23.1 Introduction

Those who enjoy mathematical elegance may share my appreciation of the Fibonacci sequence and its associated relationship, the Golden Ratio. In this chapter we're going to look at how we might represent this mathematical sequence as a collection, in the form of an STL sequence class, and then consider whether it might be better represented as an iterator, before finally coming back to seeing how a range-limited sequence is the most discoverable representation.

Unlike the other STL extensions described in this book, this one does not derive from any libraries. It is entirely pedagogical. As such, I trust you'll bear with me in some of the less practicable fancies used to illuminate the STL extension issues covered. For those who prefer real examples, worry not, this is the only such fanciful example in the whole book.

23.2 The Fibonacci Sequence

The Fibonacci sequence is a series of numbers, starting with the pair 0 and 1, where the value of each element is calculated as the sum of the two preceding it. Hence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597, 2,584, 4,181, 6,765, and so on, ad infinitum.

The ratio of each entry in the series to its next tends toward an irrational constant, known as the Golden Ratio, whose value is approximately 1.61803398875. The Golden Ratio appears to crop up in all kinds of places in the universe, from the ratio of aesthetically pleasing picture frames to the twirls of conch shells to the dimensions of the Parthenon. If you've not come across it before, I recommend you check it out.

23.3 Fibonacci as an STL Sequence

My first instinct when thinking about how to represent a mathematical sequence was to use an STL-compliant sequence, as shown in Listing 23.1. As we'll see, however, this is not as nice a fit as we might think. Since this is a notional collection—there are no elements in existence anywhere—the enumeration of the values in the sequence is carried out in the iterator, an instance of the member class const_iterator, whose element reference category is by-value temporary (Section 3.3.5).

Listing 23.1 Fibonacci_sequence Version 1 and Its Iterator Class

class Fibonacci_sequence
{
public: // Member Types
  typedef uint32_t  value_type;
  class             const_iterator;
  . . .
public: // Iteration
  const_iterator  begin() const
  {
    return const_iterator(0, 1);
  }
  const_iterator  end() const;
  . . .
};
   
class Fibonacci_sequence::const_iterator
  : public std::iterator< std::forward_iterator_tag
                        , Fibonacci_sequence::value_type, ptrdiff_t
                        , void, Fibonacci_sequence::value_type // BVT
                        >
{
public: // Member Types
  typedef const_iterator                  class_type;
  typedef Fibonacci_sequence::value_type  value_type;
public: // Construction
  const_iterator(value_type i0, value_type i1);
public: // Iteration
  class_type& operator ++();
  class_type operator ++(int);
  value_type operator *() const;
public: // Comparison
  bool equal(class_type const& rhs) const
  {
    return m_i0 == rhs.m_i0 && m_i1 == rhs.m_i1;
  }
  . . .
private: // Member Variables
  value_type  m_i0;
  value_type  m_i1;
};
   
inline bool operator ==(Fibonacci_sequence::const_iterator const& lhs
                      , Fibonacci_sequence::const_iterator const& rhs)
{
  return lhs.equal(rhs);
}
inline bool operator !=(Fibonacci_sequence::const_iterator const& lhs
                      , Fibonacci_sequence::const_iterator const& rhs)
{
  return !lhs.equal(rhs);
}

class_type& Fibonacci_sequence::const_iterator::operator ++()
{
  value_type  res   = m_i0 + m_i1;
              m_i0  = m_i1;
              m_i1  = res;
  return *this;
}
value_type Fibonacci_sequence::const_iterator::operator *() const
{
  return m_i0;
}

Each time the preincrement operator is called, the next result is calculated and moved into m_i1, after m_i1 is first moved into m_i0. The current result is held in m_i0. Note that the const_iterator could just as easily support the bidirectional iterator category, wherein the predecrement operator would subtract m_i0 from m_i1 to get the previous value in the sequence. I've not done so simply because the Fibonacci is a forward sequence.

