Why arguments of std::binary_search are forward iterators?

Question

While perusing http://en.cppreference.com/w/cpp/algorithm/binary_search I've noticed it takes forward iterator as an argument. Now I'm confused, since I thought it would rather be an random access iterator, so the binary search will be actually binary.

To satisfy my curiosity, I've written a little program:

#include <iostream>
#include <vector>
#include <forward_list>
#include <list>
#include <deque>
#include <algorithm>
#include <chrono>
#include <random>

int main()
{
    std::uniform_int_distribution<int> uintdistr(-4000000, 4000000);
    std::mt19937 twister(std::chrono::high_resolution_clock::to_time_t(std::chrono::high_resolution_clock::now()));
    size_t arr[] = { 200000, 400000, 800000, 1600000, 3200000, 6400000, 12800000 };
    for(auto size : arr)
    {
        std::list<int> my_list;
        for(size_t i = 0; i < size; i++)
            my_list.push_front(uintdistr(twister));
        std::chrono::time_point<std::chrono::high_resolution_clock> start, end;
        my_list.sort(); //fixed
        start = std::chrono::high_resolution_clock::now();

        std::binary_search(my_list.begin(), my_list.end(), 1252525);

        end = std::chrono::high_resolution_clock::now();
        long long unsigned elapsed_time = std::chrono::duration_cast<std::chrono::microseconds>(end-start).count();
        std::cout << "Test finished in " << elapsed_time << "\n";
    }
}

Compiling it with gcc 4.7.0 and running

g++ -std=c++11 test.cpp

provides following results on my machine:

Test finished in 0
Test finished in 15625
Test finished in 15625
Test finished in 46875
Test finished in 93750
Test finished in 171875
Test finished in 312500

So it looks like it doesn't actually do a binary search on a forward list. Now my questions are:

Why such a confusing name?

Why does the code like this allowed?

Why does the reference says it's "Logarithmic in the distance between first and last"?

What does the standard has to say about it?

EDIT: Now the code sorts the list before search - stupid mistake, now the results are:

Test finished in 46875
Test finished in 109375
Test finished in 265625
Test finished in 546875
Test finished in 1156250
Test finished in 2625000
Test finished in 6375000

And of course still not logarithmic ;)

Answer 1

The docs of the original SGI STL implementation, from which the standard was derived, states that

The number of comparisons is logarithmic: at most log(last - first) + 2. If ForwardIterator is a Random Access Iterator then the number of steps through the range is also logarithmic; otherwise, the number of steps is proportional to last - first.

That is, the number of comparisons is always logarithmic, while the number of advancements, which are affected by the lack of random-accessibility, can be linear. In practice, stl::advance is probably used, for which the complexity is constant if the iterator is random access, linear otherwise.

A binary search with a linear number of iterator advancements, but with a logarithmic number of comparisons makes sense if a comparison is very expensive. If, for example, you have a sorted linked-list of complicated objects, which require disk- or network-access to compare, you're probably much better off with a binary search than with a linear one.

Answer 2

Contrary to what websites may say (eg "logarithmic in distance(first, last) "), the standard actually only speaks about the comparisons (eg 25.4.3.1, lower_bound ):

Complexity: At most log2(last − first) + O(1) comparisons

The incrementing of the iterator is not included in the complexity! Note though that the standard library requires all iterator increments to have amortized constant complexity, so there'll be a cost of order O(N) coming from incrementing the iterators (but presumably this has a very small leading factor). In particular (25.4.3):

For non-random access iterators [the algorithms] execute a linear number of steps.

Answer 3

The standard specifies the sorted search algorithms ( std::lower_bound() , std::upper_bound() , and std::binary_search() ) to work in linear time for forward and binary iterators. For random access the time is logarithmic.

Note that the number of comparisons is restricted to be logarithmic, however.