```
______________________________________________________________________

25   Algorithms library                     [lib.algorithms]

______________________________________________________________________

1 This clause describes components that C++ programs may use to  perform
algorithmic  operations  on  containers  (_lib.containers_)  and other
sequences.

2 The  following  subclauses  describe  components   for   non-modifying
sequence operation, modifying sequence operations, sorting and related
operations, and algorithms from the ISO C library,  as  summarized  in
Table 1:

Table 1--Algorithms library summary

+--------------------------------------------------------------------------+
+--------------------------------------------------------------------------+
|_lib.alg.nonmodifying_ Non-modifying sequence operations                  |
|_lib.alg.modifying.operations_ Mutating sequence operations   <algorithm> |
|_lib.alg.sorting_ Sorting and related operations                          |
+--------------------------------------------------------------------------+
|_lib.alg.c.library_ C library algorithms                      <cstdlib>   |
+--------------------------------------------------------------------------+

namespace std {
// subclause _lib.alg.nonmodifying_, non-modifying sequence operations:
template<class InputIterator, class Function>
Function for_each(InputIterator first, InputIterator last, Function f);
template<class InputIterator, class T>
InputIterator find(InputIterator first, InputIterator last, const T& value);
template<class InputIterator, class Predicate>
InputIterator find_if(InputIterator first, InputIterator last,
Predicate pred);
template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);
template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);
template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1
find_first_of(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);
template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1
find_first_of(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);
template<class ForwardIterator>
template<class ForwardIterator, class BinaryPredicate>
BinaryPredicate pred);
template<class InputIterator, class T, class Size>
void count(InputIterator first, InputIterator last, const T& value,
Size& n);
template<class InputIterator, class Predicate, class Size>
void count_if(InputIterator first, InputIterator last, Predicate pred,
Size& n);
template<class InputIterator1, class InputIterator2>
pair<InputIterator1, InputIterator2>
mismatch(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2);
template<class InputIterator1, class InputIterator2, class BinaryPredicate>
pair<InputIterator1, InputIterator2>
mismatch(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, BinaryPredicate pred);
template<class InputIterator1, class InputIterator2>
bool equal(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2);
template<class InputIterator1, class InputIterator2, class BinaryPredicate>
bool equal(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, BinaryPredicate pred);

template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1 search(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);
template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1 search(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);
template<class ForwardIterator, class Size, class T>
ForwardIterator  search_n(ForwardIterator first, ForwardIterator last,
Size count, const T& value);
template<class ForwardIterator, class Size, class T, class BinaryPredicate>
ForwardIterator1 search_n(ForwardIterator first, ForwardIterator last,
Size count, const T& value,
BinaryPredicate pred);
// subclause _lib.alg.modifying.operations_, modifying sequence operations:
// _lib.alg.copy_, copy:
template<class InputIterator, class OutputIterator>
OutputIterator copy(InputIterator first, InputIterator last,
OutputIterator result);
template<class BidirectionalIterator1, class BidirectionalIterator2>
BidirectionalIterator2
copy_backward(BidirectionalIterator1 first, BidirectionalIterator1 last,
BidirectionalIterator2 result);
// _lib.alg.swap_, swap:
template<class T> void swap(T& a, T& b);
template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator2 swap_ranges(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2);
template<class ForwardIterator1, class ForwardIterator2>
void iter_swap(ForwardIterator1 a, ForwardIterator2 b);
template<class InputIterator, class OutputIterator, class UnaryOperation>
OutputIterator transform(InputIterator first, InputIterator last,
OutputIterator result, UnaryOperation op);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class BinaryOperation>
OutputIterator transform(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, OutputIterator result,
BinaryOperation binary_op);
template<class ForwardIterator, class T>
void replace(ForwardIterator first, ForwardIterator last,
const T& old_value, const T& new_value);
template<class ForwardIterator, class Predicate, class T>
void replace_if(ForwardIterator first, ForwardIterator last,
Predicate pred, const T& new_value);
template<class InputIterator, class OutputIterator, class T>
OutputIterator replace_copy(InputIterator first, InputIterator last,
OutputIterator result,
const T& old_value, const T& new_value);
template<class Iterator, class OutputIterator, class Predicate, class T>
OutputIterator replace_copy_if(Iterator first, Iterator last,
OutputIterator result,
Predicate pred, const T& new_value);

template<class ForwardIterator, class T>
void fill(ForwardIterator first, ForwardIterator last, const T& value);
template<class OutputIterator, class Size, class T>
void fill_n(OutputIterator first, Size n, const T& value);
template<class ForwardIterator, class Generator>
void generate(ForwardIterator first, ForwardIterator last, Generator gen);
template<class OutputIterator, class Size, class Generator>
void generate_n(OutputIterator first, Size n, Generator gen);
template<class ForwardIterator, class T>
ForwardIterator remove(ForwardIterator first, ForwardIterator last,
const T& value);
template<class ForwardIterator, class Predicate>
ForwardIterator remove_if(ForwardIterator first, ForwardIterator last,
Predicate pred);
template<class InputIterator, class OutputIterator, class T>
OutputIterator remove_copy(InputIterator first, InputIterator last,
OutputIterator result, const T& value);
template<class InputIterator, class OutputIterator, class Predicate>
OutputIterator remove_copy_if(InputIterator first, InputIterator last,
OutputIterator result, Predicate pred);
template<class ForwardIterator>
ForwardIterator unique(ForwardIterator first, ForwardIterator last);
template<class ForwardIterator, class BinaryPredicate>
ForwardIterator unique(ForwardIterator first, ForwardIterator last,
BinaryPredicate pred);
template<class InputIterator, class OutputIterator>
OutputIterator unique_copy(InputIterator first, InputIterator last,
OutputIterator result);
template<class InputIterator, class OutputIterator, class BinaryPredicate>
OutputIterator unique_copy(InputIterator first, InputIterator last,
OutputIterator result, BinaryPredicate pred);
template<class BidirectionalIterator>
void reverse(BidirectionalIterator first, BidirectionalIterator last);
template<class BidirectionalIterator, class OutputIterator>
OutputIterator reverse_copy(BidirectionalIterator first,
BidirectionalIterator last,
OutputIterator result);
template<class ForwardIterator>
void rotate(ForwardIterator first, ForwardIterator middle,
ForwardIterator last);
template<class ForwardIterator, class OutputIterator>
OutputIterator rotate_copy(ForwardIterator first, ForwardIterator middle,
ForwardIterator last, OutputIterator result);
template<class RandomAccessIterator>
void random_shuffle(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class RandomNumberGenerator>
void random_shuffle(RandomAccessIterator first, RandomAccessIterator last,
RandomNumberGenerator& rand);

// _lib.alg.partitions_, partitions:
template<class BidirectionalIterator, class Predicate>
BidirectionalIterator partition(BidirectionalIterator first,
BidirectionalIterator last,
Predicate pred);
template<class BidirectionalIterator, class Predicate>
