std::lower_bound(3)

NAME
std::lower_bound - std::lower_bound

Synopsis
   Defined in header <algorithm>
   template< class ForwardIt, class T >
                        (constexpr since
   ForwardIt lower_bound( ForwardIt first, C++20)
   ForwardIt last, (until C++26)
                        const T& value );
   template< class ForwardIt, class T =
   typename std::iterator_traits
    <ForwardIt>::value_type > (since C++26)
   constexpr ForwardIt lower_bound( ForwardIt
   first, ForwardIt last,
                        const T&
   value );
   template< class ForwardIt, class T, class
   Compare > (1)
                        (constexpr since
   ForwardIt lower_bound( ForwardIt first, C++20)
   ForwardIt last, (until C++26)
                        const T& value,
   Compare comp );
   template< class ForwardIt, class T =
   typename std::iterator_traits (2)
    <ForwardIt>::value_type,
             class Compare > (since C++26)
   constexpr ForwardIt lower_bound( ForwardIt
   first, ForwardIt last,
                        const T&
   value, Compare comp );
   Searches for the first element in the partitioned range [first, last) which is not
   ordered before value.
   1) The order is determined by operator<:
   Returns the first iterator iter in [first, last) where bool(*iter <
   value) is false, or last if no such iter exists.
                        (until C++20)
   If the elements elem of [first, last) are not partitioned with respect
   to the expression bool(elem < value), the behavior is undefined.
   Equivalent to std::lower_bound(first, last, value, std::less{}). (since C++20)
   2) The order is determined by comp:
   Returns the first iterator iter in [first, last) where bool(comp(*iter, value)) is
   false, or last if no such iter exists.
   If the elements elem of [first, last) are not partitioned with respect to the
   expression bool(comp(elem, value)), the behavior is undefined.

Parameters
   first, last - the partitioned range of elements to examine
   value - value to compare the elements to
                  binary predicate which returns true if the first argument is ordered
                  before the second.
                  The signature of the predicate function should be equivalent to the
                  following:
                   bool pred(const Type1 &a, const Type2 &b);
   comp - While the signature does not need to have const &, the function must
                  not modify the objects passed to it and must be able to accept all
                  values of type (possibly const) Type1 and Type2 regardless of value
                  category (thus, Type1 & is not allowed
                  , nor is Type1 unless for Type1 a move is equivalent to a copy
                  (since C++11)).
                  The type Type1 must be such that an object of type ForwardIt can be
                  dereferenced and then implicitly converted to Type1. The type Type2
                  must be such that an object of type T can be implicitly converted to
                  Type2.

Type requirements
   -
   ForwardIt must meet the requirements of LegacyForwardIterator.
   -
   Compare must meet the requirements of BinaryPredicate. It is not required to satisfy
   Compare.

Return value
   Iterator to the first element of the range [first, last) not ordered before value,
   or last if no such element is found.

Complexity
   Given \(\scriptsize N\)N as std::distance(first, last):
   1) At most \(\scriptsize \log_{2}(N)+O(1)\)log
   2(N)+O(1) comparisons with value using
   operator<
   (until C++20)
   std::less{}
   (since C++20).
   2) At most \(\scriptsize \log_{2}(N)+O(1)\)log
   2(N)+O(1) applications of the comparator comp.
   However, if ForwardIt is not a LegacyRandomAccessIterator, the number of iterator
   increments is linear in \(\scriptsize N\)N. Notably, std::map, std::multimap,
   std::set, and std::multiset iterators are not random access, and so their member
   lower_bound functions should be preferred.

Possible implementation
   See also the implementations in libstdc++ and libc++.
                        lower_bound (1)
 template<class ForwardIt, class T = typename std::iterator_traits<ForwardIt>::value_type>
 ForwardIt lower_bound(ForwardIt first, ForwardIt last, const T& value)
 {
     return std::lower_bound(first, last, value, std::less{});
 }
                        lower_bound (2)
 template<class ForwardIt, class T = typename std::iterator_traits<ForwardIt>::value_type,
          class Compare>
 ForwardIt lower_bound(ForwardIt first, ForwardIt last, const T& value, Compare comp)
 {
     ForwardIt it;
     typename std::iterator_traits<ForwardIt>::difference_type count, step;
     count = std::distance(first, last);
     while (count > 0)
     {
         it = first;
         step = count / 2;
         std::advance(it, step);
         if (comp(*it, value))
         {
             first = ++it;
             count -= step + 1;
         }
         else
             count = step;
     }
     return first;
 }

