One of the core steps in file compression like ZIP is to use the previous decoded text as a reference source. For example, the encoded stream might say "the next 219 output characters are the same as the characters from the decoded stream 5161 bytes ago." This lets you represent 219 characters with just 3 bytes or so. (There's more to ZIP than that, like Huffman compression, but I'm just talking about the reference matching.)
My question is what the strategy(ies) for the string matching algorithm is. Even looking at source code from zlib and such don't seem to give a good description of the compression matching algorithm.
The problem might be stated as: Given a block of text, say 30K of it, and an input string, find the longest reference in the 30K of text which exactly matches the front of the input string." The algorithm must be efficient when iterated, ie, the 30K block of text will be updated by deleting some bytes from the front and adding new ones to the rear and a new match performed.
I'm a lot more interested in discussions of the algorithm(s) to do this, not source code or libraries. (zlib has very good source!) I suspect there may be several approaches with different tradeoffs.
Well, I notice that you go into some detail about the problem but don't mention the information provided in section 4 of RFC 1951 (the specification for the DEFLATE Compressed Data Format, i.e. the format used in ZIP) which leads me to believe you might have missed this resource.
Their basic approach is a chained hash table using three-byte sequences as keys. As long as the chain is not empty, all the entries along it are scanned to a) eliminate false collisions, b) eliminate matches that are too old, and c) pick the longest match out of those remaining.
(Note that their recommendation is shaped by the factor of patents; it may be that they knew of a more effective technique but could not be sure that it was not covered by someone's patent. Personally, I've always wondered why one couldn't find the longest matches by examining the matches for the three-byte sequences that start at the second byte of the incoming data, the third byte, etc. and weeding out matches that don't match up. i.e., if your incoming data is "ABCDEFG..." and you've got hash matches for "ABC" at offsets 100, 302 and 416 but your only hash match for "BCD" is at offset 301, you know that unless you have two entirely coincidental overlapping hash matches -- unlikely -- then 302 is your longest match.)
Also note their recommendation of optional "lazy matching" (which ironically does more work): instead of automatically taking the longest match that starts at the first byte of the incoming data, the compressor checks for an even longer match starting at the next byte. If your incoming data is "ABCDE..." and your only matches in the sliding window are for "ABC" and for "BCDE", you're better off encoding the "A" as a literal byte and the "BCDE" as a match.
You could look at the details of the LZMA Algorithm used by 7-zip. The 7-zip author claims to have improved on the algorithm used by zlib et al.
I think you're describing a modified version of the Longest Common Substring Problem.
Related
I'm currently working for a short text compression project based on my language. But as a beginner, I also know some basic compression algorithm like LZW. But I still don't understand how smaz works. I have 2 questions:
How does smaz work?
How to build the codebook and reversed codebook?
Can any one explain it for me?
Thank you very much.
trying to answer your questions
How does smaz work?
according [1],
Smaz has a hard-wired constant built-in codebook of 254 common English
words, word fragments, bigrams, and the lowercase letters (except j,
k, q). The inner loop of the Smaz decoder is very simple:
Fetch the next byte X from the compressed file.
Is X == 254? Single byte literal: fetch the next byte L, and pass it straight through to the decoded text.
Is X == 255? Literal string: fetch the next byte L, then pass the following L+1 bytes straight through to the decoded text.
Any other value of X: lookup the X'th "word" in the codebook (that "word" can be from 1 to 5 letters), and copy that word to the decoded
text.
Repeat until there are no more compressed bytes left in the compressed file.
Because the codebook is constant, the Smaz decoder is unable to
"learn" new words and compress them, no matter how often they appear
in the original text.
This page could be helpful to understand the code.
How to build the codebook and reversed codebook?
TODO file in repository and author comments in redit poitns that the dictionary was generated by a unreleased ruby script. Also, the author explains:
btw what the Ruby program does is to consider all the possible substrings, and even all the possible separated words, and build a
table of frequencies, than adjust the weight based on the string
length, and finally hand tuning the table to compress specific things
very well. I added by hand the "http://" and ".com" token for example,
removing the final two entries.
An alternative to your project could be the shoco library which supports generation of a custom compression model based on your language.
The smaz sources is only 178 lines and just 99 lines without comments and codebook tables. You should look to see how it works.
Smaz is pretty simple compression by codebook (like LZW which you know). The library contains table with most popular terms in english (lines 5 - 51 for compression table and 56 -76 for decompression) and replace this terms with indexes in compressed string. And contrary to decompress.
