然后使用两次对 std 的调用::partition(即 O(N))将输入范围三路划分为分别小于、等于和大于所选枢轴的元素段.最后，对元素小于和大于主元的两个外部段进行递归排序:

Even for the simplest versions, quick sort is quite a bit more complicated to implement using the Standard Library than the other classic sorting algorithms. The approach below uses a few iterator utilities to locate the middle element of the input range [first, last) as the pivot, then use two calls to std::partition (which are O(N)) to three-way partition the input range into segments of elements that are smaller than, equal to, and larger than the selected pivot, respectively. Finally the two outer segments with elements smaller than and larger than the pivot are recursively sorted:

template<class FwdIt, class Compare = std::less<>>
void quick_sort(FwdIt first, FwdIt last, Compare cmp = Compare{})
{
    auto const N = std::distance(first, last);
    if (N <= 1) return;
    auto const pivot = *std::next(first, N / 2);
    auto const middle1 = std::partition(first, last, [=](auto const& elem){ 
        return cmp(elem, pivot); 
    });
    auto const middle2 = std::partition(middle1, last, [=](auto const& elem){ 
        return !cmp(pivot, elem);
    });
    quick_sort(first, middle1, cmp); // assert(std::is_sorted(first, middle1, cmp));
    quick_sort(middle2, last, cmp);  // assert(std::is_sorted(middle2, last, cmp));
}

但是，快速排序要获得正确和高效是相当棘手的，因为上述每个步骤都必须仔细检查并针对生产级代码进行优化.特别是，对于 O(N log N) 复杂度，pivot 必须导致输入数据的平衡分区，这通常不能保证 O(1) pivot，但如果将pivot设置为输入范围的O(N)中值，则可以保证这一点.

However, quick sort is rather tricky to get correct and efficient, as each of the above steps has to be carefully checked and optimized for production level code. In particular, for O(N log N) complexity, the pivot has to result into a balanced partition of the input data, which cannot be guaranteed in general for an O(1) pivot, but which can be guaranteed if one sets the pivot as the O(N) median of the input range.

细节省略:

• 上述实现特别容易受到特殊输入的影响，例如对于风琴管"输入 1, 2, 3, ..., N/2, ... 它具有 O(N^2) 复杂度3, 2, 1(因为中间总是大于所有其他元素).
• median-of-3 从 从输入范围中随机选择的元素可以防止几乎排序的输入，否则复杂性会恶化到 O(N^2).
• 三向分区(分隔小于、等于的元素到和大于枢轴)，如对 std::partition 的两次调用所示，并不是实现此结果的最有效的 O(N) 算法.
• 对于随机访问迭代器，可以通过使用std 的median pivot selection 实现有保证的O(N log N) 复杂度::nth_element(first, middle, last)，然后递归调用 quick_sort(first, middle, cmp) 和 quick_sort(middle, last, cmp).
• 然而，这种保证是有代价的，因为 std::nth_element 的 O(N) 复杂性的常数因子可能比O(1) 中位数为 3 的枢轴的复杂度，然后是对 std::partition 的 O(N) 调用(即对数据进行缓存友好的单次前向传递).
• the above implementation is particularly vulnerable to special inputs, e.g. it has O(N^2) complexity for the "organ pipe" input 1, 2, 3, ..., N/2, ... 3, 2, 1 (because the middle is always larger than all other elements).
• median-of-3 pivot selection from randomly chosen elements from the input range guards against almost sorted inputs for which the complexity would otherwise deteriorate to O(N^2).
• 3-way partitioning (separating elements smaller than, equal to and larger than the pivot) as shown by the two calls to std::partition is not the most efficient O(N) algorithm to achieve this result.
• for random access iterators, a guaranteed O(N log N) complexity can be achieved through median pivot selection using std::nth_element(first, middle, last), followed by recursive calls to quick_sort(first, middle, cmp) and quick_sort(middle, last, cmp).
• this guarantee comes at a cost, however, because the constant factor of the O(N) complexity of std::nth_element can be more expensive than that of the O(1) complexity of a median-of-3 pivot followed by an O(N) call to std::partition (which is a cache-friendly single forward pass over the data).

如果使用 O(N) 额外空间无关紧要，那么 合并排序 是一个很好的选择:它是唯一的稳定 O(N log N) 排序算法.

If using O(N) extra space is of no concern, then merge sort is an excellent choice: it is the only stable O(N log N) sorting algorithm.

