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Showing content from https://cplusplus.com/reference/algorithm/stable_sort/ below:

function template

<algorithm>

std::stable_sort 
template <class RandomAccessIterator>  void stable_sort ( RandomAccessIterator first, RandomAccessIterator last );template <class RandomAccessIterator, class Compare>  void stable_sort ( RandomAccessIterator first, RandomAccessIterator last,                     Compare comp );

Sort elements preserving order of equivalents

Sorts the elements in the range [first,last) into ascending order, like sort, but stable_sort preserves the relative order of the elements with equivalent values.

The elements are compared using operator< for the first version, and comp for the second.



Parameters
first, last
Random-access iterators to the initial and final positions of the sequence to be sorted. The range used is [first,last), which contains all the elements between first and last, including the element pointed by first but not the element pointed by last.
RandomAccessIterator shall point to a type for which swap is properly defined and which is both move-constructible and move-assignable.
comp
Binary function that accepts two elements in the range as arguments, and returns a value convertible to bool. The value returned indicates whether the element passed as first argument is considered to go before the second in the specific strict weak ordering it defines.
The function shall not modify any of its arguments.
This can either be a function pointer or a function object.

Return value none

Example 
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// stable_sort example
#include <iostream>     // std::cout
#include <algorithm>    // std::stable_sort
#include <vector>       // std::vector

bool compare_as_ints (double i,double j)
{
  return (int(i)<int(j));
}

int main () {
  double mydoubles[] = {3.14, 1.41, 2.72, 4.67, 1.73, 1.32, 1.62, 2.58};

  std::vector<double> myvector;

  myvector.assign(mydoubles,mydoubles+8);

  std::cout << "using default comparison:";
  std::stable_sort (myvector.begin(), myvector.end());
  for (std::vector<double>::iterator it=myvector.begin(); it!=myvector.end(); ++it)
    std::cout << ' ' << *it;
  std::cout << '\n';

  myvector.assign(mydoubles,mydoubles+8);

  std::cout << "using 'compare_as_ints' :";
  std::stable_sort (myvector.begin(), myvector.end(), compare_as_ints);
  for (std::vector<double>::iterator it=myvector.begin(); it!=myvector.end(); ++it)
    std::cout << ' ' << *it;
  std::cout << '\n';

  return 0;
}

compare_as_ints is a function that compares only the integral part of the elements, therefore, elements with the same integral part are considered equivalent. stable_sort preserves the relative order these had before the call.

Possible output:


using default comparison: 1.32 1.41 1.62 1.73 2.58 2.72 3.14 4.67
using 'compare_as_ints' : 1.41 1.73 1.32 1.62 2.72 2.58 3.14 4.67


Complexity If enough extra memory is available, linearithmic in the distance between first and last: Performs up to N*log2(N) element comparisons (where N is this distance), and up to that many element moves.
Otherwise, polyloglinear in that distance: Performs up to N*log22(N) element comparisons, and up to that many element swaps.

Data races The objects in the range [first,last) are modified.

Exceptions Throws if any of the element comparisons, the element swaps (or moves) or the operations on iterators throws.
Note that invalid arguments cause undefined behavior.

See also
sort
Sort elements in range (function template)
partial_sort
Partially sort elements in range (function template)
search
Search range for subsequence (function template)
reverse
Reverse range (function template)

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