BidirectionalIterator stable_partition(BidirectionalIterator first,
BidirectionalIterator last,
Predicate pred);
// subclause _lib.alg.sorting_, sorting and related operations:
// _lib.alg.sort_, sorting:
template<class RandomAccessIterator>
void sort(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void sort(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);
template<class RandomAccessIterator>
void stable_sort(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void stable_sort(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);
template<class RandomAccessIterator>
void partial_sort(RandomAccessIterator first, RandomAccessIterator middle,
RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void partial_sort(RandomAccessIterator first, RandomAccessIterator middle,
RandomAccessIterator last, Compare comp);
template<class InputIterator, class RandomAccessIterator>
RandomAccessIterator
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last);
template<class InputIterator, class RandomAccessIterator, class Compare>
RandomAccessIterator
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last,
Compare comp);
template<class RandomAccessIterator>
void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
RandomAccessIterator last, Compare comp);
// _lib.alg.binary.search_, binary search:
template<class ForwardIterator, class T>
ForwardIterator lower_bound(ForwardIterator first, ForwardIterator last,
const T& value);
template<class ForwardIterator, class T, class Compare>
ForwardIterator lower_bound(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);

template<class ForwardIterator, class T>
ForwardIterator upper_bound(ForwardIterator first, ForwardIterator last,
const T& value);
template<class ForwardIterator, class T, class Compare>
ForwardIterator upper_bound(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);
template<class ForwardIterator, class T>
pair<ForwardIterator, ForwardIterator>
equal_range(ForwardIterator first, ForwardIterator last, const T& value);
template<class ForwardIterator, class T, class Compare>
pair<ForwardIterator, ForwardIterator>
equal_range(ForwardIterator first, ForwardIterator last, const T& value,
Compare comp);
template<class ForwardIterator, class T>
bool binary_search(ForwardIterator first, ForwardIterator last,
const T& value);
template<class ForwardIterator, class T, class Compare>
bool binary_search(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);
// _lib.alg.merge_, merge:
template<class InputIterator1, class InputIterator2, class OutputIterator>
OutputIterator merge(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class Compare>
OutputIterator merge(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);
template<class BidirectionalIterator>
void inplace_merge(BidirectionalIterator first,
BidirectionalIterator middle,
BidirectionalIterator last);
template<class BidirectionalIterator, class Compare>
void inplace_merge(BidirectionalIterator first,
BidirectionalIterator middle,
BidirectionalIterator last, Compare comp);
// _lib.alg.set.operations_, set operations:
template<class InputIterator1, class InputIterator2>
bool includes(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2);
template<class InputIterator1, class InputIterator2, class Compare>
bool includes(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2, Compare comp);
template<class InputIterator1, class InputIterator2, class OutputIterator>
OutputIterator set_union(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class Compare>
OutputIterator set_union(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

template<class InputIterator1, class InputIterator2, class OutputIterator>
OutputIterator set_intersection(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class Compare>
OutputIterator set_intersection(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);
template<class InputIterator1, class InputIterator2, class OutputIterator>
OutputIterator set_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class Compare>
OutputIterator set_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);
template<class InputIterator1, class InputIterator2, class OutputIterator>
OutputIterator
set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);
template<class InputIterator1, class InputIterator2, class OutputIterator,
class Compare>
OutputIterator
set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);
// _lib.alg.heap.operations_, heap operations:
template<class RandomAccessIterator>
void push_heap(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void push_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);
template<class RandomAccessIterator>
void pop_heap(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void pop_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);
template<class RandomAccessIterator>
void make_heap(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void make_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);
template<class RandomAccessIterator>
void sort_heap(RandomAccessIterator first, RandomAccessIterator last);
template<class RandomAccessIterator, class Compare>
void sort_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

// _lib.alg.min.max_, minimum and maximum:
template<class T> const T& min(const T& a, const T& b);
template<class T, class Compare>
const T& min(const T& a, const T& b, Compare comp);
template<class T> const T& max(const T& a, const T& b);
template<class T, class Compare>
const T& max(const T& a, const T& b, Compare comp);
template<class ForwardIterator>
ForwardIterator min_element(ForwardIterator first, ForwardIterator last);
template<class ForwardIterator, class Compare>
ForwardIterator min_element(ForwardIterator first, ForwardIterator last,
Compare comp);
template<class ForwardIterator>
ForwardIterator max_element(ForwardIterator first, ForwardIterator last);
template<class ForwardIterator, class Compare>
ForwardIterator max_element(ForwardIterator first, ForwardIterator last,
Compare comp);
template<class InputIterator1, class InputIterator2>
bool lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2);
template<class InputIterator1, class InputIterator2, class Compare>
bool lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
Compare comp);
// _lib.alg.permutation.generators_, permutations
template<class BidirectionalIterator>
bool next_permutation(BidirectionalIterator first,
BidirectionalIterator last);
template<class BidirectionalIterator, class Compare>
bool next_permutation(BidirectionalIterator first,
BidirectionalIterator last, Compare comp);
template<class BidirectionalIterator>
bool prev_permutation(BidirectionalIterator first,
BidirectionalIterator last);
template<class BidirectionalIterator, class Compare>
bool prev_permutation(BidirectionalIterator first,
BidirectionalIterator last, Compare comp);
}

3 All  of  the  algorithms are separated from the particular implementa­
tions of data structures and  are  parameterized  by  iterator  types.
Because  of  this, they can work with program-defined data structures,
as long as these data structures have iterator  types  satisfying  the
assumptions on the algorithms.

4 Both   in-place   and   copying  versions  are  provided  for  certain
algorithms.1) When such a version is  provided  for  algorithm  it  is
_________________________
1) The decision whether to include a copying version was usually based
on  complexity  considerations.   When the cost of doing the operation
dominates the cost of copy, the copying version is not included.   For
example,  sort_copy  is not included since the cost of sorting is much
more significant, and users might as well do copy followed by sort.

called  algorithm_copy.   Algorithms that take predicates end with the
suffix _if (which follows the suffix _copy).

5 The Predicate parameter is used whenever an algorithm expects a  func­
tion  object that when applied to the result of dereferencing the cor­
responding iterator returns a value testable as true.  In other words,
if  an algorithm takes Predicate pred as its argument and first as its
iterator argument, it  should  work  correctly  in  the  construct  if
(pred(*first)){...}.  The function object pred is assumed not to apply
any non-constant function through the dereferenced iterator.

