Extended STL: The Fibonacci Sequence

This chapter is from the book

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);
}
```

Listing 23.2 shows the implementations of the only two nonboilerplate methods of const_iterator.

Listing 23.2. Version 1: Preincrement and Dereference Operators

```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.

23.3.3 Changing Value Type?

Perhaps a solution lies in using a different value type. Obviously, using uint64_t is only going to be a small bandage over the problem, allowing us to enumerate 93 steps and get to 7,540,113,804,746,346,429. And once we're there, we still precipitate a contract violation, indicating abuse of the sequence.

Maybe floating point is the way to go? (This implementation corresponds to Fibonacci_sequence_3.hpp on the CD.) Alas, no—32-bit float enters INF territory at 187 entries, 64-bit double at 1,478. Furthermore, since the entries in the sequence are not nicely rounded 10N values, rounding errors creep in as soon as the exponent value reaches the extent of the mantissa.

Conceivably, a BigInt type using coded decimal evaluation would be able to go infinite, but it would have correspondingly poor performance. (Readers are invited to submit such a solution. In reward I can promise the unquantifiable fame that will come from having your name on the book's Web site.)

23.3.4 Constraining Type

To avoid floating-point inaccuracies, we would like to constrain the value type to be integral. To avail ourselves of the maximum range of the type and to catch overflow, we would like to constrain the value type to be unsigned. These constraints are achieved by providing a destructor for the sequence for this very purpose, as shown in Listing 23.5.

Listing 23.5. Constraints Enforced in the Destructor

```Fibonacci_sequence::~Fibonacci_sequence() throw()
{
using stlsoft::is_integral_type; // Using using declarations . . .
using stlsoft::is_signed_type;   // . . . to fit in book. ;-)
STLSOFT_STATIC_ASSERT(0 != is_integral_type<value_type>::value);
STLSOFT_STATIC_ASSERT(0 == is_signed_type<value_type>::value);
}
```

You might think it strange to put in such constraints in a non-template class. The reason is simple: Maintenance programmers (including those who maintain their own code, hint, hint) are wont to change things without putting in all the big-picture research (i.e., reading all documentation). By putting in constraints, you are literally constraining any future changes from violating the design assumptions, or at least from doing so without extra thought.

I prefer to place constraints in the destructor of template classes because it's the method we can most rely on being instantiated. In non-template classes, I continue to use it for consistency.

23.3.5 Throw std::overflow_error?

One possible approach is to change the precondition enforcement assertion to be a legitimate runtime condition and to throw an exception. (The implementation shown in Listing 23.6 corresponds to Fibonacci_sequence_4.hpp on the CD.)

Listing 23.6. Version 4: Preincrement Operator

```class_type& Fibonacci_sequence::const_iterator::operator ++()
{
if(m_i1 < m_i0)

{

throw std::overflow_error("Exhausted integral type");

}
value_type  res   = m_i0 + m_i1;
. . . // Same as Version 2
```

Although, in strict terms, this is a legitimate approach, it really doesn't appeal. The so-called exceptional condition is not an unpredictable emergent characteristic of the system at a particular state and time but an entirely predictable and logical consequence of the relationship between the modeled concept and the type used to hold its values. Using an exception in this case just smacks of Java hackery.

I think it's clear at this point that we should either decide to represent the Fibonacci sequence as something that is genuinely infinite, with suitable indicators, or provide a mechanism for providing finite endpoints.

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