The Basics: What is Compressibility?

[Ernst]

Hello!

I'm a hobbyist who is studying data-compression. My efforts have brought me to a place where I must answer some basic questions.

What is Compressibility? What makes some data compressible or what makes it uncompressible?

I am currently studying the BWT transform so in terms of transforming data I am asking what makes for a compressible form.

This is a very basic question being asked honestly.

I have come to terms with entropy including the concept that not all people agree on it's define. Now I am looking into Compressibility.

What makes a data good and thus what is bad will be known.

Is it a reduced symbol set? Is it a pattern of bits? Is it the pattens of codes used to represent data?

In general I am exploring the idea of if I can change a data what qualities do i wish to advance in that transforming.

[SG]

On Apr 15, 2:20 am, Ernst wrote:
> Hello!
>
> I'm a hobbyist who is studying data-compression. My efforts have
> brought me to a place where I must answer some basic questions.
>
> What is Compressibility? What makes some data compressible or what
> makes it uncompressible?

I don't think there is a proper mathematical definition in information
theory of "compressible" or "compressibility". At least I don't know
any.

Instead of making up your own definition, I recommend reading a decent
introductory book about data compression. Chances are that you'll
learn things you did not even expect to learn but are vital in
furthering your understanding of data compression. From what I could
gather it seems that you focus too much on "data". By that I mean you
are blind about the "sources" of "data", about the difference between
"information" and "data".

> I have come to terms with entropy including the concept that not all
> people agree on it's define.

Entropy is a loaded term which means different things in different
contexts. What you should focus on, of course, is Information Theory.

[glen herrmannsfeldt]

SG <s.ges...@gmail.com> wrote:
> On Apr 15, 2:20 am, Ernst wrote:

>> I'm a hobbyist who is studying data-compression. My efforts have
>> brought me to a place where I must answer some basic questions.

>> What is Compressibility? What makes some data compressible or what
>> makes it uncompressible?

> I don't think there is a proper mathematical definition in information
> theory of "compressible" or "compressibility". At least I don't know
> any.

Well, some of them are easy. If you group the data into groups of
some number of bits, such as bytes or words, you can easily compute
the entropy, and so the compressibility, of those groups. Often the
entropy of one grouping, such as groups of seven bits, will also extend
other groupings, such as bytes.

Otherwise, it is usually useful to know something about the data source.
Often patterns of repetition are easy to find, and LZ-like algorithms
usually do that pretty well.

On the other hand, a file full of randomly selected English words could
be compressed by indicating the position of the word in a dictionary.
Even more if it had word usage statistics similar to English text.

Past that, you usually need to know a lot more about the source.
The motion estimation used in MPEG is not one that would be easy to
see given just the bit stream of uncompressed video. Lossy audio
compression relies on details of the human auditory system that
you would never find looking just at the bits.

> Instead of making up your own definition, I recommend reading a decent
> introductory book about data compression. Chances are that you'll
> learn things you did not even expect to learn but are vital in
> furthering your understanding of data compression. From what I could
> gather it seems that you focus too much on "data". By that I mean you
> are blind about the "sources" of "data", about the difference between
> "information" and "data".

Yes. There are many algorithms that can find most of the low entropy
in "data". Maybe one can do a little better, but not much.

The really big gains, and even medium sized gains, come when you
know more about the data source.

>> I have come to terms with entropy including the concept that not all
>> people agree on it's define.

> Entropy is a loaded term which means different things in different
> contexts. What you should focus on, of course, is Information Theory.

-- glen

[James Dow Allen]

On Apr 15, 10:40 pm, SG <s.gesem...@gmail.com> wrote:
> I don't think there is a proper mathematical definition in information
> theory of "compressible" or "compressibility". At least I don't know
> any.

To the contrary, Kolmogorov complexity provides a clear and
logical definition. Unfortunately it doesn't generally
lead to any easy computation: information content depends
on the existence of a computer program, but discovering that
program may depend on a clever programmer!

> Instead of making up your own definition, I recommend reading a decent
> introductory book about data compression.

Any specific suggestions? I think Ernst's question is a good one,
with no easy answers.

On Apr 15, 7:20 am, Ernst <Ernst_B...@sbcglobal.net> wrote:
>  What is Compressibility? What makes some data compressible ...

> or what makes it uncompressible?

A good question which lacks easy answers. It's almost
easier to answer the 2nd, opposite question: Data is
incompressible if it is random.

>  I am currently studying the BWT transform so in terms of
> transforming data I am asking what makes for a compressible form.