6 The BinaryPredicate parameter is used whenever an algorithm expects  a
function  object  that when applied to the result of dereferencing two
corresponding iterators or to dereferencing an  iterator  and  type  T
when  T is part of the signature returns a value testable as true.  In
other words, if an algorithm takes BinaryPredicate binary_pred as  its
argument  and  first1  and first2 as its iterator arguments, it should
work correctly  in  the  construct  if  (pred(*first,  *first2)){...}.
BinaryPredicate  always  takes  the  first  iterator type as its first
argument, that is, in those cases when T value is part of  the  signa­
ture,  it  should  work  correctly  in the context of if (pred(*first,
value)){...}.  binary_pred shall not apply any  non-constant  function
through the dereferenced iterators.

7 In  the  description  of the algorithms operators + and - are used for
some of the iterator categories for which  they  do  not  have  to  be
defined.  In these cases the semantics of a+n is the same is that of
{ X tmp = a;
return tmp;
}
and that of a-b is the same as of
{ Distance n;
distance(a, b, n);
return n;
}

+-------                 BEGIN BOX 1                -------+
For  the  following  algorithms: reverse, rotate, random_shuffle, sta­
ble_partition,  sort,  stable_sort  and  inplace_merge  the   iterator
requirement can be relaxed to ForwardIterator.  These algorithms could
then be dispatched upon the iterator category tags  to  use  the  most
efficient  implementation  for  each  iterator  category.  We have not
included the relaxation at this stage since it is not yet fully imple­
mented.
+-------                  END BOX 1                 -------+

25.1  Non-modifying sequence operations         [lib.alg.nonmodifying]

25.1.1  For each                                     [lib.alg.foreach]

template<class InputIterator, class Function>
Function for_each(InputIterator first, InputIterator last, Function f);

Effects:
Applies f to the result of dereferencing every iterator in the range
[first, last).
Requires:
f shall not apply any non-constant function through the dereferenced
iterator.
Returns:
f.
Complexity:
Applies f exactly last - first times.
Notes:
If f returns a result, the result is ignored.

25.1.2  Find                                            [lib.alg.find]

template<class InputIterator, class T>
InputIterator find(InputIterator first, InputIterator last,
const T& value);

template<class InputIterator, class Predicate>
InputIterator find_if(InputIterator first, InputIterator last,
Predicate pred);

Requires:
Type T is EqualityComparable (_lib.equalitycomparable_).
Returns:
The  first  iterator i in the range [first, last) for which the fol­
lowing corresponding conditions hold: *i == value, pred(*i) == true.
Returns last if no such iterator is found.
Complexity:
At most last - first applications of the corresponding predicate.

25.1.3  Find End                                    [lib.alg.find.end]

template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);

template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);

Effects:
Finds a subsequence of equal values in a sequence.
Returns:
The  last  iterator  i in the range [first1, last1 - (last2-first2))
such that for any non-negative integer n < (last2-first2), the  fol­
lowing   corresponding   conditions  hold:  *(i+n)  ==  *(first2+n),
pred(*(i+n),*(first2+n)) == true.  Returns last1 if no such iterator
is found.
Complexity:
At  most  (last2  -  first2)  *  (last1 - first1-(last2 - first2)+1)
applications of the corresponding predicate.

25.1.4  Find First                             [lib.alg.find.first.of]

template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1
find_first_of(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);

template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1
find_first_of(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);

Effects:
Finds an element that matches one of a set of values.
Returns:
The first iterator i in the range [first1, last1) such that for some
integer  j  in  the  range  [first2, last2) the following conditions
hold: *i == *j, pred(*i,*j) == true.  Returns last1 if no such iter­
ator is found.
Complexity:
At  most  (last1-first1) * (last2-first2) applications of the corre­
sponding predicate.

+-------                 BEGIN BOX 2                -------+
Should these be ForwardIterator?
+-------                  END BOX 2                 -------+

template<class ForwardIterator>

template<class ForwardIterator, class BinaryPredicate>
BinaryPredicate pred);

Returns:
The first iterator i such that both i and i + 1  are  in  the  range
[first, last) for which the following corresponding conditions hold:
*i == *(i + 1), pred(*i, *(i + 1)) == true.  Returns last if no such
iterator is found.
Complexity:
Exactly  find(first, last, value) - first applications of the corre­
sponding predicate.

25.1.6  Count                                          [lib.alg.count]

template<class InputIterator, class T>
iterator_trait<InputIterator>::distance_type
count(InputIterator first, InputIterator last, const T& value);

template<class InputIterator, class Predicate>
iterator_trait<InputIterator>::distance_type
count_if(InputIterator first, InputIterator last, Predicate pred);

Requires:
Type T is EqualityComparable (_lib.equalitycomparable_) .
Effects:
Returns the number of iterators i in the  range  [first,  last)  for
which  the  following  corresponding  conditions  hold: *i == value,
pred(*i) == true.
Complexity:
Exactly last - first applications of the corresponding predicate.

25.1.7  Mismatch                                        [lib.mismatch]

template<class InputIterator1, class InputIterator2>
pair<InputIterator1, InputIterator2>
mismatch(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2);

template<class InputIterator1, class InputIterator2,
class BinaryPredicate>
pair<InputIterator1, InputIterator2>
mismatch(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, BinaryPredicate pred);

Returns:
A pair of iterators i and j such that j == first2 + (i - first1) and
i  is  the first iterator in the range [first1, last1) for which the
following corresponding conditions hold:
!(*i == *(first2 + (i - first1))), pred(*i, *(first2 + (i - first1))) == false
Returns the pair last1 and first2 + (last1  -  first1)  if  such  an
Complexity:
At  most last1 - first1 applications of the corresponding predicate.

25.1.8  Equal                                          [lib.alg.equal]

template<class InputIterator1, class InputIterator2>
bool equal(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2);

template<class InputIterator1, class InputIterator2,
class BinaryPredicate>
bool equal(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, BinaryPredicate pred);

Returns:
true if for every iterator i in the range [first1, last1)  the  fol­
lowing  corresponding  conditions  hold:  *i  ==  *(first2  +  (i  -
first1)), pred(*i, *(first2 + (i - first1)))  ==  true.   Otherwise,
returns false.
Complexity:
At  most last1 - first1 applications of the corresponding predicate.

25.1.9  Search                                        [lib.alg.search]

template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator1
search(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);

template<class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
ForwardIterator1
search(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate pred);

Requires:
Type T is EqualityComparable (_lib.equalitycomparable_),  type  Size
is convertible to integral type (_conv.integral_, _class.conv_).
Effects:
Finds a subsequence of equal values in a sequence.
Returns:
The first iterator i in the range [first1, last1 - (last2 - first2))
such that for any non-negative integer n less than  last2  -  first2
the  following corresponding conditions hold: *(i + n) == *(first2 +
n), pred(*(i + n), *(first2 + n)) == true.  Returns last1 if no such
iterator is found.
