Wednesday, February 23, 2011

when 255 is too little and 65535 is too much

Before I begin, yes, I do realize that low-level bit twiddling is often not worth the time any more. When storage and memory is so cheap, and the sheer scale of processing has progressed so far even on a "smart" phone, little micro-optimizations may...just...not....matter. Switching to an alternative high-level algorithm/strategy better suited to the specific scenario is generally a much more promising route. However, network transmission and/or mass data record layout are two prominent exceptions to this rule. The fewer bytes that must be sent between computers on potentially high-latency and congested networks, the better, and the fewer average bytes per record that is multiplied by a potential quantity of thousands up to millions of records, the better. So there still is some life yet for bit twiddling.

One unsigned byte represents nonnegative integers up to 255 (including 0). Two unsigned bytes represent nonnegative integers up to 65535. The storage doubles, but the representational range squares. In basic-arithmetic-speak the range is "256 times its original amount". In science-speak the difference might be called "an approximate difference of 2.40824 on a logarithmic (lg) scale". In marketing-speak the difference might be called "an increase of 25,500%". The upshot is that this exponential jump could very well be overkill in particular cases. For instance, if data values never are less than zero or greater than 4000, 12 bits or one-and-half bytes is sufficient (2^12 -1 = 4095).

Yet reading and writing an arbitrary number of bits like 12 isn't well-supported, for valid performance reasons. Manipulating 12 bits in practice implies manipulating the next-greatest number of bytes, 2 or 16 bits. And that implies 4 wasted bits for "spacing" or "alignment". Why not use these bits? Two of these 12-bit unsigned integers sum to 24 bits or 3 bytes. When each of the two 12-bit number is stored in 16 bits, the fourth overall byte, consisting of the top 4 unused bits from the first number and the top 4 unused bits from the second, are waste. Avoidance of that fourth byte reduces total storage needs by 25%. Whether this substantial relative space savings amounts to a substantial absolute/raw space savings depends on the application and other factors. I'd venture that in the majority of the time it doesn't. But when it does...

Each 12-bit integer is a byte-and-half. For two, just combine their two half-bytes (4 bits) into byte number three (8 bits). The conversion procedure from two nonnegative 12-bit integers, originally stored as two 16-bit integers (i.e. unsigned short) or 4 total bytes originally:
  1. 1) Get the first output byte by truncating/coercing/casting the first 16-bit integer. This grabs the 8 low bits of the first number, leaving the remaining 4 high bits to be stored separately.
  2. 2) In the same way as #1, get the second output byte by truncating/coercing/casting the second 16-bit integer.
  3. 3) Get the remaining 4 high bits of the first integer by right-shifting it by 8 and then coercing to a byte. The result is the top 4 bits of the first integer stored in a byte as-is. All the lower 8 bits of the first integer, that were previously stored in #1 and became the first output byte, are right-shifted out completely into oblivion and zeroes take the place of the shifted bits. This byte is now the "top byte" of the two bytes in the original 16-bit storage of the integer.
  4. 4) In the same way as #3, get the remaining 4 high bits of the second integer by right-shifting by 8 and then coercing to a byte.
  5. 5) The third and last output byte will be a concatenation of the two integers' top 4 bits, which after #3 and #4 are now themselves stored as bytes in which the top 4 bits are waste. The lower 4 bits of the output byte will come from the #3 byte, in which the correct bits are already at the correct lower position. But the higher 4 bits of the output byte will come from the #4 byte, in which the correct bits are also in the lower position, which is incorrect for those 4 bits. Therefore, left-shift #4 byte by 4 to get the bits in final position.
  6. 6) Finally the two parts of the third byte, the #3 byte and #5 byte, are ready, with correct bits in correct positions and zeroes everywhere else. Thus each bit in each byte must "override" the zeroed-out bit in the other. If corresponding bits are both 0, then 0. If corresponding bits are 0 and 1, then 1. Since this is the definition of a bitwise OR (|), perform that operation on the #3 byte and #5 byte to get the third byte.
The conversion from three unsigned bytes back to two 12-bit integers, stored individually in two unsigned 16-bit variables:
  1. 1) The lower 8 bits of the first integer is the first input byte. Expand/coerce/cast the first input byte to 16-bit storage.
  2. 2) The lower 8 bits of the second integer is the second input byte. Also expand/coerce/cast to 16 bits.
  3. 3) The third byte has the top 4 bits of the first integer in lower position, and the top 4 bits of the second integer in higher position. To get the top 4 bits of the first integer as a byte all to itself, just the top 4 bits of the third byte must be replaced with zeroes. For each of those top 4 bits, 0 must result in 0 and 1 must result in 0. For each of the lower bits, the bits that must remain undisturbed, 0 must result in 0 and 1 must result in 1. A bitwise operation that sometimes turns the first bit into 0 and sometimes into 1, depending on the second bit, is AND (&). Any time the second bit of an AND is 0, the result will be 0. Any time the second bit of an AND is 1, the result will equal the first bit. Bitwise AND fits the purpose, provided all the bits in the second byte are set accordingly. This second byte for the AND operation, known as a bitmask because the zeroes in it will "mask" or hide bits while the ones in it will "pass through" bits, must simply have zeroes in the 4 higher bits and ones in the 4 lower bits. Expressed as a byte, the bitmask is "0000 1111" in binary or "0F" in hexadecimal or "15" in decimal. In any case, the result of the AND operation between the third input byte and the bitmask is the upper 4 bits of the original first 12-bit integer.
  4. 4) The higher 4 bits of the third input byte are the top 4 bits of the original second 12-bit integer. To get these 4 bits as a byte all to itself, right-shift the third input byte by 4. This pushes out the lower 4 bits of the third input byte, that was stored in #3, and leaves the higher 4 bits (with replacement zeroes in higher position).
  5. 5) The #3 byte, in which the higher 4 bits of the original first12-bit integer are in lower position within one byte, equals the original higher byte of the 16-bit storage of the original first 12-bit integer. All that's necessary is to put the #4 byte into correct 16-bit position, which is precisely one byte or 8 bits higher. Expand/coerce/cast the #3 byte into 16-bit storage and then left shift by 8.
  6. 6) In the same way as #5, expand/coerce/cast the #4 byte into 16-bit storage and then left shift by 8.
  7. 7) #1 and #5 are 16-bit unsigned integers whose bits are all in the right locations for the original first 12-bit integer stored in two bytes. And any bits in #1 and #5 that don't match are zero. This means that a bitwise OR between the two will "override" the placeholder zeroes with the real bits in the other and get the original two bytes of the first 12-bit integer.
  8. 8)  In the same way as #7, #2 and #6 after bitwise OR will be the original two bytes of the second 12-bit integer.

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