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

function template

<algorithm>

std::binary_search default (1) 
template <class ForwardIterator, class T>  bool binary_search (ForwardIterator first, ForwardIterator last,                      const T& val);
custom (2) 
template <class ForwardIterator, class T, class Compare>  bool binary_search (ForwardIterator first, ForwardIterator last,                      const T& val, Compare comp);

Test if value exists in sorted sequence

Returns true if any element in the range [first,last) is equivalent to val, and false otherwise.

The elements are compared using operator< for the first version, and comp for the second. Two elements, a and b are considered equivalent if (!(a<b) && !(b<a)) or if (!comp(a,b) && !comp(b,a)).

The elements in the range shall already be sorted according to this same criterion (operator< or comp), or at least partitioned with respect to val.

The function optimizes the number of comparisons performed by comparing non-consecutive elements of the sorted range, which is specially efficient for random-access iterators.

The behavior of this function template is equivalent to:


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template <class ForwardIterator, class T>
  bool binary_search (ForwardIterator first, ForwardIterator last, const T& val)
{
  first = std::lower_bound(first,last,val);
  return (first!=last && !(val<*first));
}

Parameters
first, last
Forward iterators to the initial and final positions of a sorted (or properly partitioned) sequence. 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.
val
Value to search for in the range.
For (1), T shall be a type supporting being compared with elements of the range [first,last) as either operand of operator<.
comp
Binary function that accepts two arguments of the type pointed by ForwardIterator (and of type T), and returns a value convertible to bool. The value returned indicates whether the first argument is considered to go before the second.
The function shall not modify any of its arguments.
This can either be a function pointer or a function object.

Return valuetrue if an element equivalent to val is found, and false otherwise.

Example 
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// binary_search example
#include <iostream>     // std::cout
#include <algorithm>    // std::binary_search, std::sort
#include <vector>       // std::vector

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

int main () {
  int myints[] = {1,2,3,4,5,4,3,2,1};
  std::vector<int> v(myints,myints+9);                         // 1 2 3 4 5 4 3 2 1

  // using default comparison:
  std::sort (v.begin(), v.end());

  std::cout << "looking for a 3... ";
  if (std::binary_search (v.begin(), v.end(), 3))
    std::cout << "found!\n"; else std::cout << "not found.\n";

  // using myfunction as comp:
  std::sort (v.begin(), v.end(), myfunction);

  std::cout << "looking for a 6... ";
  if (std::binary_search (v.begin(), v.end(), 6, myfunction))
    std::cout << "found!\n"; else std::cout << "not found.\n";

  return 0;
}

Output: 
looking for a 3... found!
looking for a 6... not found.


Complexity On average, logarithmic in the distance between first and last: Performs approximately log2(N)+2 element comparisons (where N is this distance).
On non-random-access iterators, the iterator advances produce themselves an additional linear complexity in N on average.

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

Exceptions Throws if either an element comparison or an operation on an iterator throws.
Note that invalid arguments cause undefined behavior.

See also
find
Find value in range (function template)
lower_bound
Return iterator to lower bound (function template)
upper_bound
Return iterator to upper bound (function template)
equal_range
Get subrange of equal elements (function template)

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