Let me clarify, I'm not talking about perfect compression in the sense of an algorithm that is able to compress any given source material, I realize that is impossible. What I'm trying to get at is an algorithm that is able to encode any source string of bits to it's absolute maximum compressed state, as determined by it's Shannon entropy.
I believe I have heard some things about Huffman Coding being in some sense optimal, so I believe that this encryption scheme might be based off that, but here is my issue:
Consider the bit-strings: a = "101010101010", b = "110100011010".
Using plain Shannon entropy, these bit strings should have the exact same entropy when we consider the bit strings as simply symbols of 0's and 1's, but this approach is flawed, because we can intuitively see that bitstring a has less entropy than bitstring b because it is simply a pattern of repeated 10's. With this in mind, we could get a better idea of the actual entropy of the source by calculating the Shannon entropy for the composite symbols 00, 10, 01, and 11.
This is just my understanding, and I could be totally off base, but from what I understand, for an ergodic source to be truly random, for an ergodic source with length n. the statistical probability of all n-length groups of symbols must be equally likely.
I suppose to be more specific than the question in the title, I have three main questions:
Does Huffman encoding using single bits as symbols compress a bitstring like a optimally, even with an obvious pattern that occurs when we analyze the string at the level of 2-bit symbols? If not, could one optimally compress a source by cycling through different "levels" (sorry if I'm butchering the terminology here) of Huffman coding until the best compression rate is found? Could going through different "rounds" of Huffman coding further increase the compression rate in some instances? (e.a. first go through Huffman coding with symbols that are 5 bits long, then going through Huffman coding for symbols that are 4 bits long? huff_4bits(huff_5bits(bitstring)) )
As stated by Mark, the general answer is "no", due to Kolmogorov complexity. Let me expand a bit on that.
Compression is basically two steps :
1) Model
2) Entropy
The role of the model is to "guess" the next bytes or fields to come.
Model can have any form, and there is no limit to its effectiveness.
A trivial example is a random number generator function : from an external perspective, it looks like a noise, and therefore cannot be compressed. But if you know the generation function, an infinitely long sequence can be compressed into a small set of code, the generator function.
That's why there is "no limit", and Kolmogorov complexity just states that : you can never guarantee that there is not a better way to "model" the data.
The second part is computable : Entropy is where you find the "Shannon Limit".
Given a set of symbols (typically, the output symbols from the model), which are part of an alphabet, you can compute the optimal cost, and find a way to reach the proven ultimate compression limit, which is the Shannon limit.
Huffman is optimal with regards to the Shannon limit if you accept the limitation that each symbol must be encoded using an integer number of bits. This is close but imperfect approximation. Better compression can be achieved by using fractional bits, which is what Arithmetic Coders do offer, or the more recent ANS-based Finite State Entropy coder. Both get much closer to the Shannon limit.
The Shannon limit only applies if you treat a set of symbols "individually". As soon as you try to "combine them", or find any correlations between the symbols, you are "modeling". And this is the territory of Kolmogorov Complexity, which is not computable.
No. It can be proven that there is not even an algorithm to determine how well a perfect compressor will do. See Kolmogorov Complexity.
Huffman coding (or arithmetic coding) by itself does not get close to the best compression. Other techniques need to be used to take advantage of higher order redundancies in the data.
Related
I'm trying to implement Arithmetic Code Algorithm to compress binary images (JPG images transformed to binary base using opencv). The problem is that I've to save in the compressed file, the encoded string and the symbols which I used to generate this encoded string and also their frequencies, so I can be able to decode it. The symbols take a lot of space even if I'm transforming them to ascii and if I tried to take less number of characters for each symbol the size of the encoded string becomes bigger. So I wonder if there's an efficient way to save symbols in the compressed file with minimum possible size. And I want to know the most efficient way to choose the symbols from the original file.
Thanks in advance :)
325,592,005 bytes is 310 megabytes. You managed to compress this image into 2.8+6.1=8.9 megabytes so you decreased the size by 97%. It's a good result and I wouldn't worry here. Besides 6.1 megabytes of 64 bits long symbols means that you have around 800K of them. It is much less than the maximum possible number of possible symbols i.e. 2^64 - 1. It is again a good result.
As to your question about using multiple compression algorithms. Firstly, you have to know that in the case of the loosless compression the optimal number of bits per symbol is equal to the entropy. And the arithmetic encoding is close to be optimal (see this, this or this). It means that there is no much sense in using more than 1 algorithm one after another, if one of them is the arithmetic encoding.
As to the arithmetic coding vs Huffman codes. The latter is actually the special case of the former. And as far as I know the arithmetic encoding is always at least as good as Huffman codes.
It is also worth adding one thing. If you can consider lossy compression there is actually no limit in the compression rate. Or in other words, you can compress data as much as you want as long as quality loss is still acceptable for you. However, even in this case using multiple compression algorithms is not required. The one is enough.
