# Extended STL: The Fibonacci Sequence

This chapter is from the book

## 23.5 Defining Finite Bounds

There are two clear and related solutions to this problem.

1. Have end() return an iterator in the range [begin(), ∞) whose value does not overflow the value type.
2. Allow the user to specify an upper limit for the effective range provided by the sequence, represented in the value returned by end(). This value would have to be within the valid range.

A good implementation would provide both, where solution 1 is merely the default form of solution 2. We'll examine this by looking at the user-specified limits first.

### 23.5.1 Iterators Rule After All?

Before we proceed, I must cover an issue that some readers may now be considering. Earlier I ruled out the representation of Fibonacci sequences as independent iterators. The cunning linguists among you may be considering a form that does exactly that, as in:

std::copy(Fibonacci_iterator(), Fibonacci_iterator() + 40
, std::ostream_iterator<Fibonacci_sequence::value_type>(std::cout
, " "));

In this case, the putative Fibonacci_iterator would implement the addition operator, such that the expression Fibonacci_iterator() + 40 would evaluate to an instance that would terminate the iteration of a default-constructed iterator on its fortieth increment. At first blush this seems like an adequate solution to the problem.

However, the problem is that use of the addition operator on an iterator indicates that the iterator type is a random access iterator. Further, random access iterators have constant time complexity. To be sure, we're perforce violating pure STL requirements here and there in STL extension. But such violations are never done without due care and particular attention to the effects on discoverability and the Principle of Least Surprise. For example, it's hard to imagine that users of the InetSTL findfile_sequence class (Section 21.1), an STL collection that provides iteration of remote FTP host directory contents, will assume any particular complexity guarantees, given the vagaries of Internet retrieval. However, I suggest that it's far more likely that someone would assume constant time seeing pointer arithmetic syntax on an iterator.

Further, since a user will reasonably expect to be able to type *(Fibonacci_iterator() + 40) if he or she can type Fibonacci_iterator() + 40, we'd have to implement full random access semantics. But, as far as I know, there's no constant-time function integral formula with which you can determine the Nth value of the Fibonacci sequence. (There are a couple of formulas that may be used, but they rely on the square root of five, which would rely on floating-point calculation. One of them is ((1 + sqrt(5)) / 2) - ((1 - sqrt(5)) / 2) ^ n. I'm just enough of a computer numerist to know that I know far too little about floating point to be confident of writing a 100% correct sequence using floating-point calculations.)

Thus we would have to perform a number of forward or backward calculations to arrive at the required value, which is a linear-time operation. This would be a very unobvious violation of a user's expectations and is, in my opinion, unacceptable.

(Of course, we could provide amortized constant time by storing the calculated values in an array. We could go further and provide a static member array with precalculated values. We could even use template metaprogramming and effect compile-time calculation. But the purpose of this chapter is pedagogical. Feel free to do any of these, and let me know how it goes. I'll gladly post interesting solutions on the book's Web site.)

### 23.5.2 Constructor-Bound Range

One use case of a sequence might be to retrieve the first N elements in the sequence. It would be straightforward to implement the sequence and iterator classes such that you would specify the number of elements in the sequence constructor, which would then return a bounding iterator instance via its end() method, as shown in Listing 23.7. (This corresponds to Fibonacci_sequence_5.hpp on the CD.)

#### Listing 23.7. Version 5: Constructor and end() Method

public: // Construction
explicit Fibonacci_sequence(size_t n) // Max # entries to enumerate
: m_numEntries(n)
{}
. . .
public: // Iteration
const_iterator begin();
const_iterator end()
{
return const_iterator(m_numEntries); // Define end of sequence
}
. . .

You could use this as follows:

Fibonacci_sequence fs(25);

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

This could be implemented by adding an additional m_stepIndex member to const_iterator, which would be incremented each time operator ++() is called, and by evaluating equality (in equal()) by comparing the m_stepIndex members of the comparands (Listing 23.8).

