So you're decomposing a given number into a unique(?) sum of F=.Fibonaccis represented by a Boolean with a 1 for each F? That is, the Boolean does not waste space by representing non-F numbers: the positions 0 1 2 3 4.. represent 1 2 3 5 8...?
It looks potentially efficient if you have very large numbers to represent. On Thu, Jul 18, 2019 at 8:26 PM 'Jon Hough' via Programming < [email protected]> wrote: > The encoding follows the following algorithm: > > "To encode an integer N: > > Find the largest Fibonacci number equal to or less than N; subtract this > number from N, keeping track of the remainder. > If the number subtracted was the ith Fibonacci number F(i), put a 1 in > place i−2 in the code word (counting the left most digit as place 0). > Repeat the previous steps, substituting the remainder for N, until a > remainder of 0 is reached. > Place an additional 1 after the rightmost digit in the code word. > To decode a code word, remove the final "1", assign the remaining the > values 1,2,3,5,8,13... (the Fibonacci numbers) to the bits in the code > word, and sum the values of the "1" bits." > see: https://en.wikipedia.org/wiki/Fibonacci_coding > > > It might not be the most elegant or terse verb, but I think there is not > much complexity in it. You are right that there is no need to use char > sequence, could just use integer list, which may or may not be fast. I > really did that for debugging purposes, to easily read the representation > while encoding. On Friday, July 19, 2019, 1:13:18 AM GMT+9, Raul Miller > <[email protected]> wrote: > > I'm not quite sure I understand your encoding scheme. > > Specifically, you're encoding into a sequence of characters, but when > decoding, you're not using (8#2)#:a.i.y and it looks like you've > cooked up some mechanism to compensate for how |."1#: behaves ... > > In other words, I think that the fibonacci encoding aspect isn't the > problem here. > > So I think I would start with either a careful description of your > "encode to character" mechanism (how can I know if you've implemented > what you intended without that? I'd rather not try to reverse engineer > your thoughts and assumptions if that's not necessary) or I would > start with an implementation which isn't so complicated. > > Thanks, > > -- > Raul > > On Thu, Jul 18, 2019 at 10:46 AM 'Jon Hough' via Programming > <[email protected]> wrote: > > > > Positive integers can be encoded using Fibonacci numbers. > > For the sake of easer, assume Fibonacci numbers are 1,2,3,5,... (only > one 1). > > We can encode a positive integer uniquely using non-consecutive > Fibonacci numbers. For example > > > > 4 = 1 + 3 (note: 1 and 3 a re non-consecutive) > > 5 = 5 (already a Fib number) > > 6 = 1 + 5 > > 7 = 2 + 5 > > 8 = 8 > > > > These representation can be encoded using bits. 1's where a Fibonacci > number is in the representation, 0s when not. > > 4 = 1 + 3 = 101 > > 5 = 0001 > > 6 = 1001 > > 7 = 0101 > > 8 = 00001 > > ... > > > > We can append a 1 on the end of the encoding a a delimiter, so that > during decoding we know easily where to stop. This way integers can be > packed together. > > > > Here are some J verbs for encoding and decoding positive integers. > > > > > > NB. =================================================================== > > genfibs=: 3 : 0 > > r=. 1 2 > > i1=. 1 > > i2=. 2 > > c=. 2 > > while. y > c=. c+1 do. > > t=. i1 + i2 > > i1=. i2 > > i2=. t > > r=. r, x: t > > end. > > ) > > > > > > Fibs=: genfibs 1400 > > > > encode=: 3 : 0 > > d=. > (>@:,&:>)/ (<@encode1)"0 y > > r=. d,'0' #~ (#d) -~ 8 * >. 8 %~ # d > > pack r > > ) > > > > encode1=: 3 : 0 > > n=. x: y > > r=. '' > > k=: '' > > fl=. x: Fibs{~ I. Fibs <: y > > i=. <:#fl > > while. n do. > > r=. r,'1' > > n=. n- i{fl > > k=: k,i{fl > > i=. i-1 > > while. (i >: 0) *. (n<i{fl) do. > > r=. r,'0' > > i=. i-1 > > end. > > end. > > (|.r),'1' > > ) > > > > > > > > pack=: 3 : 0 > > a.{~ #. _8 ] \"."0 y > > ) > > > > decode=: 3 : 0 > > i=. , {:|."1 (8 # 2) ,:|."1 #: a.i. y > > n=. '' > > while. 1 e. 1 1 E. i do. > > idx=. {. I. 1 1 E. i > > n=. n, +/ Fibs {~ I. i {~ i. >: idx > > i=. (2+idx) }. i > > end. > > n > > ) > > > > NB. =================================================================== > > > > The encode verb outputs the Fib-representation as bytes. > > encode 4 > > � > > > > #: a.i. encode 4 > > 1 0 1 1 0 0 0 0 > > > > > > decode encode 449239438124834384923493383837734733747181x > > 449239438124834384923493383837734733747181 > > > > > > decode encode 1 2 3 4 5 6 7 > > 1 2 3 4 5 6 7 > > > > Fibonacci encoding is not really a good compression encoding, but has > some good error handling properties. > > I am not happy with the decode verb, as I wanted to write it without a > while loop. Difficult though as there are some > > awkaward consecutive bits that might appear in the encoding, e.g. > > ...0 1 1 1 0... > > Using 1 1 E. with this breaks decoding unless done carefully. Easier > just to use a while loop and eat the input every iteration. > > > > refs: https://en.wikipedia.org/wiki/Fibonacci_coding > > https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > -- Devon McCormick, CFA Quantitative Consultant ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
