In this next part of the big STL algorithm tutorial, we cover the binary search operations. I use plural because there isn't simply std::binary_search
available for us, but other algorithms as well:
binary_seach
equal_range
lower_bound
upper_bound
binary_seach
std::binary_seach
helps us  guess what  to search for an element in a container. As the first two parameters, you have to pass two iterators defining the input range.
Give that we haven't discussed algorithms for a while, here are a few reminders:
 the two iterators must point to the same container, otherwise, the behaviour is undefined
 the compiler has no way to validate this requirement, it's up to the caller
Binary search operations have an additional requirement for their input ranges, they must be sorted, otherwise, the behaviour is undefined.
The first time when I learned about this I felt a bit perplexed. Isn't that a bit too much? Shouldn't the algorithm take care of this? Maybe just sort it when needed.
If you think about it a little bit longer, it makes perfect sense. One of the main principles of (C and) C++ is that you should only pay for what you use. The name binary_seach
is pretty straightforward. It will search for elements with a given mathematical algorithm. Why should it sort anything? Sorting doesn't come for free. If you need to sort your container, use std::sort
or something similar, if you need to check first if the input range is sorted, use is_sorted
or is_sorted_until
.
If binary_seach
would do anything else, it would be a problem for those who use it as it is supposed to be used. Given that checking this semantic requirement has a cost, it's preferable to simply declare that the outcome is undefined if we pass in unsorted elements.
That's enough about the input range.
As a third parameter we have to pass in the value that we are searching for and there is an optional 4th element that is the comparator. As we got used to it, it can be a lambda, a function pointer or a function object (a functor). It's a binary function, it accepts two elements and returns a value that is convertible to a bool
.
In any case, binary::search
will return a boolean, true
in case an element is found in the input range, false
otherwise.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {6, 8, 1, 5, 9, 4, 7, 2, 3};
std::cout << std::boolalpha;
std::cout << "Looking for 1 in the unsorted container\n";
std::cout << std::binary_search(numbers.begin(), numbers.end(), 1) << '\n';
std::sort(numbers.begin(), numbers.end());
std::cout << "Looking for 1 once the container is sorted\n";
std::cout << std::binary_search(numbers.begin(), numbers.end(), 1) << '\n';
auto is_equal = [](int lhs, int rhs){return lhs == rhs;};
std::cout << "Looking for 1 once the container is sorted with custom comparator\n";
std::cout << std::binary_search(numbers.begin(), numbers.end(), 1, is_equal) << '\n';
}
But what if you need the element that you were looking for? We learnt about the different find*
algorithms earlier that you can use to return find one element, but if you look for a range of elements, you have other options than nesting std::find
into a loop.
equal_range
equal_range
returns a pair of iterators giving a handle to all the matching items.
The first iterator points to the first element that is not less than the searched for value and the second element points to the first element that is greater than that value. We are going to check what are the different scenarios, but first, we have to briefly talk about the inputs.
The input parameters are the very same as for binary_seach
:
 at the first two positions we define the input range
 then the value we are looking for
 and finally the optional comparator
Once again, the input range should be fully sorted.
So let's get back to the different scenarios.
Value not found and it's bigger than any elements
When the value cannot be found and it is bigger than any elements, both first
and last
point right after the container.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {1, 2, 2, 3, 4, 5, 5, 5, 7};
std::cout << "Size of numbers: " << numbers.size() << '\n';
const auto [first, last] = std::equal_range(numbers.begin(), numbers.end(), 8);
std::cout << "First distance from numbers.begin(): " << std::distance(numbers.begin(), first) << std::endl;
std::cout << "Value of first: " << *first << std::endl;
std::cout << "First distance from numbers.last(): " << std::distance(numbers.begin(), last) << std::endl;
std::cout << "Value of last: " << *last << std::endl;
}
/*
Size of numbers: 9
First distance from numbers.begin(): 9
Value of first: 0
First distance from numbers.last(): 9
Value of last: 0
*/
Value not found and it's smaller than any elements
When the value cannot be found and it is smaller than any elements, both first
and last
point at the first element.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {1, 2, 2, 3, 4, 5, 5, 5, 7};
std::cout << "Size of numbers: " << numbers.size() << '\n';
const auto [first, last] = std::equal_range(numbers.begin(), numbers.end(), 0);
std::cout << "First distance from numbers.begin(): " << std::distance(numbers.begin(), first) << std::endl;
std::cout << "Value of first: " << *first << std::endl;
std::cout << "First distance from numbers.last(): " << std::distance(numbers.begin(), last) << std::endl;
std::cout << "Value of last: " << *last << std::endl;
}
/*
Size of numbers: 9
First distance from numbers.begin(): 0
Value of first: 1
First distance from numbers.last(): 0
Value of last: 1
Value not found, there are smaller and bigger items in the container
When the value cannot be found, but there are smaller and bigger elements as well in the container, both first
and last
point at the first element greater than the searched value.