#### Listing 23.8. Version 5: const_iterator

class Fibonacci_sequence::const_iterator
: . . . // As shown previously
{
public: // Construction
const_iterator(value_type i0, value_type i1)
: m_i0(i0)
, m_i1(i1)
, m_stepIndex(0)
{}
const_iterator(size_t stepIndex)
: m_i0(0)
, m_i1(0)
, m_stepIndex(stepIndex)
{}
public: // Iteration
class_type& operator ++()
{
. . . // Perform the advancement summations as before
++m_stepIndex;
return *this;
}
public: // Comparison
bool equal(class_type const& rhs) const
{
return m_stepIndex == rhs.m_stepIndex;
}
. . .

This is a nice solution, and it also allows us to meaningfully support the empty() method. However, there's an equally valid use case, that of constraining the enumeration within a given integral range, for example, to enumerate all entries less than the value 1,000,000,000. The sequence might look like that shown in Listing 23.9. (This implementation corresponds to Fibonacci_sequence_6.hpp on the CD.)

#### Listing 23.9. Version 6: Constructor and end() Method

class Fibonacci_sequence
{
. . .
public: // Construction

explicit Fibonacci_sequence(value_type limit); // Value ceiling

: m_limit(limit)

{}
. . .
public: // Iteration
const_iterator begin();
const_iterator end()

{

return const_iterator(m_limit); // Define sequence ceiling

}
. . .

Comparison would be conducted by the somewhat abstruse implementation of equal() shown in Listing 23.10. (There's an overflow bug in here, which I've left since this is a pedagogical class. Try setting the limit to 1,836,311,904 for a 32-bit unsigned value type. If readers want to implement the full testing for overflow, I'll be happy to post any correct solutions on the book's Web site.)

#### Listing 23.10. Version 6: const_iterator::equal() Method

bool
Fibonacci_sequence::const_iterator::equal(class_type const& rhs)
const
{
if( 0 != m_i1 &&
0 != rhs.m_i1)
{
// Both definitely normal iterable instances
return m_i0 == rhs.m_i0 && m_i1 == rhs.m_i1;
}
else if(0 != m_threshold &&
0 != rhs.m_threshold)
{
// Both definitely threshold sentinel instances
return m_threshold == rhs.m_threshold;
}
else
{
// Heterogeneous mix of the two types
if(0 == m_threshold)
{
return m_i0 >= rhs.m_threshold;
}
else
{
return rhs.m_i0 >= m_threshold;
}
}
}

A more flexible class would accommodate both these usage models. But doing so presents the sticky problem of how to unambiguously construct an instance of the sequence for either use case. One solution would be to use a two-parameter constructor, as follows:

. . .
public: // Construction
Fibonacci_sequence(size_t n, value_type limit);
. . .

The parameter for the end-marker type not used would be given a stock value, for example:

Fibonacci_sequence(0, 10000); // This uses a limit of 10,000
Fibonacci_sequence(20, 0);    // This gives a sequence of 20 entries

Obviously, this is an inelegant and highly error-prone approach. A slightly less revolting alternative would be to use an enumeration to indicate the type of end marker required and use a value_type parameter for both the threshold and the number of entries:

. . .
public: // Member Constants
enum LimitType { thresholdLimit, countLimit };
public: // Construction
Fibonacci_sequence(value_type limit, LimitType type);
. . .

### 23.5.3 True Typedefs

The best solution uses true typedefs (Section 12.3), which facilitate the unambiguous overloading of essentially similar or even identical types. The final implementation of the Fibonacci_sequence does this, as shown in Listing 23.11. (This corresponds to Fibonacci_sequence_7.hpp on the CD.) Note the use of precondition enforcements in both constructors. A valid design alternative would be to throw std::out_of_range (since the user's value is not predictable).