Complexity:
At most (last1 - first1) * (last2 - first2) applications of the cor­
responding predicate.

template<class ForwardIterator, class Size, class T>
ForwardIterator
search_n(ForwardIterator first, ForwardIterator last, Size count,
const T& value);

template<class ForwardIterator, class Size, class T,
class BinaryPredicate>
ForwardIterator
search_n(ForwardIterator first, ForwardIterator last, Size count,
const T& value, BinaryPredicate pred);

Effects:
Finds a subsequence of equal values in a sequence.
Returns:
The first iterator i in the range [first, last -  count)  such  that
for  any non-negative integer n less than count the following corre­
sponding conditions hold: *(i + n) == value, pred(*(i + n),value) ==
true.  Returns last if no such iterator is found.
Complexity:
At  most  (last1 - first1) * count applications of the corresponding
predicate.

25.2  Mutating sequence operations      [lib.alg.modifying.operations]

25.2.1  Copy                                            [lib.alg.copy]

template<class InputIterator, class OutputIterator>
OutputIterator copy(InputIterator first, InputIterator last,
OutputIterator result);

Effects:
Copies  elements.  For each non-negative integer n < (last - first),
performs *(result + n) = *(first + n).
Returns:
result + (last - first).
Requires:
result shall not be in the range [first, last).
Complexity:
Exactly last - first assignments.

template<class BidirectionalIterator1, class BidirectionalIterator2>
BidirectionalIterator2
copy_backward(BidirectionalIterator1 first,
BidirectionalIterator1 last,
BidirectionalIterator2 result);

Effects:
Copies elements in the range [first, last) into the range [result  -
(last  -  first),  result)  starting from last - 1 and proceeding to
first  .  2) For each positive integer n <= (last - first), Performs
_________________________

*(result - n) = *(last - n).
Requires:
result shall not be in the range [first, last).
Returns:
result - (last - first).
Complexity:
Exactly last - first assignments.

25.2.2  Swap                                            [lib.alg.swap]

template<class T> void swap(T& a, T& b);

Requires:
Type T is Assignable (_lib.container.requirements_).
Effects:
Exchanges values stored in two locations.

template<class ForwardIterator1, class ForwardIterator2>
ForwardIterator2
swap_ranges(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2);

Effects:
For each  non-negative  integer  n  <  (last1  -  first1)  performs:
swap(*(first1 + n), *(first2 + n)).
Requires:
The  two  ranges  [first1,  last1)  and  [first2,  first2 + (last1 -
first1)) shall not overlap.
Returns:
first2 + (last1 - first1).
Complexity:
Exactly last1 - first1 swaps.

template<class ForwardIterator1, class ForwardIterator2>
void iter_swap(ForwardIterator1 a, ForwardIterator2 b);

Effects:
Exchanges the values pointed to by the two iterators a and b.

25.2.3  Transform                                  [lib.alg.transform]

_________________________
2) copy_backward (_lib.copy.backward_) should be used instead of  copy
when last is in the range [result - (last - first), result).

template<class InputIterator, class OutputIterator,
class UnaryOperation>
OutputIterator
transform(InputIterator first, InputIterator last,
OutputIterator result, UnaryOperation op);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class BinaryOperation>
OutputIterator
transform(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, OutputIterator result,
BinaryOperation binary_op);

Effects:
Assigns through every iterator i in  the  range  [result,  result  +
(last1  -  first1)) a new corresponding value equal to op(*(first1 +
(i - result)) or binary_op(*(first1 + (i - result), *(first2 + (i  -
result))).
Requires:
op and binary_op shall not have any side effects.
Returns:
result + (last1 - first1).
Complexity:
Exactly last1 - first1 applications of op or binary_op
Notes:
result  may  be  equal  to  first  in case of unary transform, or to
first1 or first2 in case of binary transform.

25.2.4  Replace                                      [lib.alg.replace]

template<class ForwardIterator, class T>
void replace(ForwardIterator first, ForwardIterator last,
const T& old_value, const T& new_value);

template<class ForwardIterator, class Predicate, class T>
void replace_if(ForwardIterator first, ForwardIterator last,
Predicate pred, const T& new_value);

Requires:
Type  T  is  Assignable  (_lib.container.requirements_)  (and,   for
replace(), EqualityComparable (_lib.equalitycomparable_).
Effects:
Substitutes elements referred by the iterator i in the range [first,
last) with new_value, when the  following  corresponding  conditions
hold: *i == old_value, pred(*i) == true.
Complexity:
Exactly last - first applications of the corresponding predicate.

template<class InputIterator, class OutputIterator, class T>
OutputIterator
replace_copy(InputIterator first, InputIterator last,
OutputIterator result,
const T& old_value, const T& new_value);

template<class Iterator, class OutputIterator, class Predicate, class T>
OutputIterator
replace_copy_if(Iterator first, Iterator last,
OutputIterator result,
Predicate pred, const T& new_value);

Requires:
Type   T  is  Assignable  (_lib.container.requirements_)  (and,  for
replace_copy(), EqualityComparable (_lib.equalitycomparable_).   The
ranges  [first,  last)  and  [result, result+(last-first)) shall not
overlap.
Effects:
Assigns to every iterator i in the range [result, result +  (last  -
first))  either  new_value  or  *(first + (i - result)) depending on
whether the following corresponding conditions hold:
*(first + (i - result)) == old_value, pred(*(first + (i -  result)))
== true.
Returns:
result + (last - first).
Complexity:
Exactly last - first applications of the corresponding predicate.

25.2.5  Fill                                            [lib.alg.fill]

template<class ForwardIterator, class T>
void fill(ForwardIterator first, ForwardIterator last, const T& value);

template<class OutputIterator, class Size, class T>
void fill_n(OutputIterator first, Size n, const T& value);

Requires:
Type  T  is  Assignable (_lib.container.requirements_), Size is con­
vertible to an integral type (_conv.integral_, _class.conv_).
Effects:
Assigns value through all the iterators in the range [first, last)or
[first, first + n).
Complexity:
Exactly last - first (or n) assignments.

25.2.6  Generate                                    [lib.alg.generate]

template<class ForwardIterator, class Generator>
void generate(ForwardIterator first, ForwardIterator last,
Generator gen);

template<class OutputIterator, class Size, class Generator>
void generate_n(OutputIterator first, Size n, Generator gen);

Effects:
Invokes  the function object gen and assigns the return value of gen
though all the iterators in the range [first, last) or [first, first
+ n).
Requires:
gen  takes  no  arguments,  Size  is convertible to an integral type
(_conv.integral_, _class.conv_).
Complexity:
Exactly last - first (or n) invocations of gen and assignments.

