Negafibonacci coding: Difference between revisions

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In [[mathematics]], '''negafibonacci coding''' is a [[universal code (data compression)|universal code]] which encodes nonzero integers into binary [[Code word (communication)|code word]]s. It is similar to [[Fibonacci coding]], except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end.

{{Unreferenced|date=September 2009}}

{{Cleanup|date=January 2009}}

{{numeral systems}}

In [[mathematics]], '''negaFibonacci coding''' is a [[universal code (data compression)|universal code]] which encodes nonzero integers into binary [[code word]]s. It is similar to [[Fibonacci coding]], except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end. The code for the integers from -11 to 11 is given below.

== Encoding method ==

xx negaFibonacci representation negaFibonacci code

-11 101000 0001011

-10 101001 1001080

-9 100010 0100011

-8 100000 0000011

-7 100001 1000011

-6 100100 0010011

-5 100101 1010011

-4 1010 01011

-3 1000 00011

-2 1001 10011

-1 10 011

0 0 (cannot be encoded)

1 1 11

2 100 0011

3 101 1011

4 10010 010011

5 10000 000011

6 10001 100011

7 10100 001011

8 10101 101011

9 1001010 01010011

10 1001000 00010011

11 1001001 10010011

The following steps describe how to encode a nonzero integer <math> x </math>. Note that <math> f </math> denotes the Negafibonacci sequence.

The Fibonacci code is closely related to ''negaFibonacci representation'', a positional [[numeral system]] sometimes used by mathematicians. The negaFibonacci code for a particular nonzero integer is exactly that of the integer's negaFibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negaFibonacci code for all negative numbers has an odd

number of digits, while those of all positive numbers have an even number of digits.

# If <math> x </math> is positive, compute the greatest odd negative integer <math> n </math> such that the sum of the odd negative terms of the Negafibonacci sequence from −1 to <math> n </math> with a step of −2, is greater than or equal to <math> x </math>: <br /> <math> n \in \{ - \left( 2k + 1 \right) , k \in [0, \infty [ \} , \quad \sum_{i=-1, \; i \; odd}^{n-2} f(i) < x \leq \sum_{i=-1, \; i \; odd}^{n} f(i). </math> <br /> If <math> x </math> is negative, compute the greatest even negative integer <math> n </math> such that the sum of the even negative terms of the Negafibonacci sequence from 0 to <math> n </math> with a step of −2, is less than or equal to <math> x </math>: <br /> <math> n \in \{ - 2k , k \in [2, \infty [ \} , \quad \sum_{i=-2, \; i \; even}^{n-2} f(i) > x \geq \sum_{i=-2, \; i \; even}^{n} f(i) </math>

To encode a nonzero integer ''X'':

# Add a 1 at the <math> |n|^{\text{th}} </math> bit of the binary word. Subtract <math> f(n) </math> from <math> x </math>.

# Calculate the largest (or smallest) encodeable number with ''N'' bits by summing the odd (or even) [[negafibonacci]] numbers from 1 to ''N''.

# Repeat the process from step 1 with the new value of ''x'', until it reaches 0.

# When it is determined that ''N'' bits is just enough to contain ''X'', subtract the ''Nth'' negaFibonacci number from ''X'', keeping track of the remainder, and put a one in the ''Nth'' bit of the output.

# Add a 1 on the left of the resulting binary word to finish the encoding.

# Working downward from the ''Nth'' bit to the first one, compare each of the corresponding negaFibonacci numbers to the remainder. Subtract it from the remainder if the absolute value of the difference is less, AND if the next higher bit does not already have a one in it. A one is placed in the appropriate bit if the subtraction is made, or a zero if not.

# Put a one in the ''N+1th'' bit to finish.

To decode aan tokenencoded in thebinary codeword, remove the lastleftmost "1" from the binary word, since it is used only to denote the end of the encoded number. Then assign the remaining bits the values of the Negafibonacci sequence from −1 (1,-1 −1, 2,-3 −3, 5,-8 −8, 13... (the [[negafibonacci]] numbers), and addsum the "1"all bitsthe values associated with a 1.

== Negafibonacci representation ==

==See also==

<!--[[Negafibonacci representation redirects here; please update the redirect if this section is retitled, split, or moved.-->

{{numeral systems}}

Negafibonacci coding is closely related to '''negafibonacci representation''', a positional [[numeral system]] sometimes used by mathematicians. The negafibonacci code for a particular nonzero integer is exactly that of the integer's negafibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negafibonacci code for all negative numbers has an odd number of digits, while those of all positive numbers have an even number of digits.

== Table ==

The code for the integers from −11 to 11 is given below.

{| class="wikitable"

|-

!Number

!Negafibonacci representation

!Negafibonacci code

|-

| −11 ||101000 ||0001011

|-

| −10 ||101001 ||1001011

|-

| −9 ||100010 ||0100011

|-

| −8 ||100000 ||0000011

|-

| −7 ||100001 ||1000011

|-

| −6 ||100100 ||0010011

|-

| −5 ||100101 ||1010011

|-

| −4 ||1010 ||01011

|-

| −3 ||1000 ||00011

|-

| −2 ||1001 ||10011

|-

| −1 ||10 ||011

|-

| 0 ||0 ||(cannot be encoded)

|-

| 1 ||1 ||11

|-

| 2 ||100 ||0011

|-

| 3 ||101 ||1011

|-

| 4 ||10010 ||010011

|-

| 5 ||10000 ||000011

|-

| 6 ||10001 ||100011

|-

| 7 ||10100 ||001011

|-

| 8 ||10101 ||101011

|-

| 9 ||1001010 ||01010011

|-

| 10 ||1001000 ||00010011

|-

| 11 ||1001001 ||10010011

|}

== See also ==

* [[Fibonacci numbers]]

* [[Golden ratio base]]

* [[Zeckendorf's theorem]]

== References ==

{{No footnotes|date=September 2022}}

{{Reflist}}

=== Works cited ===

{{Refbegin}}

* {{Cite conference |last=Knuth |first=Donald |year=2008 |title=Negafibonacci Numbers and the Hyperbolic Plane |conference=Annual meeting of the Mathematical Association of America |___location=San Jose, California}}

* {{Cite book |last=Knuth |first=Donald |title=[[The Art of Computer Programming]], Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams |year=2009 |publisher=Addison-Wesley |isbn=978-0-321-58050-4}} In the [https://www-cs-faculty.stanford.edu/~uno/fasc1a.ps.gz pre-publication draft of section 7.1.3] see in particular pp.&nbsp;36–39.

* {{Cite book |last=Margenstern |first=Maurice |url=https://books.google.com/books?id=eEgvfic3A4kC&pg=PA79 |title=Cellular Automata in Hyperbolic Spaces |publisher=Archives contemporaines |year=2008 |isbn=9782914610834 |series=Advances in unconventional computing and cellular automata |volume=2 |page=79}}

{{Refend}}

{{Compression Methods}}

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[[Category:Lossless compression algorithms]]

[[Category:Fibonacci numbers]]

[[Category:Data compression]]

[[fr:Codage de Fibonacci]]