#### Listing 23.11. Version 7: Class Declaration and Traits Class

template <typename T>
struct Fibonacci_traits;

template <>
struct Fibonacci_traits<uint32_t>
{
static const uint32_t maxThreshold  = 2971215073;
static const size_t   maxLimit      = 47;
};
template <>
struct Fibonacci_traits<uint64_t>
{
static const uint64_t maxThreshold  = 12200160415121876738;
static const size_t   maxLimit      = 93;
};

class Fibonacci_sequence
{
public: // Member Types
typedef ?? uint32_t or uint64_t ??         value_type;
typedef Fibonacci_traits<value_type>       traits_type;
typedef true_typedef<size_t, unsigned>     limit;
typedef true_typedef<value_type, signed>   threshold;
class                                      const_iterator;
public: // Construction
explicit Fibonacci_sequence(limit l = limit(traits_type::maxLimit))
: m_limit(l.base_type_value())
, m_threshold(0)
{
STLSOFT_MESSAGE_ASSERT( "Sequence limit exceeded"
, l <= traits_type::maxLimit());
}
explicit Fibonacci_sequence(threshold t)
: m_limit(0)
, m_threshold(t.base_type_value())
{
STLSOFT_MESSAGE_ASSERT( "Sequence threshold exceeded"
, t <= traits_type::maxThreshold());
}
public: // Iteration
const_iterator  begin() const;
const_iterator  end() const
{
return (0 == m_limit)
? const_iterator(m_threshold)
: const_iterator(m_limit, 0);
}
public: // Size
bool   empty() const
{
return 0 == m_limit && 0 == m_threshold;
}
size_t max_size() const
{
return traits_type::maxLimit;
}
private: // Member Variables
const size_t      m_limit;
const value_type  m_threshold;
};

Note the use of the traits. Although they're not required by the definition of the sequence as it stands, they serve two important purposes. First, they provide a clear and obvious place for the limit and threshold magic numbers to reside, as well as making them largely self-documenting. Second, should you choose to use a 32- or 64-bit value type, the change involves just a single line.

The iterator class can now be defined as shown in Listing 23.12.

#### Listing 23.12. Version 7: const_iterator

class Fibonacci_sequence::const_iterator
: . . . // As shown previously
{
public: // Member Types
typedef Fibonacci_sequence::value_type      value_type;
typedef Fibonacci_sequence::const_iterator  class_type;
private: // Construction
friend class Fibonacci_sequence;
const_iterator();
const_iterator(Fibonacci_sequence::limit lim);
const_iterator(Fibonacci_sequence::threshold t);
. . . // Iteration and Comparison methods as before
};

With this definition, all the following are well defined (and thereby value constrained):

typedef Fibonacci_sequence  fibseq_t;
fibseq_t  fs(fibseq_t::limit(0));              // Empty sequence
fibseq_t  fs(fibseq_t::limit(1));              // 1 value
fibseq_t  fs(fibseq_t::limit(10));             // 10 values
fibseq_t  fs(fibseq_t::limit(47));             // 47 values
fibseq_t  fs;                                  // 47 values
fibseq_t  fs(fibseq_t::threshold(0));          // Empty sequence
fibseq_t  fs(fibseq_t::threshold(1));          // 1 value
fibseq_t  fs(fibseq_t::threshold(2));          // 3 values
fibseq_t  fs(fibseq_t::threshold(47));         // 10 values
fibseq_t  fs(fibseq_t::threshold(100));        // 12 values
fibseq_t  fs(fibseq_t::threshold(1000000000)); // 45 values

Equally important, the following are not well defined, and the user knows this because he or she can evaluate them against the member constants Fibonacci_sequence::traits_ type::maxLimit and Fibonacci_sequence::traits_type::maxThreshold. Furthermore, because of the enforcements placed in the constructor bodies, the user finds out immediately when something is wrong, rather than at a later point during enumeration when the values overflow.

fibseq_t  fs(fibseq_t::limit(50));             // Breaks ctor precond
fibseq_t  fs(fibseq_t::threshold(2971215075)); // Breaks ctor precond

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.