25.2.7  Remove                                        [lib.alg.remove]

template<class ForwardIterator, class T>
ForwardIterator remove(ForwardIterator first, ForwardIterator last,
const T& value);

template<class ForwardIterator, class Predicate>
ForwardIterator remove_if(ForwardIterator first, ForwardIterator last,
Predicate pred);

Requires:
Type T is EqualityComparable (_lib.equalitycomparable_).
Effects:
Eliminates all the elements referred to by iterator i in  the  range
[first, last) for which the following corresponding conditions hold:
*i == value, pred(*i) == true.
Returns:
The end of the resulting range.
Notes:
Stable:  the relative order of the elements that are not removed  is
the same as their relative order in the original range.
Complexity:
Exactly last - first applications of the corresponding predicate.

template<class InputIterator, class OutputIterator, class T>
OutputIterator
remove_copy(InputIterator first, InputIterator last,
OutputIterator result, const T& value);

template<class InputIterator, class OutputIterator, class Predicate>
OutputIterator
remove_copy_if(InputIterator first, InputIterator last,
OutputIterator result, Predicate pred);

Requires:
Type T is EqualityComparable (_lib.equalitycomparable_).  The ranges
[first, last) and [result, result+(last-first)) shall not overlap.
Effects:
Copies all the elements referred to by the iterator i in  the  range
[first,  last)  for  which the following corresponding conditions do
not hold: *i == value, pred(*i) == true.
Returns:
The end of the resulting range.
Complexity:
Exactly last - first applications of the corresponding predicate.
Notes:
Stable:  the relative order of the elements in the  resulting  range
is the same as their relative order in the original range.

25.2.8  Unique                                        [lib.alg.unique]

template<class ForwardIterator>
ForwardIterator unique(ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class BinaryPredicate>
ForwardIterator unique(ForwardIterator first, ForwardIterator last,
BinaryPredicate pred);

Effects:
Eliminates all but the first element from every consecutive group of
equal elements referred to by the iterator i in  the  range  [first,
last)  for  which the following corresponding conditions hold: *i ==
*(i - 1) or pred(*i, *(i - 1)) == true
Returns:
The end of the resulting range.
Complexity:
Exactly (last - first) - 1 applications of the corresponding  predi­
cate.

template<class InputIterator, class OutputIterator>
OutputIterator
unique_copy(InputIterator first, InputIterator last,
OutputIterator result);

template<class InputIterator, class OutputIterator,
class BinaryPredicate>
OutputIterator
unique_copy(InputIterator first, InputIterator last,
OutputIterator result, BinaryPredicate pred);

Requires:
The ranges [first, last) and [result, result+(last-first)) shall not
overlap.
Effects:
Copies only the first element from every consecutive group of  equal
elements  referred  to  by the iterator i in the range [first, last)

for which the following corresponding conditions hold: *i ==  *(i  -
1) or pred(*i, *(i - 1)) == true
Returns:
The end of the resulting range.
Complexity:
Exactly last - first applications of the corresponding predicate.

25.2.9  Reverse                                      [lib.alg.reverse]

template<class BidirectionalIterator>
void reverse(BidirectionalIterator first, BidirectionalIterator last);

Effects:
For each non-negative integer i <= (last - first)/2, applies swap to
all pairs of iterators first + i, (last - i) - 1.
Complexity:
Exactly (last - first)/2 swaps.

template<class BidirectionalIterator, class OutputIterator>
OutputIterator
reverse_copy(BidirectionalIterator first,
BidirectionalIterator last, OutputIterator result);

Effects:
Copies the range [first, last) to the range [result, result +  (last
-  first)) such that for any non-negative integer i < (last - first)
the following assignment takes place:

*(result + (last - first) - i) = *(first + i)

Requires:
The ranges [first, last) and [result, result + (last - first)) shall
not overlap.
Returns:
result + (last - first).
Complexity:
Exactly last - first assignments.

25.2.10  Rotate                                       [lib.alg.rotate]

template<class ForwardIterator>
void rotate(ForwardIterator first, ForwardIterator middle,
ForwardIterator last);

Effects:
For each non-negative integer i < (last - first), places the element
from the position first + i into position first + (i + (last -  mid­
dle)) % (last - first).
Notes:
This is a left rotate.

Requires:
[first, middle) and [middle, last) are valid ranges.
Complexity:
At most last - first swaps.

template<class ForwardIterator, class OutputIterator>
OutputIterator
rotate_copy(ForwardIterator first, ForwardIterator middle,
ForwardIterator last, OutputIterator result);

Effects:
Copies  the range [first, last) to the range [result, result + (last
- first)) such that for each non-negative integer i < (last - first)
the following assignment takes place:
*(first + i) =  *(result + (i + (middle - first)) % (last - first))

+-------                 BEGIN BOX 3                -------+
Should this be:
*(result + i) =  *(first + (i + (middle - first)) % (last - first))
+-------                  END BOX 3                 -------+

Returns:
result + (last - first).
Requires
The ranges [first, last) and [result, result + (last - first)) shall
not overlap.
Complexity:
Exactly last - first assignments.

25.2.11  Random shuffle                       [lib.alg.random.shuffle]

template<class RandomAccessIterator>
void random_shuffle(RandomAccessIterator first,
RandomAccessIterator last);

template<class RandomAccessIterator, class RandomNumberGenerator>
void random_shuffle(RandomAccessIterator first,
RandomAccessIterator last,
RandomNumberGenerator& rand);

Effects:
Shuffles the elements in the range [first, last) with  uniform  dis­
tribution.
Complexity:
Exactly (last - first) - 1 swaps.
Notes:
random_shuffle()  can  take  a  particular  random number generating
function object rand such that rand(n) (where n is a positive  argu­
ment of type RandomAccessIterator::distance) returns a randomly cho­
sen value of type RandomAccessIterator::distance)  in  the  interval
[0, n).

+-------                 BEGIN BOX 4                -------+
Can  it  accept  an  argument  that  yields  a  result of a type that,
although different from RandomAccessIterator::distance,  can  be  con­
verted to it?
+-------                  END BOX 4                 -------+

25.2.12  Partitions                               [lib.alg.partitions]

template<class BidirectionalIterator, class Predicate>
BidirectionalIterator
partition(BidirectionalIterator first,
BidirectionalIterator last, Predicate pred);

Effects:
Places all the elements in the range [first, last) that satisfy pred
before all the elements that do not satisfy it.
Returns:
An iterator i such that for any iterator j in the range [first,  i),
pred(*j)  ==  true,  and  for any iterator k in the range [i, last),
pred(*j) == false.
Complexity:
At most (last - first)/2 swaps.  Exactly last -  first  applications
of the predicate is done.