I have 3 main questions:
Let's say I have a large text file. (1)Is replacing the words with their rank an effective way to compress the file? (Got answer to this question. This is a bad idea.)
Also, I have come up with a new compression algorithm. I read some existing compression models that are used widely and I found out they use some pretty advanced concepts like statistical redundancy and probabilistic prediction. My algorithm does not use all these concepts and is a rather simple set of rules that need to be followed while compressing and decompressing. (2)My question is am I wasting my time trying to come up with a new compression algorithm without having enough knowledge about existing compression schemes?
(3)Furthermore, if I manage to successfully compress a string can I extend my algorithm to other content like videos, images etc.?
(I understand that the third question is difficult to answer without knowledge about the compression algorithm. But I am afraid the algorithm is so rudimentary and nascent I feel ashamed about sharing it. Please feel free to ignore the third question if you have to)
Your question doesn't make sense as it stands (see answer #2), but I'll try to rephrase and you can let me know if I capture your question. Would modeling text using the probability of individual words make for a good text compression algorithm? Answer: No. That would be a zeroth order model, and would not be able to take advantage of higher order correlations, such as the conditional probability of a given word following the previous word. Simple existing text compressors that look for matching strings and varied character probabilities would perform better.
Yes, you are wasting your time trying to come up with a new compression algorithm without having enough knowledge about existing compression schemes. You should first learn about the techniques that have been applied over time to model data, textual and others, and the approaches to use the modeled information to compress the data. You need to study what has already been researched for decades before developing a new approach.
The compression part may extend, but the modeling part won't.
Do you mean like having a ranking table of words sorted by frequency and assign smaller "symbols" to those words that are repeated the most, therefore reducing the amount of information that needs to be transmitted?
That's basically how Huffman Coding works, the problem with compression is that you always hit a limit somewhere along the road, of course, if the set of things that you try to compress follows a particular pattern/distribution then it's possible to be really efficient about it, but for general purposes (audio/video/text/encrypted data that appears to be random) there is no (and I believe that there can't be) "best" compression technique.
Huffman Coding uses frequency on letters. You can do the same with words or with letter frequency in more dimensions, i.e. combinations of letters and their frequency.
I'm revamping my programming skills and implemented the Huffman algorithm. For now, I'm just considering [a-z] with no special characters. The probability values for a-z have been used from wikipedia.
When I run it, I get roughly 2x compression for random paragraphs.
But for this calculation I assume original letters require 8 bits each (ASCII).
But if I think about it, to represent 26 items, i just need 5 bits. If I calculate based on this fact, then compression factor drops to almost 1.1
So my question is, how is the compression factor determined in real world applications?
2nd question - if I write an encoder / decoder which uses 5 bits for representing a-z ( say a=0, b=1, etc) - is this also a considered a valid "compression" algorithm?
You have essentially the right answer, which is that you can't expect a lot of compression if all that you're working with is the letter frequencies of the English language.
The correct way to calculate the gain resulting from knowledge of the letter frequencies is to consider the entropy of a 26-symbol alphabet of equal probabilities with the entropy of the letters in English.
(I wish stackoverflow allowed TeX equations like math.stackexchange.com does. Then I could write decent equations here. Oh well.)
The key formula is -p log(p), where p is the probability of that symbol and the log is in base 2 to get the answer in bits. You calculate this for each symbol and then sum over all symbols.
Then in an ideal arithmetic coding scheme, an equiprobable set of 26-symbols would be coded in 4.70 bits per symbol. For the distribution in English (using the probabilities from the Wikipedia article), we get 4.18 bits per symbol. A reduction of only about 11%.
So that's all the frequency bias by itself can buy you. (It buys you a lot more in Scrabble scores, but I digress.)
We can also look at the same thing in the approximate space of Huffman coding, where each code is an integral number of bits. In this case you would not assume five bits per letter (with six codes wasted). Applying Huffman coding to 26 symbols of equal probability gives six codes that are four bits in length and 20 codes that are five bits in length. This results in 4.77 bits per letter on average. Huffman coding using the letter frequencies occurring in English gives an average of 4.21 bits per letter. A reduction of 12%, which is about the same as the entropy calculation.
There are many ways that real compressors do much better than this. First, they code what is actually in the file, using the frequencies of what's there instead of what they are across the English language. This makes it language independent, optimizes for the actual contents, and doesn't even code symbols that are not present. Second, you can break up the input into pieces and make a new code for each. If the pieces are big enough, then the overhead of transmitting a new code is small, and the gain is usually larger to optimize on a smaller chunk. Third, you can look for higher order effects. Instead of the frequency of single letters, you can take into account the previous letter and look at the probability of the next letter given its predecessor. Now you have 26^2 probabilities (for just letters) to track. These can also be generated dynamically for the actual data at hand, but now you need more data to get a gain, more memory, and more time. You can go to third order, fourth order, etc. for even greater compression performance at the cost of memory and time.