It makes sense as that's the first element that is not less than the looked for value and also the first that is greater than it.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {1, 2, 2, 3, 4, 5, 5, 5, 7};
std::cout << "Size of numbers: " << numbers.size() << '\n';
const auto [first, last] = std::equal_range(numbers.begin(), numbers.end(), 6);
std::cout << "First distance from numbers.begin(): " << std::distance(numbers.begin(), first) << std::endl;
std::cout << "Value of first: " << *first << std::endl;
std::cout << "First distance from numbers.last(): " << std::distance(numbers.begin(), last) << std::endl;
std::cout << "Value of last: " << *last << std::endl;
}
/*
Size of numbers: 9
First distance from numbers.begin(): 8
Value of first: 7
First distance from numbers.last(): 8
Value of last: 7
Value found
This is the nominal case and it behaves as it's expected. first
is the last one smaller than the looked for value and last
is the first one that is greater.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {1, 2, 2, 3, 4, 5, 5, 5, 7};
std::cout << "Size of numbers: " << numbers.size() << '\n';
const auto [first, last] = std::equal_range(numbers.begin(), numbers.end(), 5);
std::cout << "First distance from numbers.begin(): " << std::distance(numbers.begin(), first) << std::endl;
std::cout << "Value of first: " << *first << std::endl;
std::cout << "First distance from numbers.last(): " << std::distance(numbers.begin(), last) << std::endl;
std::cout << "Value of last: " << *last << std::endl;
}
/*
Size of numbers: 9
First distance from numbers.begin(): 5
Value of first: 5
First distance from numbers.last(): 8
Value of last: 7
Having seen all these examples, we can observe that in case equal_range
couldn't find the value it looked for, then both iterators will point at the same place, otherwise not. This means that we don't need binary_search
to validate first the existence of the element when we look for the range, we can simply check if the two iterators are pointing to the same place or not
lower_bound
and upper_bound
While equal_range
returns a pair of iterators lower_bound
and upper_bound
returns only one:

lower_bound
returns an iterator pointing to the first item not less than the searched value 
upper_bound
returns an iterator pointing to the first element greater than the searched value
The parameters, these functions take are really the same as we saw before:
 at the first two positions we define the input range
 then the value we are looking for
 and finally the optional comparator
Now let's see a couple of examples.
#include <iostream>
#include <algorithm>
#include <vector>
int main() {
std::vector numbers {1, 2, 2, 3, 4, 5, 5, 5, 7};
std::for_each(numbers.begin(), numbers.end(), [](int num) {std::cout << num << " ";});
std::cout << '\n';
std::cout << "Size of numbers: " << numbers.size() << '\n';
std::cout << '\n';
{
std::cout << "========\n";
const auto value = 5;
std::cout << "Looking for " << value << ", that is inside the container\n";
auto lower = std::lower_bound(numbers.begin(), numbers.end(), value);
auto upper = std::upper_bound(numbers.begin(), numbers.end(), value);
std::cout << "lower's distance from numbers.begin(): " << std::distance(numbers.begin(), lower) << std::endl;
std::cout << "Value of lower: " << *lower << std::endl;
std::cout << "upper's distance from numbers.begin(): " << std::distance(numbers.begin(), upper) << std::endl;
std::cout << "Value of upper: " << *upper << std::endl;
}
{
std::cout << "========\n";
const auto value = 0;
std::cout << "Looking for " << value << ", that is smaller than the smallest item of the container\n";
const auto lower = std::lower_bound(numbers.begin(), numbers.end(), value);
const auto upper = std::upper_bound(numbers.begin(), numbers.end(), value);
std::cout << "lower's distance from numbers.begin(): " << std::distance(numbers.begin(), lower) << std::endl;
std::cout << "Value of lower: " << *lower << std::endl;
std::cout << "upper's distance from numbers.begin(): " << std::distance(numbers.begin(), upper) << std::endl;
std::cout << "Value of upper: " << *upper << std::endl;
}
{
std::cout << "========\n";
const auto value = 9;
std::cout << "Looking for " << value << ", that is larger than the largest item of the container\n";
const auto lower = std::lower_bound(numbers.begin(), numbers.end(), value);
const auto upper = std::upper_bound(numbers.begin(), numbers.end(), value);
std::cout << "lower's distance from numbers.begin(): " << std::distance(numbers.begin(), lower) << std::endl;
std::cout << "Value of lower: " << *lower << std::endl;
std::cout << "upper's distance from numbers.begin(): " << std::distance(numbers.begin(), upper) << std::endl;
std::cout << "Value of upper: " << *upper << std::endl;
}
{
std::cout << "========\n";
const auto value = 6;
std::cout << "Looking for " << value << ", that is not in the container that contains both smaller and larger values than " << value << '\n';
const auto lower = std::lower_bound(numbers.begin(), numbers.end(), value);
const auto upper = std::upper_bound(numbers.begin(), numbers.end(), value);
std::cout << "lower's distance from numbers.begin(): " << std::distance(numbers.begin(), lower) << std::endl;
std::cout << "Value of lower: " << *lower << std::endl;
std::cout << "upper's distance from numbers.begin(): " << std::distance(numbers.begin(), upper) << std::endl;
std::cout << "Value of upper: " << *upper << std::endl;
}
}
/*
1 2 2 3 4 5 5 5 7
Size of numbers: 9
========
Looking for 5, that is inside the container
lower's distance from numbers.begin(): 5
Value of lower: 5
upper's distance from numbers.begin(): 8
Value of upper: 7
========
Looking for 0, that is smaller than the smallest item of the container
lower's distance from numbers.begin(): 0
Value of lower: 1
upper's distance from numbers.begin(): 0
Value of upper: 1
========
Looking for 9, that is larger than the largest item of the container
lower's distance from numbers.begin(): 9
Value of lower: 0
upper's distance from numbers.begin(): 9
Value of upper: 0
========
Looking for 6, that is not in the container that contains both smaller and larger values than 6
lower's distance from numbers.begin(): 8
Value of lower: 7
upper's distance from numbers.begin(): 8
Value of upper: 7
*/
I don't share any comments on the results, essentially they are the same as for equal_range
, you can find a deeper explanation at that section.
Conclusion
This time, we learned about binary search algorithms. We saw how to check if an item can be found in the container and also how to get its or even their position (in case there are multiple items with the same value).
Next time we'll discover merging algorithms.
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