template<class BidirectionalIterator, class Predicate>
BidirectionalIterator
stable_partition(BidirectionalIterator first,
BidirectionalIterator last, Predicate pred);

Effects:
Places all the elements in the range [first, last) that satisfy pred
before all the elements that do not satisfy it.
Returns:
An iterator i such that for any iterator j in the range [first,  i),
pred(*j)  ==  true,  and  for any iterator k in the range [i, last),
pred(*j) == false.  The relative  order  of  the  elements  in  both
groups is preserved.
Complexity:
At  most  (last  - first) * log(last - first) swaps, but only linear
number of swaps if there is enough extra  memory.   Exactly  last  -
first applications of the predicate.

25.3  Sorting and related operations                 [lib.alg.sorting]

1 All  the operations in clause _lib.alg.sorting_ have two versions: one
that takes a function object of type Compare  and  one  that  uses  an
operator<.

2 Compare  is  used as a function object which returns true if the first
argument is less than the second, and false otherwise.   Compare  comp

is  used  throughout for algorithms assuming an ordering relation.  It
is assumed that comp will not apply any non-constant function  through
the dereferenced iterator.

3 For  all  algorithms  that  take Compare, there is a version that uses
operator< instead.  That is, comp(*i, *j) == true defaults to *i <  *j
==  true.   For the algorithms to work correctly, comp has to induce a
strict weak ordering on the values.

4 The term strict refers to the requirement of an  irreflexive  relation
(!comp(x, x)  for  all  x), and the term weak to requirements that are
not as strong as those for a total ordering, but stronger  than  those
for  a  partial  ordering.  If we define equiv(a, b) as !comp(a, b) &&
!comp(b, a), then the requirements are that comp  and  equiv  both  be
transitive  relations:

--comp(a, b) && comp(b, c) implies comp(a, c)

--equiv(a, b)  &&  equiv(b, c)  implies equiv(a, c) [Note: Under these
conditions, it can be shown that

--equiv is an equivalence relation

--comp induces a well-defined  relation  on  the  equivalence  classes
determined by equiv

--The induced relation is a strict total ordering.   --end note]

5 A  sequence  is  sorted  with  respect to a comparator comp if for any
iterator i pointing to the sequence and  any  non-negative  integer  n
such  that  i  +  n  is a valid iterator pointing to an element of the
sequence, comp(*(i + n), *i) == false.

6 In the descriptions of the functions that deal with ordering relation­
ships  we  frequently use a notion of equivalence to describe concepts
such as stability.  The equivalence to which we refer is not necessar­
ily  an  operator==, but an equivalence relation induced by the strict
weak ordering.  That is, two elements a and b are  considered  equiva­
lent if and only if !(a < b) && !(b < a).

25.3.1  Sorting                                         [lib.alg.sort]

25.3.1.1  sort                                              [lib.sort]

template<class RandomAccessIterator>
void sort(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void sort(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

Effects:
Sorts the elements in the range [first, last).
Complexity:
Approximately  NlogN  (where  N  == last - first) comparisons on the
average.3)

25.3.1.2  stable_sort                                [lib.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);

Effects:
Sorts the elements in the range [first, last).
Complexity:
It does at most N(logN)2 (where N == last - first)  comparisons;  if
enough extra memory is available, it is NlogN.
Notes:
Stable:  the relative order of the equivalent elements is preserved.

25.3.1.3  partial_sort                              [lib.partial.sort]

template<class RandomAccessIterator>
void partial_sort(RandomAccessIterator first,
RandomAccessIterator middle,
RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void partial_sort(RandomAccessIterator first,
RandomAccessIterator middle,
RandomAccessIterator last,
Compare comp);

Effects:
Places the first middle -  first  sorted  elements  from  the  range
[first,  last) into the range [first, middle).  The rest of the ele­
ments in the range [middle,  last)  are  placed  in  an  unspecified
order.
Complexity:
It  takes approximately (last - first) * log(middle - first) compar­
isons.

_________________________
3)  If  the   worst   case   behavior   is   important   stable_sort()
(_lib.stable.sort_)  or  partial_sort() (_lib.partial.sort_) should be
used.

25.3.1.4  partial_sort_copy                    [lib.partial.sort.copy]

template<class InputIterator, class RandomAccessIterator>
RandomAccessIterator
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last);

template<class InputIterator, class RandomAccessIterator,
class Compare>
RandomAccessIterator
partial_sort_copy(InputIterator first, InputIterator last,
RandomAccessIterator result_first,
RandomAccessIterator result_last,
Compare comp);

Effects:
Places the first  min(last  -  first,  result_last  -  result_first)
sorted   elements  into  the  range  [result_first,  result_first  +
min(last - first, result_last - result_first)).
Returns:
The smaller of: result_last or result_first + (last - first)
Complexity:
Approximately (last - first) * log(min(last - first,  result_last  -
result_first)) comparisons.

25.3.2  Nth element                              [lib.alg.nth.element]

template<class RandomAccessIterator>
void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
RandomAccessIterator last,  Compare comp);

1 After nth_element the element in the position pointed to by nth is the
element that would be in that position if the whole range were sorted.
Also  for  any iterator i in the range [first, nth) and any iterator j
in the range [nth, last) it holds that: !(*i > *j) or comp(*j, *i)  ==
false.
Complexity:
Linear on average.

25.3.3  Binary search                          [lib.alg.binary.search]

1 All  of  the  algorithms in this section are versions of binary search
and assume that the sequence being searched is in order  according  to
the  implied or explicit comparison function.  They work on non-random
access iterators minimizing the number of comparisons, which  will  be

logarithmic for all types of iterators.  They are especially appropri­
ate for random access iterators, since these algorithms do a logarith­
mic number of steps through the data structure.  For non-random access
iterators they execute a linear number of steps.

25.3.3.1  lower_bound                                [lib.lower.bound]

template<class ForwardIterator, class T>
ForwardIterator
lower_bound(ForwardIterator first, ForwardIterator last,
const T& value);

template<class ForwardIterator, class T, class Compare>
ForwardIterator
lower_bound(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_).
Effects:
Finds the first position into which value can  be  inserted  without
violating the ordering.
Returns:
The  furthermost iterator i in the range [first, last) such that for
any iterator j in the range [first, i) the  following  corresponding
conditions hold: *j < value or comp(*j, value) == true

+-------                 BEGIN BOX 5                -------+
Should the range of i be changed to: [first, last]?
+-------                  END BOX 5                 -------+

Complexity:
At most log(last - first) + 1 comparisons.

25.3.3.2  upper_bound                                [lib.upper.bound]

template<class ForwardIterator, class T>
ForwardIterator
upper_bound(ForwardIterator first, ForwardIterator last,
const T& value);

template<class ForwardIterator, class T, class Compare>
ForwardIterator
upper_bound(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_).
