How to implement an STL-style iterator and avoid common pitfalls? has a handy chart that details the specs of § 24.2.2 of the C++11 standard. Basically, the iterators have tags that describe the valid operations, and the tags have a hierarchy. Below is purely symbolic, these classes don’t actually exist as such.

iterator {
    iterator(const iterator&);
    iterator& operator=(const iterator&);
    iterator& operator++(); //prefix increment
    reference operator*() const;
    friend void swap(iterator& lhs, iterator& rhs); //C++11 I think

input_iterator : public virtual iterator {
    iterator operator++(int); //postfix increment
    value_type operator*() const;
    pointer operator->() const;
    friend bool operator==(const iterator&, const iterator&);
    friend bool operator!=(const iterator&, const iterator&); 
//once an input iterator has been dereferenced, it is 
//undefined to dereference one before that.

output_iterator : public virtual iterator {
    reference operator*() const;
    iterator operator++(int); //postfix increment
//dereferences may only be on the left side of an assignment
//once an output iterator has been dereferenced, it is 
//undefined to dereference one before that.

forward_iterator : input_iterator, output_iterator {
//multiple passes allowed

bidirectional_iterator : forward_iterator {
    iterator& operator--(); //prefix decrement
    iterator operator--(int); //postfix decrement

random_access_iterator : bidirectional_iterator {
    friend bool operator<(const iterator&, const iterator&);
    friend bool operator>(const iterator&, const iterator&);
    friend bool operator<=(const iterator&, const iterator&);
    friend bool operator>=(const iterator&, const iterator&);

    iterator& operator+=(size_type);
    friend iterator operator+(const iterator&, size_type);
    friend iterator operator+(size_type, const iterator&);
    iterator& operator-=(size_type);  
    friend iterator operator-(const iterator&, size_type);
    friend difference_type operator-(iterator, iterator);

    reference operator[](size_type) const;

contiguous_iterator : random_access_iterator { //C++17
}; //elements are stored contiguously in memory.

You can either specialize std::iterator_traits<youriterator>, or put the same typedefs in the iterator itself, or inherit from std::iterator (which has these typedefs). I prefer the second option, to avoid changing things in the std namespace, and for readability, but most people inherit from std::iterator.

struct std::iterator_traits<youriterator> {        
    typedef ???? difference_type; //almost always ptrdiff_t
    typedef ???? value_type; //almost always T
    typedef ???? reference; //almost always T& or const T&
    typedef ???? pointer; //almost always T* or const T*
    typedef ???? iterator_category;  //usually std::forward_iterator_tag or similar

Note the iterator_category should be one of std::input_iterator_tag, std::output_iterator_tag, std::forward_iterator_tag, std::bidirectional_iterator_tag, or std::random_access_iterator_tag, depending on which requirements your iterator satisfies. Depending on your iterator, you may choose to specialize std::next, std::prev, std::advance, and std::distance as well, but this is rarely needed. In extremely rare cases you may wish to specialize std::begin and std::end.

Your container should probably also have a const_iterator, which is a (possibly mutable) iterator to constant data that is similar to your iterator except it should be implicitly constructable from a iterator and users should be unable to modify the data. It is common for its internal pointer to be a pointer to non-constant data, and have iterator inherit from const_iterator so as to minimize code duplication.

My post at Writing your own STL Container has a more complete container/iterator prototype.

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