For example, string the end would compressed by 58% becouse if terms the would be one byte index in compression table. So 7 bytes lenght string became 4 bytes length string.
I'm having a hard time to understand the LZW algorithm. I'm examining the pseudocode supplied on wikipedia (http://en.wikipedia.org/w/index.php?title=Lempel-Ziv-Welch&oldid=245292660) and there's one part in the decompressor code that I don't understand:
else if (k == currSizeDict)
entry = w + w[0];
Can someone explain to me a scenario where this would happen?
This problem is explained very well here: https://www.cs.duke.edu/csed/curious/compression/lzw.html. The basic idea is that since LZW only requires the compressed string and a dictionary containing all elements of the alphabet (rather than a dictionary containing all encoded patterns), it's necessary to reconstruct all encodings of more complex patterns on the fly while decoding. This results in a situation where it's possible to run into encodings that aren't in the dictionary. Interestingly enough, as the link above points out, this can only happen when the encoded string begins and ends with the same character.
Suppose I have the following string:
ABCADCADCADABC
I want to compress it by finding repeating substrings.
What's an algorithm that gives the optimal compression?
In the above example it should return
AB*1 CAD*3 ABC*1
For comparison, a greedy algorithm might return
ABC*1 ADC*2 AD*1 ABC*1
Depending on whether you prefer fast and simple or high compression ratio you could take a look into the Lempel-Ziv-Welch (LZW) or Lempel-Ziv-Markov chain (LZMA) algorithms. They both keep dictionaries of recurring strings.
This sounds like a job for suffix arrays/trees!
http://en.wikipedia.org/wiki/Suffix_array
You can use a suffix array built over your string to figure out patterns that repeat. For instance, we can build a suffix array over your example as follows (I'm using $ as always coming after every letter, you can sort it so that $ comes before every letter ... either way will work):
ABCADCADCADABC$
ABC$
ADABC$
ADCADABC$
ADCADCADABC$
BCADCADCADABC$
BC$
CADABC$
CADCADABC$
CADCADCADABC$
C$
DABC$
DCADABC$
DCADCADABC$
$
From this, we can more easily see the common patterns in the string. Using the information in this suffix array representation, we can see that CAD is repeated 3x in a local area, and we'd likely use this as our choice for compression. ADC and DCA and so on are not as attractive because they compress less of the string.
http://en.wikipedia.org/wiki/Suffix_tree
Suffix trees are more efficient ways of doing the same task. Once you wrap your head around how to do something using suffix arrays, it's not too far of a jump to go onto suffix trees. In fact, this is used in popular compression algorithms including LZW 1 and BWT (Bzip) 2.
It may not be practically relevant, but for the particular question you ask there is a dynamic programming solution. If you have computed the optimum way to compress the strings of length 1, 2, 3...n-1 starting from the first character, then you can compute the optimum way to compress the string of length n starting from the first character by looking at the last k characters for each possibility k and seeing if they form a multiple of a simple string. If so, compute the cost of compressing the first n-k characters and then expressing the last k characters using a multiple of a string.
So in your example you would finish up by noticing that ABC was a multiple of itself, and that if you expressed this as ABC*1 you could use the answer you had already worked out for the first 11 characters of AB CAD*3 to produce AB*1 CAD*3 ABC*1
Better still would be:
ABCAD(6,3)(3,11)
where (n,d) is a length and distance back of a match. So (6,3) copies six bytes starting from three bytes back. While that may sound a little odd, by the time it gets three bytes in, the next three bytes it needs have been copied. So CADCAD is appended. The (3,11) causes ABC to be appended.
This is called LZ77 compression. It is what is implemented by zip, gzip, and zlib using the deflate compressed data format. That format not only references previous string matches, but also uses Huffman compression on the literals (e.g. ABCAD) as well as the lengths and distances.
We are working with a number of unix based filesystems, all of which share a similar set of restrictions on that certain characters can't be used in the username fields. One of those restrictions is no "#" , "_", or "." in the names. Being unix there are a number of other restrictions.
So the question is if there is a good known algorithm that can take an email address and turn that into a predictable unix filename. We would need to reverse this at some point to get the email.
I've considered doing thing like "."->"DOT", "#"->"AT", etc. But there are size limitations and other things that are generally problematic. I could also optimize by being able to map the #xyz.com part of the email to a special char or something. Each implementation would only have at most 3 domains it would need to support. I'm hoping someone has found a solution without a huge number of tradeoffs.