Effects:
Finds  the  furthermost  position  into  which value can be inserted
without violating the ordering.

Returns:
The furthermost iterator i in the range [first, last) such that  for
any  iterator  j in the range [first, i) the following corresponding
conditions hold: !(value < *j) or comp(value, *j) == false

+-------                 BEGIN BOX 6                -------+
Should the range of i be changed to: [first, last]?
+-------                  END BOX 6                 -------+

Complexity:
At most log(last - first) + 1 comparisons.

25.3.3.3  equal_range                                [lib.equal.range]

template<class ForwardIterator, class T>
pair<ForwardIterator, ForwardIterator>
equal_range(ForwardIterator first,
ForwardIterator last, const T& value);

template<class ForwardIterator, class T, class Compare>
pair<ForwardIterator, ForwardIterator>
equal_range(ForwardIterator first,
ForwardIterator last, const T& value,
Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_).
Effects:
Finds the largest subrange  [i,  j)  such  that  the  value  can  be
inserted  at any iterator k in it without violating the ordering.  k
satisfies the corresponding conditions: !(*k < value) &&  !(value  <
*k) or comp(*k, value) == false && comp(value, *k) == false.
Complexity:
At most 2 * log(last - first) + 1 comparisons.

25.3.3.4  binary_search                            [lib.binary.search]

template<class ForwardIterator, class T>
bool binary_search(ForwardIterator first, ForwardIterator last,
const T& value);

template<class ForwardIterator, class T, class Compare>
bool binary_search(ForwardIterator first, ForwardIterator last,
const T& value, Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_).
Returns:
true if there is an iterator i in the range [first last) that satis­
fies the corresponding conditions: !(*i < value) && !(value < *i) or
comp(*i, value) == false && comp(value, *i) == false.

Complexity:
At most log(last - first) + 2 comparisons.

25.3.4  Merge                                          [lib.alg.merge]

template<class InputIterator1, class InputIterator2,
class OutputIterator>
OutputIterator
merge(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class Compare>
OutputIterator
merge(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

Effects:
Merges  two  sorted  ranges [first1, last1) and [first2, last2) into
the range [result, result + (last1 - first1) + (last2 - first2)).

1 The resulting range shall not overlap  with  either  of  the  original
ranges.
Returns:
result + (last1 - first1) + (last2 - first2).
Complexity:
At most (last1 - first1) + (last2 - first2) - 1 comparisons.
Notes:
Stable:   for  equivalent  elements  in the two ranges, the elements
from the first range always precede the elements from the second.

template<class BidirectionalIterator>
void inplace_merge(BidirectionalIterator first,
BidirectionalIterator middle,
BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
void inplace_merge(BidirectionalIterator first,
BidirectionalIterator middle,
BidirectionalIterator last, Compare comp);

Effects:
Merges two sorted consecutive ranges [first,  middle)  and  [middle,
last), putting the result of the merge into the range [first, last).
Complexity:
When enough additional memory is available, (last - first) - 1  com­
parisons.   If  no additional memory is available, an algorithm with
complexity NlogN (where N is equal to last - first) may be used.

Notes:
Stable:  for equivalent elements in the  two  ranges,  the  elements
from the first range always precede the elements from the second.

25.3.5  Set operations on sorted              [lib.alg.set.operations]
structures

1 This section defines all the basic set  operations  on  sorted  struc­
tures.  They even work with multisets (_lib.multiset_) containing mul­
tiple copies of equivalent elements.  The semantics of the set  opera­
tions  are  generalized  to  multisets  in  a standard way by defining
union() to contain the maximum number of occurrences of every element,
intersection() to contain the minimum, and so on.

25.3.5.1  includes                                      [lib.includes]

template<class InputIterator1, class InputIterator2>
bool includes(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2);

template<class InputIterator1, class InputIterator2, class Compare>
bool includes(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
Compare comp);

Returns:
true  if  every element in the range [first2, last2) is contained in
the range [first1, last1).  Returns false otherwise.
Complexity:
At most 2 * ((last1 - first1) + (last2 - first2)) - 1 comparisons.

25.3.5.2  set_union                                    [lib.set.union]

template<class InputIterator1, class InputIterator2,
class OutputIterator>
OutputIterator
set_union(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class Compare>
OutputIterator
set_union(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

Effects:
Constructs a sorted union of the elements from the two ranges;  that
is,  the  set  of  elements  that  are present in one or both of the
ranges.

Requires:
The resulting range shall not overlap with either  of  the  original
ranges.
Returns:
The end of the constructed range.
Complexity:
At most 2 * ((last1 - first1) + (last2 - first2)) - 1 comparisons.
Notes:
Stable:   if  an element is present in both ranges, the one from the
first range is copied.

25.3.5.3  set_intersection                      [lib.set.intersection]

template<class InputIterator1, class InputIterator2,
class OutputIterator>
OutputIterator
set_intersection(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class Compare>
OutputIterator
set_intersection(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

Effects:
Constructs a sorted  intersection  of  the  elements  from  the  two
ranges; that is, the set of elements that are present in both of the
ranges.
Requires:
The resulting range shall not overlap with either  of  the  original
ranges.
Returns:
The end of the constructed range.
Complexity:
At most 2 * ((last1 - first1) + (last2 - first2)) - 1 comparisons.
Notes:
Stable,  that  is,  if an element is present in both ranges, the one
from the first range is copied.

25.3.5.4  set_difference                          [lib.set.difference]

template<class InputIterator1, class InputIterator2,
class OutputIterator>
OutputIterator
set_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class Compare>
OutputIterator
set_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

Effects:
Copies the elements of the range [first1, last1) which are not  pre­
sent  in the range [first2, last2) to the range beginning at result.
The elements in the constructed range are sorted.
Requires:
The resulting range shall not overlap with either  of  the  original
ranges.
Returns:
The end of the constructed range.
Complexity:
At most 2 * ((last1 - first1) + (last2 - first2)) - 1 comparisons.

25.3.5.5  set_symmetric_difference      [lib.set.symmetric.difference]

template<class InputIterator1, class InputIterator2,
class OutputIterator>
OutputIterator
set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result);

template<class InputIterator1, class InputIterator2,
class OutputIterator, class Compare>
OutputIterator
set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
OutputIterator result, Compare comp);

Effects:
Copies the elements of the range [first1, last1) which are not  pre­
sent  in  the  range  [first2, last2), and the elements of the range
[first2, last2) which are not present in the range  [first1,  last1)
to  the  range beginning at result.  The elements in the constructed
range are sorted.
Requires:
The resulting range shall not overlap with either  of  the  original
ranges.

