I'd favour putting a count of the number of significant characters in the last byte in the last 2 bits of the that byte. (Which may require a zero last byte.)

The advantage is that it makes comparisions slightly easier...

1. strings which compress to the same number of bytes can be compared directly.

2. strings which compress to different numbers of bytes can be compared directly up to the last byte of the shorter. If that doesn't settle the matter, then you need to use the last bits of the shorter to select a mask...

I'd favour putting a count of the number of significant characters in the last byte in the last 2 bits of the that byte. (Which may require a zero last byte.)

Sorry guys, but I don't see any way of making a trailing indicator allow for string comparison.

• If you do this way, the bigger strings sort before shorter strings:

  input      Using a significant bit count
             in the last 2 bits of the last byte
  
  ACGT       1b 00
  ACGTA      1b 01
  ACGTAA     1b 02
  ACGTAAA    1b 03
  ACGTAAAA   1b 00 00
  ACGTAAAAA  1b 00 01
  [download]

• This way is a non starter, things sort all over the place:

  input      Using the significant bit count
             in the first byte
  
  ACGT       00 1b 00
  ACGTA      01 1b 00
  ACGTAA     02 1b 00
  ACGTAAA    03 1b 00
  ACGTAAAA   00 1b 00 00
  ACGTAAAAA  01 1b 00 00
  [download]

• With a trailing full length, the last comes first and the first last, with those in the middle keeping their correct relative ordering:

  input      packed with trailing  
             network (BE) length   
  
  ACGT       1b 00 00 00 04        
  ACGTA      1b 00 00 00 00 05     
  ACGTAA     1b 00 00 00 00 06     
  ACGTAAA    1b 00 00 00 00 07     
  ACGTAAAA   1b 00 00 00 00 08     
  ACGTAAAAA  1b 00 00 00 00 00 09
  [download]

• The only method that I can see working is prefixing the output length in network (BE) order to the front:

  input      packed with leading  
             network (BE) length   
  
  ACGT       00 00 00 04 1b       
  ACGTA      00 00 00 05 1b 00
  ACGTAA     00 00 00 06 1b 00   
  ACGTAAA    00 00 00 07 1b 00   
  ACGTAAAA   00 00 00 08 1b 00   
  ACGTAAAAA  00 00 00 09 1b 00
  [download]

  Update: Even that doesn't work! These two sort in the wrong order.

  in: ACGTACGTACGTACGTATT  packed( 9): 000000131b1b1b1b3c  
  in: ACGTACGTACGTACGTAAAA packed(10): 000000141b1b1b1b0000
  [download]

  And the penalty of that for what is intended as a compression mechanism is prohibitive for short sequences. You might lower the overhead by using a short instead of a long, but 64k is not enough for real genome work.

That said, the sorting and comparing of packed ACGT (other than simple equality), is a fairly non-useful thing anyway. Sequence and subsequence representations don't have any intrisic ordering.

The more frequent operation is to search one sequence for the presence of another (sub)sequence, and doing that with packed sequences means you're only checking every fourth position, rather than every position. Performing shifts or rotates on bitstring greater than register size is a prohibitively expensive operation. It far outweights any gains you might get from searching quarter sized strings, as you have to perform 4 searches anyway, and you need the expensive shift operation inbetween each search.

I think the only useful use for packed ACGT (2bit) format is to reduce storage overhead. It's almost certainly quicker to just unpack the sequences for searching. In which case, the prefix byte of the significant bits in the last byte is a good compromise between compression level and implementation ease.

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Expanding on what I suggested:

1. strings which compress to the same number of bytes can be compared directly.

   As per your example:

     ACGT       1b 00
     ACGTA      1b 01
     ACGTAA     1b 02
     ACGTAAA    1b 03
   

2. strings which compress to different numbers of bytes can be compared directly up to the last byte of the shorter. If that doesn't settle the matter, then you need to use the last bits of the shorter to select a mask...

   Taking your example:

     ACGTAAA    1b 03
     ACGTAAAA   1b 00 00
   

   the strings match up to (but, for the avoidance of doubt, not including) the last byte of the shorter, so the '3' selects the mask 0xFC, under that mask the last corresponding bytes are equal, so the longer string is the larger. Similarly:

     ACGT TGCA AGA    1b F4 23
     ACGT TGCA AGAC   1b F4 21 00
   

   Actually, it's not strictly necessary to apply the mask to both strings, or indeed to use any mask other than 0xFC -- but it seemed tidier.

There is some futzing about to be done to compare the compressed forms, though the decision is likely to be made before the last byte of the shorter.

Of course, if the comparison is only for searching for equality, then it doesn't really matter where you put the length mod 4 (except that if there's little variation in length, putting it at the end keeps it out of the way). Alphabetic order does appeal to a sense of neatness, though.

As far as implementation of packing/unpacking goes, placing the length mod 4 at the end means that all the fiddling about happens at the end, rather than at both ends. I have modified the code I posted previously to do this and included that below. I look forward to improvements :-)

Mind you, if performance is the issue, I'd cast this into C.

Read more... See the code (2 kB)

With a trailing full length, the last comes first and the first last, with those in the middle keeping their correct relative ordering

So, don't use the actual length, use length mod 4.

input	packed
ACGT	1b 00
ACGTA	1b 01
ACGTAA	1b 02
ACGTAAA	1b 03
ACGTAAAA	1b 00 00
ACGTAAAAA	1b 00 01

string length +1 modulo 4 is the same as the number of significant characters in the last byte. I.e. that is an implementation detail I left out on purpose.

If you mean comparision for equality, then putting the count into the first byte will be a lot faster for 75% of different-length strings and not much slower for same-length strings. Only if you want to know the first position at which they differ is the last byte better.

The advantage I see for last byte is that you can more easily append or cut from the end of the compressed string

Sure. If you're only interested in equality, then having the length component at the front will help -- unless the strings are generally the same length, in which case it makes things worse.

For an "alphabetic" comparision the length is only significant if the strings are equal up to the end of shorter, so I reckon the trick is to put the length information at the far end.