There are other schemes to pre-process the data by, for example, doing run-length encoding, looking for matching strings, applying block transforms, tokenizing XML, delta-coding audio or images, etc., etc. to further expose redundancies for an entropy coder to then take advantage of. I alluded to arithmetic coding, which can be used instead of Huffman to code very probable symbols in less than a bit and all symbols to fractional bit accuracy for better performance in the entropy step.
Back to your question of what constitutes compression, you can begin with any starting point you like, e.g. one eight-bit byte per letter, make assertions about your input, e.g. all lower case letters (accepting that if the assertion is false, the scheme fails), and then assess the compression effectiveness. So long as you use all of the same assumptions when comparing two different compression schemes. You must be careful that anything that is data dependent must also be considered part of the compressed data. E.g. a custom Huffman code derived from a block of data must be sent with that block of data.
If you ran an unrestricted Huffman-coding compression on the same text you'd get the same result, so I think it's reasonable to say that you're getting 2x compression over an ASCII encoding of the same text. I would be more inclined to say that your program is getting the expected compression, but currently has a limitation that it can't handle arbitrary input, and other simpler compression schemes to get compression over ASCII as well if that limitation is in place.
Why not extend your algorithm to handle arbitrary byte values? That way it's easier to make a true heads-up comparison.
It's not 5 bits for 26 character it's log(26) / log(2) = 4,7 bits. This is the maximum entropy but you need to know the specific entropy. For the german language it's 4,0629. When you know that you can use the formula R=Hmax - H. Look here: http://de.wikipedia.org/wiki/Entropie_(Informationstheorie)
http://en.wikipedia.org/wiki/Information_theory#Entropy
I'm looking to encode user_ids in a long list of call records. The parts of these records that takes up the most space are the symbols for the caller and receiver. I will create a map that assigns the most active callers shorter symbols---this will help keep the overall size of the files (and therefore the I/O time) down.
I know in advance how many times each symbol will be used---in other words I know the relative probability distribution. Furthermore, it is not important that the codes that are produced be "prefix free" such as Huffman codes. So what's the best encoding scheme, i.e., the one that will deliver the most compression and for which a quick implementation exists?
An answer should not only point to a compression scheme, it should also point to an implementation of that encoding scheme.
For general-purpose lossless encoding with a known probability distribution, aside from Huffman coding, the other "textbook" answer is arithmetic coding.
In practice, there are a variety of implementations. See these general-purpose coders. Each has different properties. Without further information, we can't give you a more precise answer.
#conradlee: re "In what cases is arithmetic coding better than Huffman coding?" In terms of compression, nearly always. If you have a symbol,S, with a probability, Ps, then the ideal number of bits to code it with, bs, is -log(Ps)/log(2). For example, if Ps is 1/3 then bs is ~ 1.585 bits. With Huffman you have to round up or down to the nearest whole number of bits (so the compression ratio will decrease). Arithmetic encoding will store it with a fractional number of bits.
This is a theoretical question, so expect that many details here are not computable in practice or even in theory.
Let's say I have a string s that I want to compress. The result should be a self-extracting binary (can be x86 assembler, but it can also be some other hypothetical Turing-complete low level language) which outputs s.
Now, we can easily iterate through all possible such binaries and programs, ordered by size. Let B_s be the sub-list of these binaries who output s (of course B_s is uncomputable).
As every set of positive integers must have a minimum, there must be a smallest program b_min_s in B_s.
For what languages (i.e. set of strings) do we know something about the size of b_min_s? Maybe only an estimation. (I can construct some trivial examples where I can always even calculate B_s and also b_min_s, but I am interested in more interesting languages.)
This is Kolmogorov complexity, and you are correct that it's not computable. If it were, you could create a paradoxical program of length n that printed a string with Kolmogorov complexity m > n.
Clearly, you can bound b_min_s for given inputs. However, as far as I know most of the efforts to do so have been existence proofs. For instance, there is an ongoing competition to compress English Wikipedia.
Claude Shannon estimated the information density of the English language to be somewhere between 0.6 and 1.3 bits per character in his 1951 paper Prediction and Entropy of Printed English (PDF, 1.6 MB. Bell Sys. Tech. J (3) p. 50-64).
The maximal (avarage) compression rate possible is 1:1.
The number of possible inputs is equal to the number of outputs.
It has to be to be able to map the output back to the input.
To be able to store the output you need container at the same size as the minimal container for the input - giving 1:1 compression rate.
Basically, you need enough information to rebuild your original information. I guess the other answers are more helpful for your theoretical discussion, but just keep this in mind.