Notes
   Although std::lower_bound only requires [first, last) to be partitioned, this
   algorithm is usually used in the case where [first, last) is sorted, so that the
   binary search is valid for any value.
   Unlike std::binary_search, std::lower_bound does not require operator< or comp to be
   asymmetric (i.e., a < b and b < a always have different results). In fact, it does
   not even require value < *iter or comp(value, *iter) to be well-formed for any
   iterator iter in [first, last).
             Feature-test macro Value Std Feature
   __cpp_lib_algorithm_default_value_type 202403 (C++26) List-initialization for
                        algorithms (1,2)

Example
// Run this code
 #include <algorithm>
 #include <cassert>
 #include <complex>
 #include <iostream>
 #include <vector>
 struct PriceInfo { double price; };
 int main()
 {
     const std::vector<int> data{1, 2, 4, 5, 5, 6};
     for (int i = 0; i < 8; ++i)
     {
         // Search for first element x such that i ≤ x
         auto lower = std::lower_bound(data.begin(), data.end(), i);
         std::cout << i << " ≤ ";
         lower != data.end()
             ? std::cout << *lower << " at index " << std::distance(data.begin(), lower)
             : std::cout << "not found";
         std::cout << '\n';
     }
     std::vector<PriceInfo> prices{{100.0}, {101.5}, {102.5}, {102.5}, {107.3}};
     for (const double to_find : {102.5, 110.2})
     {
         auto prc_info = std::lower_bound(prices.begin(), prices.end(), to_find,
             [](const PriceInfo& info, double value)
             {
                 return info.price < value;
             });
         prc_info != prices.end()
             ? std::cout << prc_info->price << " at index " << prc_info - prices.begin()
             : std::cout << to_find << " not found";
         std::cout << '\n';
     }
     using CD = std::complex<double>;
     std::vector<CD> nums{{1, 0}, {2, 2}, {2, 1}, {3, 0}};
     auto cmpz = [](CD x, CD y) { return x.real() < y.real(); };
     #ifdef __cpp_lib_algorithm_default_value_type
         auto it = std::lower_bound(nums.cbegin(), nums.cend(), {2, 0}, cmpz);
     #else
         auto it = std::lower_bound(nums.cbegin(), nums.cend(), CD{2, 0}, cmpz);
     #endif
     assert((*it == CD{2, 2}));
 }

Output:
 0 ≤ 1 at index 0
 1 ≤ 1 at index 0
 2 ≤ 2 at index 1
 3 ≤ 4 at index 2
 4 ≤ 4 at index 2
 5 ≤ 5 at index 3
 6 ≤ 6 at index 5
 7 ≤ not found
 102.5 at index 2
 110.2 not found
   Defect reports
   The following behavior-changing defect reports were applied retroactively to
   previously published C++ standards.
      DR Applied to Behavior as published Correct behavior
                       Compare was required to satisfy only a partitioning is
   LWG 270 C++98 Compare and T was required required;
                       to be LessThanComparable (strict weak heterogeneous comparisons
                       ordering required) permitted
                       at most \(\scriptsize corrected to \(\scriptsize
   LWG 384 C++98 \log(N)+1\)log(N)+1 comparisons were \log_{2}(N)+O(1)\)log
                       allowed 2(N)+1
                       if any iterator iter exists in
                       [first, last) such that
   LWG 2150 C++98 bool(comp(*iter, value)) is false, no iterator after
                       std::lower_bound iter can be returned
                       could return any iterator in
                       [iter, last)

See also
   equal_range returns range of elements matching a specific key
                       (function template)
   partition divides a range of elements into two groups
                       (function template)
   partition_point locates the partition point of a partitioned range
   (C++11) (function template)
                       returns an iterator to the first element greater than a certain
   upper_bound value
                       (function template)
                       returns an iterator to the first element not less than the given
   lower_bound key
                       (public member function of std::set<Key,Compare,Allocator>)
                       returns an iterator to the first element not less than the given
   lower_bound key
                       (public member function of std::multiset<Key,Compare,Allocator>)
   ranges::lower_bound returns an iterator to the first element not less than the given
   (C++20) value
                       (niebloid)