使用标准算法很容易实现:使用一些迭代器实用程序来定位输入范围 [first, last) 的中间位置，并将两个递归排序的段与 std 组合起来::inplace_merge:

It is simple to implement using Standard algorithms: use a few iterator utilities to locate the middle of the input range [first, last) and combine two recursively sorted segments with a std::inplace_merge:

template<class BiDirIt, class Compare = std::less<>>
void merge_sort(BiDirIt first, BiDirIt last, Compare cmp = Compare{})
{
    auto const N = std::distance(first, last);
    if (N <= 1) return;                   
    auto const middle = std::next(first, N / 2);
    merge_sort(first, middle, cmp); // assert(std::is_sorted(first, middle, cmp));
    merge_sort(middle, last, cmp);  // assert(std::is_sorted(middle, last, cmp));
    std::inplace_merge(first, middle, last, cmp); // assert(std::is_sorted(first, last, cmp));
}

合并排序需要双向迭代器，瓶颈是 std::inplace_merge.请注意，在对链表进行排序时，归并排序仅需要 O(log N) 额外空间(用于递归).后一种算法由标准库中的 std::list<T>::sort 实现.

Merge sort requires bidirectional iterators, the bottleneck being the std::inplace_merge. Note that when sorting linked lists, merge sort requires only O(log N) extra space (for recursion). The latter algorithm is implemented by std::list<T>::sort in the Standard Library.

堆排序很容易实现，执行一个O(N log N) 就地排序，但不稳定.

Heap sort is simple to implement, performs an O(N log N) in-place sort, but is not stable.

第一个循环，O(N)heapify"阶段，将数组放入堆中.第二个循环，O(N log N)排序"阶段，反复提取最大值并恢复堆顺序.标准库使这变得非常简单:

The first loop, O(N) "heapify" phase, puts the array into heap order. The second loop, the O(N log N) "sortdown" phase, repeatedly extracts the maximum and restores heap order. The Standard Library makes this extremely straightforward:

template<class RandomIt, class Compare = std::less<>>
void heap_sort(RandomIt first, RandomIt last, Compare cmp = Compare{})
{
    lib::make_heap(first, last, cmp); // assert(std::is_heap(first, last, cmp));
    lib::sort_heap(first, last, cmp); // assert(std::is_sorted(first, last, cmp));
}

如果你认为使用 std::make_heap 和 std::sort_heap 是作弊"，你可以更深入一层，自己编写这些函数std::push_heap 和 std::pop_heap 分别为:

In case you consider it "cheating" to use std::make_heap and std::sort_heap, you can go one level deeper and write those functions yourself in terms of std::push_heap and std::pop_heap, respectively:

namespace lib {

// NOTE: is O(N log N), not O(N) as std::make_heap
template<class RandomIt, class Compare = std::less<>>
void make_heap(RandomIt first, RandomIt last, Compare cmp = Compare{})
{
    for (auto it = first; it != last;) {
        std::push_heap(first, ++it, cmp); 
        assert(std::is_heap(first, it, cmp));           
    }
}

template<class RandomIt, class Compare = std::less<>>
void sort_heap(RandomIt first, RandomIt last, Compare cmp = Compare{})
{
    for (auto it = last; it != first;) {
        std::pop_heap(first, it--, cmp);
        assert(std::is_heap(first, it, cmp));           
    } 
}

}   // namespace lib

标准库将 push_heap 和 pop_heap 都指定为复杂度 O(log N).但是请注意，在 [first, last) 范围内的外部循环会导致 make_heap 的复杂度为 O(N log N)，而 >std::make_heap 只有 O(N) 复杂度.对于 heap_sort 的整体 O(N log N) 复杂度来说，这并不重要.

The Standard Library specifies both push_heap and pop_heap as complexity O(log N). Note however that the outer loop over the range [first, last) results in O(N log N) complexity for make_heap, whereas std::make_heap has only O(N) complexity. For the overall O(N log N) complexity of heap_sort it doesn't matter.

省略细节:O(N)实现<代码>make_heap

这里有四个实时示例(C++14、C++11, C++98 和 Boost, C++98) 在各种输入 (并不意味着详尽或严格).请注意 LOC 的巨大差异:C++11/C++14 需要大约 130 LOC，C++98 和 Boost 190 (+50%) 和 C++98 超过 270 (+100%).

Here are four Live Examples (C++14, C++11, C++98 and Boost, C++98) testing all five algorithms on a variety of inputs (not meant to be exhaustive or rigorous). Just note the huge differences in the LOC: C++11/C++14 need around 130 LOC, C++98 and Boost 190 (+50%) and C++98 more than 270 (+100%).