Because the sequence is infinite, end() is defined to return an instance of const_

iterator whose value is such that it will never compare equal() to a valid iterator. (The implementation shown in Listing 23.3 corresponds to Fibonacci_sequence_1.hpp on the CD.)

Listing 23.3 Version 1: end() Method

class Fibonacci_sequence
{
  . . .
  const_iterator  end() const
  {
    return const_iterator(0, 0);
  }
  . . .

Let's now use this definition of the sequence:

Fibonacci_sequence                  fs;
Fibonacci_sequence::const_iterator  b = fs.begin();
   
for(size_t i = 0; i < 10; ++i, ++b)
{
  std::cout << i << ": " << *b << std::endl;
}

This works a treat, giving the first ten elements in the Fibonacci sequence: 0–34. However, as we well know, iterators like to work with algorithms and usually take them in pairs, for example:

std::copy(fs.begin(), fs.end()
    , std::ostream_iterator<Fibonacci_sequence::value_type>(std::cout
                                                           , " "));

Unfortunately, there are two problems with this statement. First, it runs forever, which represents somewhat of an inconvenience when you want to use your computer for something worthwhile, such as updating it with the latest virus definitions and operating system patches to fill up that last 12GB of disk you were saving for your database of fine European chocolatiers. You might wonder whether we will be saved when the overflowed arithmetic happens on a result whose value modulo 0x10000000 is 0. Although this does eventually occur—after 3,221,225,426 iterations, as it happens—the iterator still does not compare equal to the end() iterator because its m_i1 member is nonzero. Since it is not possible for both members to be 0 at one time, the code will loop forever.

Second, after the forty-seventh iteration, the results returned are no longer members of the Fibonacci sequence but pseudo junk values as a consequence of overflow of our 32-bit value type. As we know, computers don't generally like to live in the infinite, and integral types are particularly antipathetic to unconstrained ranges.

23.3.1 Interface of an Infinite Sequence

We'll deal with the first problem first. Since the Fibonacci sequence is infinite, one option would be to make the Fibonacci_sequence infinite also. This is easily effected by removing the end() method. The sequence is now quite literally one without end. Now users of the class cannot make the mistake, shown earlier, of passing an ostensibly bounded [begin(), end()) range to an algorithm since there is no end.

In my opinion, this is the most appealing form from a conceptual point of view because the public interface of the sequence is representing its semantics most clearly. However, it's not terribly practical because, as we've already seen, overflow occurs after a soberingly finite number of steps. For infinite sequences whose values are bound within a representable range, this would be a good candidate approach, but it's not suitable for the Fibonacci sequence.

Note that this reasoning also rules out the possible alternative implementations of Fibonacci sequences as independent iterator classes or as generator functions.

23.3.2 Put a Contract on It

Let's now take the sensible step of putting some contract programming protection into the preincrement operator before we attempt to use the sequence. (The implementation shown in Listing 23.4 corresponds to Fibonacci_sequence_2.hpp on the CD.)

Listing 23.4 Version 2: Preincrement Operator

class_type& Fibonacci_sequence::const_iterator::operator ++()
{
  STLSOFT_MESSAGE_ASSERT("Exhausted integral type", m_i0 <= m_i1);
  value_type  res   = m_i0 + m_i1;
              m_i0  = m_i1;
              m_i1  = res;
  return *this;
}

In executing the std::copy statement shown previously, we find that the assertion is fired on the increment after output of the value 2,971,215,073. At this point, the previous value was 1,836,311,903, so we would expect m_i1 to be 4,807,526,976. However, that exceeds the maximum value representable in a 32-bit unsigned integer (4,294,967,295), so the result is truncated (to 512,559,680), and the assertion fires. Hence, although we've managed to iterate 48 items, the last increment left the iterator in an invalid state, an unincrementable state, so there are only actually 47 viable enumerable values from a 32-bit representation.

I want to stress the distinction between providing a usable interface and guarding against misuse, well exemplified in this case. Thus far, our Fibonacci sequence does not have a usable interface—since its failure is a matter of surprise—but now, with the introduction of the assertion, it does have protection against its misuse.