The BWT is definitely worth study.
I don't think I've seen a good explanation of how
it achieves its "magic." If you can wrap your mind
into such an understanding you've furthered your goal.

(BTW -- warning: self-plug -- I whipped a simple BWT
compressor together many years ago and see that its
source code is still the 2nd hit when Googling
"Burrows–Wheeler transform" )

Of course there are plenty of simpler examples of
rearranging data to make it easier to compress --
the zigzag ordering in Jpeg is one.
Separating data into statistics bins is another.

> In general I am exploring the idea of if I can change
> a data what qualities do i wish to advance in that transforming.

Be aware that, theoretically, you can't make data more compressible.
What you can do is make the existing compressibility easier
to achieve.

James

[Thomas Richter]

Am 16.04.2013 08:12, schrieb James Dow Allen:
>
> On Apr 15, 10:40 pm, SG<s.gesem...@gmail.com> wrote:
>> I don't think there is a proper mathematical definition in information
>> theory of "compressible" or "compressibility". At least I don't know
>> any.
>
> To the contrary, Kolmogorov complexity provides a clear and
> logical definition. Unfortunately it doesn't generally
> lead to any easy computation: information content depends
> on the existence of a computer program, but discovering that
> program may depend on a clever programmer!

Kolmogorov complexity has it's own set of problems. It's always relative
to a specific Turing machine, thus if you have the freedom to design the
machine, the complexity of any finite sequence is - by this definition -
trivial.

Well, probably that's only a theoretical problem - in practical
applications, the machine is given. Just that the K-C does not provide
you with an answer how to reach the ideal code length, but in fact even
gives you the (negative) answer that the ideal code is not computable.

Point being: You cannot point at a specific(!) sequence and say "this
one is compressible!". For every finite sequence a machine exists that
compresses it to nothing, so not a useful definition.

>> Instead of making up your own definition, I recommend reading a decent
>> introductory book about data compression.
>
> Any specific suggestions? I think Ernst's question is a good one,
> with no easy answers.

There of course a couple of books giving you a good introduction into
known techniques, such as Salomon's "Handbook of data compression". But
as soon as you know more about the sources and can construct good
models, it's getting more and more specific: Compressing audio, images,
video... is quite another business.

Thus, sequences are "more compressible" if you know more on the source
that created them. But this definition is cyclic - data is compressible
if you can compress it better. Wow, what an insight!

> Of course there are plenty of simpler examples of
> rearranging data to make it easier to compress --
> the zigzag ordering in Jpeg is one.
> Separating data into statistics bins is another.
>
>> In general I am exploring the idea of if I can change
>> a data what qualities do i wish to advance in that transforming.
>
> Be aware that, theoretically, you can't make data more compressible.
> What you can do is make the existing compressibility easier
> to achieve.

I would rather say: You can built better models for the data to compress.

You know, it's the same old game as with "Lena-compress". The ideal
model for image compression still gets good compression results in
typical benchmarks. (-;

[Thomas Richter]

Am 15.04.2013 17:40, schrieb SG:

> Entropy is a loaded term which means different things in different
> contexts. What you should focus on, of course, is Information Theory.

Why? Entropy has a very definite meaning, and in fact, the two meanings
(the thermodynamical one and the information theory one) agree up to a
factor.

However, what one should be aware of is that entropy (or entropy-rate)
is the property of a random process, and not the property of a given
sequence of symbols. That said, you cannot point at a specific
realization of a random process (a sequence) and say "this sequence has
an entropy... of so and so many bits". It simply doesn't work this way.
You can implicitly define a model that might have created this sequence,
and then compute the entropy of this model.

[SG]

On Apr 15, 8:57 pm, glen herrmannsfeldt wrote:

> SG <s.gesem...@gmail.com> wrote:
> > On Apr 15, 2:20 am, Ernst wrote:
> >> I'm a hobbyist who is studying data-compression. My efforts have
> >> brought me to a place where I must answer some basic questions.
> >> What is Compressibility? What makes some data compressible or what
> >> makes it uncompressible?
> > I don't think there is a proper mathematical definition in information
> > theory of "compressible" or "compressibility". At least I don't know
> > any.
>
> Well, some of them are easy. If you group the data into groups of
> some number of bits, such as bytes or words, you can easily compute
> the entropy, and so the compressibility, of those groups. Often the
> entropy of one grouping, such as groups of seven bits,  will also extend
> other groupings, such as bytes.

"the entropy" of what? -- Of an assumed source model based on
estimated probabilities. And that goes back to what I tried to say
about "sources" and "information". ;-)

SG

[SG]

On Apr 16, 8:12 am, James Dow Allen wrote:

> On Apr 15, 10:40 pm, SG wrote:
>
> > I don't think there is a proper mathematical definition in information
> > theory of "compressible" or "compressibility". At least I don't know
> > any.
>
> To the contrary, Kolmogorov complexity provides a clear and
> logical definition.