UPDATE:
-The two target filesystems are AFS and NFS.
-Base64 doesn't work as it has not compatible characters. "/"
-Readable is preferable.
Seems like the best answer would be to replace the #xyz.com domain to a single non-standard character, and then have a function that could shrink the first part of a name to something that fits in the username length restrictions of the various filesystems. But what is a good function for that?
You could try a modified version of the URL percent (%) encoding scheme used on for URIs.
If the percent symbol isn't allowed on your particular filesystem(s), simply replace it with a different, allowed character (and remember to encode any occurrences of that character properly).
Using this method:
mail.address#server.com
Would become:
mail%2Eaddress%40server%2Ecom
Or, if you had to substitute (for example), the letter a instead of the % symbol:
ma61ila2Ea61ddressa40servera2Ecom
Not exactly humanly-readable perhaps, but easily enough processed through an encoding algorithm. For the best space efficiency, your escape character should be a character allowed by the filesystem, yet one that is not likely to appear frequently in an address.
This encoding scheme has the advantage that there is no size increase for most normal characters. The string length will ONLY go up for characters not supported by the filesystem.
Check out base64. Encoding and decoding is well defined.
I'd prefer this over rolling my own format any day.
Hmm, from your question I'm not totally clear on this point, but since you wanted some conversion I'm assuming that you want something that is at least human readable?
Each OS may have different restrictions, but are you close enough to the platforms that you would be able to find out/test what is acceptable in a username? If you could find three 'special' characters that you could use just to do a replace on '#', '.', '_' you would be good to go. (Is that comprehensive? if not you would need to make sure you know all of them otherwise you could clash.) I searched a bit trying to find whether there was a POSIX standard, but wasn't able to find anything, so that's why I think if you can just test what's valid that would be the most direct route.
With even one special character, you could do URL encoding, either with '%' if it's available, or whatever you choose if not, say '!", then { '#'->'!40", '_'->'!5F', '.'-> '!2E' }. (The spec [RFC1738] http://www.rfc-editor.org/rfc/rfc1738.txt) defines the characters as US-ASCII so you can just find a table, e.g. in wikipedia's ASCII article and look up the correct hex digits there.) Or, you could just do your own simple mapping since you don't need the whole ASCII set, you could just do a map with two characters per escaped character and have, say, '!a','!u','!p' for at, underscore, period.
If you have two special characters, say, '%', and '!', you could delimit text that represents the character, say, %at!, &us!, and '&pd!'. (This is pretty much html-style encoding, but instead of '&' and ';' you are using the available ones, and you're making up your own mnemonics.) Another idea is that you could use runs of a symbol to determine the translated character, where each new character flops which symbol is being used. (This conveniently stops the run if we need to put two of the disallowed characters next to each other.) So assume '%' and '!', with period being 1, underscore 2, and at-sign being three, 'mickey._sample_#fake.out' would become 'mickey%!!sample%%!!!fake%out'. There are other variations but this one is easy to code.
If none of this is an option (e.g. no symbols at all, just [a-zA-Z0-9]), then really I think the Base64 answer sounds about right. Really once we're getting to anything other than a simple replacement (and even that) it's already getting hard to type if that's the goal. But if you really need to try to keep the email mostly readable, what you do is implement some sort of escaping. I'm thinking use '0' as your escape character, so now '0' becomes '00', '#' becomes '01', '.' becomes '02', and '_' becomes '03'. So now, 'mickey01._sample_#fake.out'would become 'mickey0010203sample0301fake02out'. Not beautiful but it should work; since we escaped any raw 0's, just always make sure you define a mapping for whatever you choose as your escape char and you should be fine..
That's all I can think of atm. :) Definitely if there's no need for these usernames to be readable in the raw it seems like apparently Base64 won't work, since it can produce slashes. Heck, ok, just the 2-digit US-ASCII hex value for each character and you're done...] is a good way to go; there's lots of nice debugged, heavily field-tested code out there for it and it solves your problem quite handily. :)
Given...