Returns:
The end of the constructed range.
Complexity:
At most 2 * ((last1 - first1) + (last2 - first2)) - 1 comparisons.

25.3.6  Heap operations                      [lib.alg.heap.operations]

1 A heap is a particular organization of elements in a range between two
random access iterators [a, b).  Its two key properties are:

(1)*a is the largest element in the range and

(2)*a may be  removed  by  pop_heap(),  or  a  new  element  added  by
push_heap(), in O(logN) time.

2 These properties make heaps useful as priority queues.

3 make_heap()  converts a range into a heap and sort_heap() turns a heap
into a sorted sequence.

25.3.6.1  push_heap                                    [lib.push.heap]

template<class RandomAccessIterator>
void push_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void push_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

Requires:
The range [first, last - 1) shall be a valid heap.
Effects:
Places the value in the location last - 1 into  the  resulting  heap
[first, last).
Complexity:
At most log(last - first) comparisons.

25.3.6.2  pop_heap                                      [lib.pop.heap]

template<class RandomAccessIterator>
void pop_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void pop_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

Requires:
The range [first, last) shall be a valid heap.
Effects:
Swaps the value in the location first with the value in the location
last - 1 and makes [first, last - 1) into a heap.

Complexity:
At most 2 * log(last - first) comparisons.

25.3.6.3  make_heap                                    [lib.make.heap]

template<class RandomAccessIterator>
void make_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void make_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

Effects:
Constructs a heap out of the range [first, last).
Complexity:
At most 3 * (last - first) comparisons.

25.3.6.4  sort_heap                                    [lib.sort.heap]

template<class RandomAccessIterator>
void sort_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
void sort_heap(RandomAccessIterator first, RandomAccessIterator last,
Compare comp);

Effects:
Sorts elements in the heap [first, last).
Complexity:
At most NlogN comparisons (where N == last - first).
Notes:
Not stable.

25.3.7  Minimum and maximum                          [lib.alg.min.max]

template<class T> const T& min(const T& a, const T& b);
template<class T, class Compare>
const T& min(const T& a, const T& b, Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_) and CopyCon­
structible (_lib.copyconstructible_).
Returns:
The smaller value.
Notes:
Returns the first argument when the arguments are equivalent.

template<class T> const T& max(const T& a, const T& b);
template<class T, class Compare>
const T& max(const T& a, const T& b, Compare comp);

Requires:
Type T is LessThanComparable (_lib.lessthancomparable_) and CopyCon­
structible (_lib.copyconstructible_).
Returns:
The larger value.
Notes:
Returns the first argument when the arguments are equivalent.

template<class ForwardIterator>
ForwardIterator min_element(ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class Compare>
ForwardIterator min_element(ForwardIterator first, ForwardIterator last,
Compare comp);

Returns:
The first iterator i in the range [first, last) such  that  for  any
iterator  j  in  the range [first, last) the following corresponding
conditions hold: !(*j < *i) or comp(*j, *i) == false
Complexity:
Exactly max((last - first) - 1, 0) applications of the corresponding
comparisons.

template<class ForwardIterator>
ForwardIterator max_element(ForwardIterator first, ForwardIterator last);
template<class ForwardIterator, class Compare>
ForwardIterator max_element(ForwardIterator first, ForwardIterator last,
Compare comp);

Returns:
The  first  iterator  i in the range [first, last) such that for any
iterator j in the range [first, last)  the  following  corresponding
conditions hold: !(*i < *j) or comp(*i, *j) == false.
Complexity:
Exactly max((last - first) - 1, 0) applications of the corresponding
comparisons.

25.3.8  Lexicographical comparison            [lib.alg.lex.comparison]

template<class InputIterator1, class InputIterator2>
bool
lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2);

template<class InputIterator1, class InputIterator2, class Compare>
bool
lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
InputIterator2 first2, InputIterator2 last2,
Compare comp);

Returns:
true if the sequence of  elements  defined  by  the  range  [first1,
last1)  is  lexicographically  less  than  the  sequence of elements
defined by the range [first2, last2).
Returns false otherwise.
Complexity:
At most min((last1 - first1), (last2 - first2)) applications of  the
corresponding comparison.
Notes:
If  two  sequences have the same number of elements and their corre­
sponding elements are equivalent, then neither sequence  is  lexico­
graphically less than the other.  If one sequence is a prefix of the
other, then the shorter sequence is lexicographically less than  the
longer  sequence.   Otherwise, the lexicographical comparison of the
sequences yields the same result as the comparison of the first cor­
responding pair of elements that are not equivalent.
for (i = first1, j = first2;
i != last1 && j != last2 && !(*i < *j) && !(*j < *i);
++i, ++j);
return j == last2 ? false : i == last1 || *i < *j;

25.3.9  Permutation generators        [lib.alg.permutation.generators]

template<class BidirectionalIterator>
bool next_permutation(BidirectionalIterator first,
BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
bool next_permutation(BidirectionalIterator first,
BidirectionalIterator last, Compare comp);

Effects:
Takes  a  sequence defined by the range [first, last) and transforms
it into the next permutation.  The  next  permutation  is  found  by
assuming  that  the  set  of  all  permutations is lexicographically
sorted with respect to operator< or comp.   If  such  a  permutation
exists, it returns true.  Otherwise, it transforms the sequence into
the smallest permutation, that is, the ascendingly sorted  one,  and
returns false.
Complexity:
At most (last - first)/2 swaps.

template<class BidirectionalIterator>
bool prev_permutation(BidirectionalIterator first,
BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
bool prev_permutation(BidirectionalIterator first,
BidirectionalIterator last, Compare comp);

Effects:
Takes  a  sequence defined by the range [first, last) and transforms
it into the previous permutation.  The previous permutation is found
by  assuming  that  the set of all permutations is lexicographically
sorted with respect to operator< or comp.
Returns:
true if such a permutation exists.   Otherwise,  it  transforms  the
sequence  into  the  largest  permutation, that is, the descendingly
sorted one, and returns false.
Complexity:
At most (last - first)/2 swaps.

25.4  C library algorithms                         [lib.alg.c.library]

1 Header <cstdlib> (partial, Table 2):

+-----------------------------+
|   Type          Name(s)     |
+-----------------------------+
|Functions:   bsearch   qsort |
+-----------------------------+

2 The contents are the same as the Standard C library.

[Note: For the Standard C library function:

void qsort(void* base, size_t nmemb, size_t size,
int (*compar)(const void*, const void*));

the  function  argument  compar  shall   have   extern   "C"   linkage
(_dcl.link_).  Also, since compar() may throw an exception, qsort() is
allowed to propagate the exception  (_lib.res.on.exception.handling_).
--end note]