Right. I did not consider this to be part of information theory. But I
guess it is ... well ... Wikipedia says it's a subfield of it --
algorithmic information theory. Maybe I'm biased because I consider
this topic to be so "dead-end-y".

> Unfortunately it doesn't generally
> lead to any easy computation: information content depends
> on the existence of a computer program, but discovering that
> program may depend on a clever programmer!

Right. Has there ever come something practical about starting from the
Kolmogorov's complexity definition? If so, let me know.

> > I recommend reading a decent
> > introductory book about data compression.
>
> Any specific suggestions?  I think Ernst's question is a good one,
> with no easy answers.

I always liked K. Sayhood's "Introduction to Data Compression". It
covers lots of topics starting from the mathematical basics (sources,
entropy, Kraft-McMillen inequality) up to things like vector
quantization, transform coding, subband coding, JPEG, CELP, etc.

SG

[SG]

On Apr 16, 9:19 am, Thomas Richter wrote:
> Am 15.04.2013 17:40, schrieb SG:
> > Entropy is a loaded term which means different things in different
> > contexts. What you should focus on, of course, is Information Theory.
>
> Why? Entropy has a very definite meaning, and in fact, the two meanings
> (the thermodynamical one and the information theory one) agree up to a
> factor.

I don't know much about thermodynamics. I believe you if you say that
in these two contexts the agreement is up to a factor. Anyhow, my
wording was also a little poor. What I meant to say that Ernst should
look up entropy in the context of information theory -- not
economoncs, not sociology, not ... ;-)

> [...] That said, you cannot point at a specific

> realization of a random process (a sequence) and say "this sequence has
> an entropy... of so and so many bits". It simply doesn't work this way.

Right. This is what I meant by focus w.r.t. data versus information /
source model.

SG

[James Dow Allen]

On Apr 16, 2:09 pm, Thomas Richter <t...@math.tu-berlin.de> wrote:
> Am 16.04.2013 08:12, schrieb James Dow Allen:
> > To the contrary, Kolmogorov complexity provides a clear and
> > logical definition.  Unfortunately it doesn't generally
> > lead to any easy computation: information content depends
> > on the existence of a computer program, but discovering that
> > program may depend on a clever programmer!
>
> Kolmogorov complexity has it's own set of problems. It's always relative
> to a specific Turing machine, thus if you have the freedom to design the
> machine, the complexity of any finite sequence is - by this definition -
> trivial.

No. Just as with the Nelson million-dollar challenge, the
bits required for the program (and any special architecture)
must be counted in the cost.
More or less, up to a O(1) factor, various Turing-equivalent
machines are ... wait for it ... equivalent!

> Point being: You cannot point at a specific(!) sequence and say "this
> one is compressible!". For every finite sequence a machine exists that
> compresses it to nothing, so not a useful definition.

Wrong again. See above.

> Thus, sequences are "more compressible" if you know more on the source
> that created them. But this definition is cyclic - data is compressible
> if you can compress it better. Wow, what an insight!

It may be obvious. But it is still a key insight.

James

[Thomas Richter]

Am 16.04.2013 18:04, schrieb James Dow Allen:
> On Apr 16, 2:09 pm, Thomas Richter<t...@math.tu-berlin.de> wrote:
>> Am 16.04.2013 08:12, schrieb James Dow Allen:
>>> To the contrary, Kolmogorov complexity provides a clear and
>>> logical definition. Unfortunately it doesn't generally
>>> lead to any easy computation: information content depends
>>> on the existence of a computer program, but discovering that
>>> program may depend on a clever programmer!
>>
>> Kolmogorov complexity has it's own set of problems. It's always relative
>> to a specific Turing machine, thus if you have the freedom to design the
>> machine, the complexity of any finite sequence is - by this definition -
>> trivial.
>
> No. Just as with the Nelson million-dollar challenge, the
> bits required for the program (and any special architecture)
> must be counted in the cost.
> More or less, up to a O(1) factor, various Turing-equivalent
> machines are ... wait for it ... equivalent!

I afraid "the bits for the program" do not make sense. Yes, if you have
a *universal Turing machine*, you can ask how many bits the program has
that simulates the specific Turing machine I had in mind, but this
specific information is again specific to the universal machine you
started with.

A Turing machine may or may not have a "program" on its input tape - a
universal one surely can - but *at least* it has an internal state
machine that defines how it reacts on the bits on its input tape (the
defining property, of course).

