261 lines
13 KiB
Markdown
261 lines
13 KiB
Markdown
[alg.binary.search]
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# 26 Algorithms library [[algorithms]](./#algorithms)
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## 26.8 Sorting and related operations [[alg.sorting]](alg.sorting#alg.binary.search)
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### 26.8.4 Binary search [alg.binary.search]
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#### [26.8.4.1](#general) General [[alg.binary.search.general]](alg.binary.search.general)
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[1](#general-1)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9395)
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All of the algorithms in [alg.binary.search] are versions of binary search and
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assume that the sequence being searched
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is partitioned with respect to an expression
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formed by binding the search key to an argument of the comparison function[.](#general-1.sentence-1)
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They work on non-random access iterators minimizing the number of comparisons,
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which will be logarithmic for all types of iterators[.](#general-1.sentence-2)
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They are especially appropriate for random access iterators,
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because these algorithms do a logarithmic number of steps
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through the data structure[.](#general-1.sentence-3)
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For non-random access iterators they execute a linear number of steps[.](#general-1.sentence-4)
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#### [26.8.4.2](#lower.bound) lower_bound [[lower.bound]](lower.bound)
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[ð](#lib:lower_bound)
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`template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type>
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constexpr ForwardIterator
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lower_bound(ForwardIterator first, ForwardIterator last,
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const T& value);
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template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type,
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class Compare>
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constexpr ForwardIterator
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lower_bound(ForwardIterator first, ForwardIterator last,
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const T& value, Compare comp);
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template<[forward_iterator](iterator.concept.forward#concept:forward_iterator "24.3.4.11 Concept forward_iterator [iterator.concept.forward]") I, [sentinel_for](iterator.concept.sentinel#concept:sentinel_for "24.3.4.7 Concept sentinel_for [iterator.concept.sentinel]")<I> S, class Proj = identity,
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class T = projected_value_t<I, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<I, Proj>> Comp = ranges::less>
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constexpr I ranges::lower_bound(I first, S last, const T& value, Comp comp = {},
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Proj proj = {});
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template<[forward_range](range.refinements#concept:forward_range "25.4.6 Other range refinements [range.refinements]") R, class Proj = identity,
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class T = projected_value_t<iterator_t<R>, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<iterator_t<R>, Proj>> Comp =
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ranges::less>
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constexpr borrowed_iterator_t<R>
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ranges::lower_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
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`
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[1](#lower.bound-1)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9436)
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Let comp be less{} andproj be identity{} for overloads with no parameters by those names[.](#lower.bound-1.sentence-1)
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[2](#lower.bound-2)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9441)
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*Preconditions*: The elements e of [first, last)
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are partitioned with respect to the expression
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bool(invoke(comp, invoke(proj, e), value))[.](#lower.bound-2.sentence-2)
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[3](#lower.bound-3)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9447)
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*Returns*: The furthermost iterator i in the range [first, last]
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such that for every iterator j in the range [first, i),bool(invoke(comp, invoke(proj, *j), value)) is true[.](#lower.bound-3.sentence-1)
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[4](#lower.bound-4)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9453)
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*Complexity*: At most log2(last - first)+O(1) comparisons and projections[.](#lower.bound-4.sentence-1)
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#### [26.8.4.3](#upper.bound) upper_bound [[upper.bound]](upper.bound)
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[ð](#lib:upper_bound)
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`template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type>
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constexpr ForwardIterator
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upper_bound(ForwardIterator first, ForwardIterator last,
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const T& value);
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template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type,
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class Compare>
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constexpr ForwardIterator
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upper_bound(ForwardIterator first, ForwardIterator last,
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const T& value, Compare comp);
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template<[forward_iterator](iterator.concept.forward#concept:forward_iterator "24.3.4.11 Concept forward_iterator [iterator.concept.forward]") I, [sentinel_for](iterator.concept.sentinel#concept:sentinel_for "24.3.4.7 Concept sentinel_for [iterator.concept.sentinel]")<I> S, class Proj = identity,
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class T = projected_value_t<I, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<I, Proj>> Comp = ranges::less>
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constexpr I ranges::upper_bound(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
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template<[forward_range](range.refinements#concept:forward_range "25.4.6 Other range refinements [range.refinements]") R, class Proj = identity,
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class T = projected_value_t<iterator_t<R>, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<iterator_t<R>, Proj>> Comp =
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ranges::less>
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constexpr borrowed_iterator_t<R>
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ranges::upper_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
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`
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[1](#upper.bound-1)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9486)
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Let comp be less{} andproj be identity{} for overloads with no parameters by those names[.](#upper.bound-1.sentence-1)
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[2](#upper.bound-2)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9491)
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*Preconditions*: The elements e of [first, last)
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are partitioned with respect to the expression
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!bool(invoke(comp, value, invoke(proj, e)))[.](#upper.bound-2.sentence-2)
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[3](#upper.bound-3)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9497)
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*Returns*: The furthermost iterator i in the range [first, last]
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such that for every iterator j in the range [first, i),!bool(invoke(comp, value, invoke(proj, *j))) is true[.](#upper.bound-3.sentence-1)
