Q21: What are Gray codes, and why are they used?

     The correct spelling is "Gray"---not  "gray",  "Grey",  or  "grey"---
     since Gray codes are named after the Frank Gray  who  patented  their
     use for shaft encoders in 1953  [1].   Gray  codes  actually  have  a
     longer history, and the inquisitive reader may want to  look  up  the
     August, 1972,  issue  of  Scientific  American,  which  contains  two
     articles of interest: one on the origin  of  binary  codes  [2],  and
     another by Martin  Gardner  on  some  entertaining  aspects  of  Gray
     codes [3].  Other references containing descriptions  of  Gray  codes
     and more modern, non-GA, applications include the second  edition  of
     Numerical  Recipes  [4],  Horowitz  and  Hill  [5],  Kozen  [6],  and
     Reingold [7].

     A Gray code represents  each  number  in  the  sequence  of  integers
     {0...2^N-1} as a binary string of length N  in  an  order  such  that
     adjacent integers have Gray code representations that differ in  only
     one bit position.  Marching through the  integer  sequence  therefore
     requires flipping just one bit at a time.  Some  call  this  defining
     property of Gray codes the "adjacency property" [8].

     Example (N=3):  The binary coding of {0...7} is {000, 001, 010,  011,
     100, 101, 110, 111}, while one Gray coding is {000,  001,  011,  010,
     110, 111, 101, 100}.  In essence, a Gray code takes a binary sequence
     and shuffles  it  to  form  some  new  sequence  with  the  adjacency
     property.  There exist,  therefore,  multiple  Gray  codings  or  any
     given N.  The example shown here belongs to a  class  of  Gray  codes
     that goes by the fancy name "binary-reflected Gray codes".  These are
     the most  commonly  seen  Gray  codes,  and  one  simple  scheme  for
     generationg such a Gray code sequence says, "start with all bits zero
     and successively flip the right-most bit that produces a new string."

     Hollstien [9] investigated the use of GAs for optimizing functions of
     two variables and claimed that  a  Gray  code  representation  worked
     slightly better than the binary representation.  He attrributed  this
     difference to the adjacency property of Gray codes.   Notice  in  the
     above example that the step from three to four requires the  flipping
     of all the bits in the binary representation.  In  general,  adjacent
     integers in the binary representaion often lie many bit flips  apart.
     This  fact makes it less likely that a MUTATION operator  can  effect
     small changes for a binary-coded INDIVIDUAL.

     A Gray code representation seems to  improve  a  MUTATION  operator's
     chances of making incremental improvements, and a  close  examination
     suggests why.  In  a  binary-coded  string  of  length  N,  a  single
     mutation in the most significant  bit  (MSB)  alters  the  number  by
     2^(N-1).  In a Gray-coded string, fewer mutations lead  to  a  change
     this large.  The user of Gray codes does, however, pay  a  price  for
     this feature: those "fewer mutations" lead to  much  larger  changes.
     In the Gray code illustrated above, for example, a single mutation of
     the left-most bit changes a zero to a seven and vice-versa, while the
     largest change a single mutation can make to a corresponding  binary-
     coded INDIVIDUAL is always four.  One might still view this aspect of
     Gray codes with some favor:  most  mutations  will  make  only  small
     changes, while the occasional  mutation  that  effects  a  truly  big
     change may initiate EXPLORATION of an  entirely  new  region  in  the
     space of CHROMOSOMEs.

     The algorithm for converting between the binary-reflected  Gray  code
     described above  and  the  standard  binary  code  turns  out  to  be
     surprisingly simple to state.  First label the bits of a binary-coded
     string B[i], where larger i's represent more  significant  bits,  and
     similarly label the corresponding Gray-coded string G[i].  We convert
     one to the other as follows:  Copy the most  significant  bit.   Then
     for each smaller i  do  either  G[i] = XOR(B[i+1], B[i])---to convert
     binary to  Gray---or B[i] = XOR(B[i+1], G[i])---to  convert  Gray  to

     One may easily implement the above algorithm in C.   Imagine  you  do
     something like

	  typedef unsigned short ALLELE;

     and then use type "allele" for each bit in your CHROMOSOME, then  the
     following two functions will convert between  binary  and  Gray  code
     representations.  You must pass them the address  of  the  high-order
     bits for each of the two strings  as  well  as  the  length  of  each
     string.  (See  the  comment  statements  for  examples.)   NB:  These
     functions assume a chromosome arranged  as  shown  in  the  following

     index:    C[9]                                                    C[0]
     Char C:   |  1  |  1  |  0  |  0  |  1  |  0  |  1  |  0  |  0  |  0  |
	       ^^^^^                                                 ^^^^^
	       high-order bit                                  low-order bit

     /* Gray <==> binary conversion routines */
     /* written by Dan T. Abell, 7 October 1993 */
     /* please send any comments or suggestions */
     /* to dabell@quark.umd.edu */

     void gray_to_binary (Cg, Cb, n)
     /* convert chromosome of length n from */
     /* Gray code to binary representation */
     allele    *Cg,*Cb;
     int  n;
	  int j;

	  *Cb = *Cg;               /* copy the high-order bit */
	  for (j = 0; j < n; j++) {
	       Cb--; Cg--;         /* for the remaining bits */
	       *Cb= *(Cb+1)^*Cg;   /* do the appropriate XOR */

     void binary_to_gray(Cb, Cg, n)
     /* convert chromosome of length n from */
     /* binary to Gray code representation */
     allele    *Cb, *Cg;
     int  n;
	  int j;

	  *Cg = *Cb;               /* copy the high-order bit */
	  for (j = 0; j < n; j++) {
	       Cg--; Cb--;         /* for the remaining bits */
	       *Cg= *(Cb+1)^*Cb;   /* do the appropriate XOR */


     [1]  F.  Gray,  "Pulse  Code  Communication", U. S. Patent 2 632 058,
     March 17, 1953.

     [2] F. G. Heath, "Origins of the Binary  Code",  Scientific  American
     v.227,n.2 (August, 1972) p.76.

     [3]   Martin   Gardner,  "Mathematical  Games",  Scientific  American
     v.227,n.2 (August, 1972) p.106.

     [4] William H. Press, et al., Numerical Recipes in C, Second  Edition
     (Cambridge University Press, 1992).

     [5]  Paul  Horowitz and Winfield Hill, The Art of Electronics, Second
     Edition (Cambridge University Press, 1989).

     [6] Dexter Kozen, The Design and Analysis  of  Algorithms  (Springer-
     Verlag, New York, NY, 1992).

     [7]  Edward  M.  Reingold, et al., Combinatorial Algorithms (Prentice
     Hall, Englewood Cliffs, NJ, 1977).

     [8] David E. Goldberg, GENETIC ALGORITHMs  in  Search,  OPTIMIZATION,
     and Machine Learning (Addison-Wesley, Reading, MA, 1989).

     [9]  R.  B.  Hollstien,  Artificial  Genetic  Adaptation  in Computer
     Control Systems (PhD thesis, University of Michigan, 1971).
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