- the limited set of characters allowed in various file systems
- the desire to keep the encoded email address short (both for human readability and for possible concerns with file system limitations)
...a possible approach may be a two steps encoding logic whereby the email is
first compressed using a lossless compression algorithm such as Lempel-Ziv, effectively turning it into a "binary" form, stored in a shorter array of bytes
then this array of bytes is encoded using a Base64-like algorithm
The idea is to minimize the size of the binary representation, so that the expansion associated with the storage inefficiency of the encoding -which can only store roughly 6 bits (and probably a bit less) per character-, doesn't cause the encoded string to be too long.
Without getting overly sophisticated for the compression nor the encoding, such a system would likely produce encoded strings that are maybe 4/5 of the input string size (the email address): the compression should easily half the size, but the encoding, say Base32, would grow the binary form size by 8/5.
Efforts in improving the compression ratio may allow the selection of more "wasteful" encoding schemes (with smaller character sets) and this may help making the output more human-readable and also more broadly safe on various flavors of file systems. For example whereby a Base64 seems optimal. space-wise, using only uppercase letter (base 26) may ensure portability of the underlying scheme to file systems where the file names are not case sensitive.
Another benefit of the initial generic compression is that few, if any, assumptions need to be made about the syntax of valid input key (email addresses here).
Ideas for compression:
LZ seems like a good choice, 'though one may consider primin its initial buffer with common patterns found in email addresses (example ".com" or even "a.com", "b.com" etc.). This initial buffer would ensure several instances of "citations" per compressed email address, hence a better compression ratio overall). To further squeeze a few bytes, maybe LZH or other LZ-variations could be used.
Aside from the priming of the buffer mentioned above, another customization may be to use a shorter buffer than typical LZ algorithms, since the string we have to compress (email address instances) are themselves very short and would not benefit from say a 512 bytes buffer. (Shorter buffer sizes allow shorter codes for the citations)
Ideas for encoding:
Base64 is not suitable as-is because of the slash (/), plus (+) and equal (=) characters. Alternate characters could be used to replace these; dash (-) comes to mind, but finding three charcters, allowed by all "flavors" of the targeted file systems may be a stretch.
Never the less, Base64 and its 4 output characters per 3 payload bytes ratio provide what is probably the barely achievable upper limit of storage efficiency [for an acceptable character set].
At the lower end of this efficiency, is maybe an ASCII representation of the Hexadeciamal values of the bytes in the array. This format with a doubling of the payload bytes may be acceptable, length-wise, and is interesting because of its simplicity (there is a direct and simple relation between each nibble (4 bits) in the input and characters in the encoded string.
Base32 whereby A thru Z encode 0 thru 25 and 0 thru 5 encode 26 thru 31, respectively, essentially variation of Base64 with an 8 output characters per 5 payload bytes ratio may be a very viable compromise.
I want to find (not generate) 2 text strings such that, after removing all non letters and ucasing, one string can be translated to the other by simple substitution.
The motivation for this comes from a project I known of that is testing methods for attacking cyphers via probability distributions. I'd like to find a large, coherent plain text that, once encrypted with a simple substitution cypher, can be decrypted to something else that is also coherent.
This ends up as 2 parts, find the longest such strings in a corpus, and get that corpus.
The first part seems to me to be amiable to some sort of attack with a B-tree keyed off the string after a substitution that makes the sequence of first occurrences sequential.
HELLOWORLDTHISISIT
1233454637819a9b98
A little optimization based on knowing the maximum value and length of the string based on each depth of the tree and the rest is just coding.
The Other part would be quite a bit more involved; how to generate a large corpus of text to search? some kind of internet spider would seem to be the ideal approach as it would have access to the largest amount of text but how to strip it to just the text?
The question is; Any ideas on how to do this better?
Edit: the cipher that was being used is an insanely basic 26 letter substitution cipher.
p.s. this is more a thought experiment then a probable real project for me.
There are 26! different substitution ciphers. That works out to a bit over 88 bits of choice:
>>> math.log(factorial(26), 2)
88.381953327016262
The entropy of English text is something like 2 bits per character at least. So it seems to me you can't reasonably expect to find passages of more than 45-50 characters that are accidentally equivalent under substitution.
For the large corpus, there's the Gutenberg Project and Wikipedia, for a start. You can download an dump of all the English Wikipedia's XML files from their website.
I think you're asking a bit much to generate a substitution that is also "coherent". That is an AI problem for the encryption algorithm to figure out what text is coherent. Also, the longer your text is the more complicated it will be to create a "coherent" result... quickly approaching a point where you need a "key" as long as the text you are encrypting. Thus defeating the purpose of encrypting it at all.