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[4](#upper.bound-4)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9503)
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*Complexity*: At most log2(last - first)+O(1) comparisons and projections[.](#upper.bound-4.sentence-1)
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#### [26.8.4.4](#equal.range) equal_range [[equal.range]](equal.range)
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[ð](#lib:equal_range)
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`template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type>
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constexpr pair<ForwardIterator, ForwardIterator>
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equal_range(ForwardIterator first,
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ForwardIterator last, const T& value);
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template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type,
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class Compare>
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constexpr pair<ForwardIterator, ForwardIterator>
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equal_range(ForwardIterator first,
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ForwardIterator last, const T& value,
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Compare comp);
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template<[forward_iterator](iterator.concept.forward#concept:forward_iterator "24.3.4.11 Concept forward_iterator [iterator.concept.forward]") I, [sentinel_for](iterator.concept.sentinel#concept:sentinel_for "24.3.4.7 Concept sentinel_for [iterator.concept.sentinel]")<I> S, class Proj = identity,
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class T = projected_value_t<I, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<I, Proj>> Comp = ranges::less>
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constexpr subrange<I>
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ranges::equal_range(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
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template<[forward_range](range.refinements#concept:forward_range "25.4.6 Other range refinements [range.refinements]") R, class Proj = identity,
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class T = projected_value_t<iterator_t<R>, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<iterator_t<R>, Proj>> Comp =
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ranges::less>
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constexpr borrowed_subrange_t<R>
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ranges::equal_range(R&& r, const T& value, Comp comp = {}, Proj proj = {});
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`
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[1](#equal.range-1)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9538)
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Let comp be less{} andproj be identity{} for overloads with no parameters by those names[.](#equal.range-1.sentence-1)
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[2](#equal.range-2)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9543)
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*Preconditions*: The elements e of [first, last)
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are partitioned with respect to the expressionsbool(invoke(comp, invoke(proj, e), value)) and!bool(invoke(comp, value, invoke(proj, e)))[.](#equal.range-2.sentence-1)
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Also, for all elements e of [first, last),bool(comp(e, value)) implies !bool(comp(value, e)) for the overloads in namespace std[.](#equal.range-2.sentence-2)
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[3](#equal.range-3)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9553)
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*Returns*:
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- [(3.1)](#equal.range-3.1)
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For the overloads in namespace std:{lower_bound(first, last, value, comp),
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upper_bound(first, last, value, comp)}
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- [(3.2)](#equal.range-3.2)
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For the overloads in namespace ranges:{ranges::lower_bound(first, last, value, comp, proj),
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ranges::upper_bound(first, last, value, comp, proj)}
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[4](#equal.range-4)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9570)
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*Complexity*: At most2âlog2(last - first)+O(1) comparisons and projections[.](#equal.range-4.sentence-1)
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#### [26.8.4.5](#binary.search) binary_search [[binary.search]](binary.search)
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[ð](#lib:binary_search)
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`template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type>
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constexpr bool
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binary_search(ForwardIterator first, ForwardIterator last,
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const T& value);
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template<class ForwardIterator, class T = iterator_traits<ForwardIterator>::value_type,
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class Compare>
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constexpr bool
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binary_search(ForwardIterator first, ForwardIterator last,
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const T& value, Compare comp);
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template<[forward_iterator](iterator.concept.forward#concept:forward_iterator "24.3.4.11 Concept forward_iterator [iterator.concept.forward]") I, [sentinel_for](iterator.concept.sentinel#concept:sentinel_for "24.3.4.7 Concept sentinel_for [iterator.concept.sentinel]")<I> S, class Proj = identity,
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class T = projected_value_t<I, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<I, Proj>> Comp = ranges::less>
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constexpr bool ranges::binary_search(I first, S last, const T& value, Comp comp = {},
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Proj proj = {});
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template<[forward_range](range.refinements#concept:forward_range "25.4.6 Other range refinements [range.refinements]") R, class Proj = identity,
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class T = projected_value_t<iterator_t<R>, Proj>,
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[indirect_strict_weak_order](indirectcallable.indirectinvocable#concept:indirect_strict_weak_order "24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]")<const T*, projected<iterator_t<R>, Proj>> Comp =
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ranges::less>
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constexpr bool ranges::binary_search(R&& r, const T& value, Comp comp = {},
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Proj proj = {});
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`
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[1](#binary.search-1)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9605)
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Let comp be less{} andproj be identity{} for overloads with no parameters by those names[.](#binary.search-1.sentence-1)
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[2](#binary.search-2)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9610)
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*Preconditions*: The elements e of [first, last)
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are partitioned with respect to the expressionsbool(invoke(comp, invoke(proj, e), value)) and!bool(invoke(comp, value, invoke(proj, e)))[.](#binary.search-2.sentence-1)
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Also, for all elements e of [first, last),bool(comp(e, value)) implies !bool(comp(value, e)) for the overloads in namespace std[.](#binary.search-2.sentence-2)
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[3](#binary.search-3)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9620)
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*Returns*: true if and only if
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for some iterator i in the range [first, last),!bool(invoke(comp, invoke(proj, *i), value)) &&!bool(invoke(comp, value, invoke(proj, *i))) is true[.](#binary.search-3.sentence-1)
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[4](#binary.search-4)
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[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/algorithms.tex#L9628)
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*Complexity*: At most log2(last - first)+O(1) comparisons and projections[.](#binary.search-4.sentence-1)
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