Is The Effectiveness Of Huffman Coding Limited?

My problem is that I have a开发者_JS百科 100,000+ different elements and as I understand it Huffman works by assigning the most common element a 0 code, and the next 10, the next 110, 1110, 11110 and so on. My question is, if the code for the nth element is n-bits long then surely once I have passed the 32nd term it is more space efficient to just sent 32-bit data types as they are, such as ints for example? Have I missed something in the methodology?

Many thanks for any help you can offer. My current implementation works by doing

code = (code << 1) + 2;

to generate each new code (which seems to be correct!), but the only way I could encode over 100,000 elements would be to have an int[] in a makeshift new data type, where to access the value we would read from the int array as one continuous long symbol... that's not as space efficient as just transporting a 32-bit int? Or is it more a case of Huffmans use being with its prefix codes, and being able to determine each unique value in a continuous bit stream unambiguously?

Thanks

Your understanding is a bit off - take a look at http://en.wikipedia.org/wiki/Huffman_coding. And you have to pack the encoded bits into machine words in order to get compression - Huffman encoded data can best be thought of as a bit-stream.

You seem to understand the principle of prefix codes.

Could you tell us a little more about these 100,000+ different elements you mention?

The fastest prefix codes -- universal codes -- do, in fact, involve a series of bit sequences that can be pre-generated without regard to the actual symbol frequencies. Compression programs that use these codes, as you mentioned, associate the most-frequent input symbol to the shortest bit sequence, the next-most-frequent input symbol to the next-shorted bit sequence, and so on.

What you describe is one particular kind of prefix code: unary coding. Another popular variant of the unary coding system assigns elements in order of frequency to the fixed codes "1", "01", "001", "0001", "00001", "000001", etc.

Some compression programs use another popular prefix code: Elias gamma coding. The Elias gamma coding assigns elements in order of frequency to the fixed set of codewords

1
010
011
00100
00101
00110
00111
0001000
0001001
0001010
0001011
0001100
0001101
0001110
0001111
000010000
000010001
000010010
...

The 32nd Elias gamma codeword is about 10 bits long, about half as long as the 32nd unary codeword. The 100,000th Elias gamma codeword will be around 32 bits long.

If you look carefully, you can see that each Elias gamma codeword can be split into 2 parts -- the first part is more or less the unary code you are familiar with. That unary code tells the decoder how many more bits follow afterward in the rest of that particular Elias gamma codeword.

There are many other kinds of prefix codes. Many people (confusingly) refer to all prefix codes as "Huffman codes".

When compressing some particular data file, some prefix codes do better at compression than others. How do you decide which one to use? Which prefix code is the best for some particular data file?

The Huffman algorithm -- if you neglect the overhead of the Huffman frequency table -- chooses exactly the best prefix code for each data file. There is no singular "the" Huffman code that can be pre-generated without regard to the actual symbol frequencies. The prefix code choosen by the Huffman algorithm is usually different for different files.

The Huffman algorithm doesn't compress very well when we really do have 100,000+ unique elements -- the overhead of the Huffman frequency table becomes so large that we often can find some other "suboptimal" prefix code that actually gives better net compression. Or perhaps some entirely different data compression algorithm might work even better in your application.

The "Huffword" implementation seems to work with around 32,000 or so unique elements, but the overwhelming majority of Huffman code implementations I've seen work with around 257 unique elements (the 256 possible byte values, and the end-of-text indicator).

You might consider somehow storing your data on a disk in some raw "uncompressed" format. (With 100,000+ unique elements, you will inevitably end up storing many of those elements in 3 or more bytes). Those 257-value implementations of Huffman compression will be able to compress that file; they re-interpret the bytes of that file as 256 different symbols.

My question is, if the code for the nth element is n-bits long then surely once I have passed the 32nd term it is more space efficient to just sent 32-bit data types as they are, such as ints for example? Have I missed something in the methodology?

One of the more counter-intuitive features of prefix codes is that some symbols (the rare symbols) are "compressed" into much longer bit sequences. If you actually have 2^8 unique symbols (all possible 8 bit numbers), it is not possible to gain any compression if you force the compressor to use prefix codes limited to 8 bits or less. By allowing the compressor to expand rare values -- to use more than 8 bits to store a rare symbol that we know can be stored in 8 bits -- that frees up the compressor to use less than 8 bits to store the more-frequent symbols.

related: Maximum number of different numbers, Huffman Compression