Now the question: How do you measure the "size of the state machine in
bits"?

Answer: You can't. You can only by describing this state machine as a
program of a *second universal* Turing machine that simulates the state
machine of the first one. But that doesn't solve the problem. It only
moves it forwards one step.

So, IOWs, "the K-complexity of a sequence S" makes no sense again. It is
up to a machine U - the pair (S,U) is well-defined (though not
computable in general for infinite S - a issue of practicability).

However, one result we have: Independent of U, (S,U) is either always
finite, or infinite. And in the latter case, we have an "incomputable
sequence". (Not necessarily incompressible, for whatever this word might
mean).

> More or less, up to a O(1) factor, various Turing-equivalent
> machines are ... wait for it ... equivalent!

Yes, of course, up to O(1). That's what I said.

>> Point being: You cannot point at a specific(!) sequence and say "this
>> one is compressible!". For every finite sequence a machine exists that
>> compresses it to nothing, so not a useful definition.
>
> Wrong again. See above.

Of course. I can define a Turing machine whose internal state machine is
built such that, on reading a single bit from its input tape, generates
the requested output, whatever this output may be, as long as it is
finite (so must the state machine). So, (S,U) = 1 for this machine. Note
that I said *finite* sequence. It's a different issue when considering
the limit N->\infty, N = sequence length, and for example counting the
number of operations the machine has to perform, as function of N. Or to
put it in different words, the size of any finite sequence can be
absorbed into the O(1) part by taking the freedom of defining U. (-:

I would again say that this is a fairly trivial and probably fairly
useless observation. One again cannot expect to point at a particular
sequence and say "the complexity of this sequence is Q bits". It doesn't
work this way. You either need to state this in terms of a limit for
large sequences, or relative to a specific machine.

Greetings,
Thomas

[glen herrmannsfeldt]

SG <s.ges...@gmail.com> wrote:

(snip)

>> Well, some of them are easy. If you group the data into groups of
>> some number of bits, such as bytes or words, you can easily compute
>> the entropy, and so the compressibility, of those groups. Often the
>> entropy of one grouping, such as groups of seven bits,  will also extend
>> other groupings, such as bytes.

> "the entropy" of what? -- Of an assumed source model based on
> estimated probabilities. And that goes back to what I tried to say
> about "sources" and "information". ;-)

Yes, I didn't say it very well.

If you only consider the frequency of bit groups (such as bytes)
and nothing else, you can compute how compressible the data stream
is based only on that. That is most useful when the data is a stream
of uncorrelated data, for example coin flips or dice rolls.
(Loaded dice rolls will be more compressible than fair dice.)

Other than that specific case, one should not expect to write a
general compression algorithm without knowing more about the source.
So, yes, the difference between "sources" and "information".

-- glen

[lawco...@gmail.com]

so far so good these posts has provided reasonable 'snapshot' of current latest state of art as understood as of then , likely may turn out accepted universal be very mistaken with time .

here is a small helpful step forward :

http://home.wxs.nl/~gkorthof/kortho44a.htm

LawCounsels

[James Dow Allen]

On Apr 18, 7:08 pm, lawcouns...@gmail.com wrote:
> here is a small helpful step forward :
>
> http://home.wxs.nl/~gkorthof/kortho44a.htm
>
> LawCounsels

Mr. Counsel's link raises an important point,
though it may be moot how it relates to data
compression in the usual sense.

Meaning arises from patterns, i.e. *order*,
so it cannot relate well to "information"
in the sense used in data compression, which
is maximized when there is maximal *disorder* !

I read a good 1990's journal article related,
but don't remember enough to Google. :-}
One point I remember is
> Since the goal of human language is to
> communicate, high information content is
> appropriate. Yet the language must have
> high redundancy. If it didn't, babies
> would never be able to learn it.

James

[LawCounsels]

.... also in the article were mentioned beginning of various
'cracks' / paradoxes , the trend already started then re Kolmogorov
NOT a correct logical measure of quantitative a.k.a Shannon
information ... including latest re Wolfram's rule 30 & 110
'randomness' not easy reconciled with Kolmogorov

[Ernst]

Thank you all so far who replied.

I have been an experimenter and I now have mathematics to publish. I intend to take some Classes at the Junior College so I can be competitive in writing things down. I hope to let you all in on the discovery soon but I have to learn to write the proof first. Crawl before walking for sure.

So this question and it's answers are now my guideline. I will spend for books and I will read them so thank you all for those links.
I didn't need to advance to general competency in data compression to experiment but it's different now. I must become "literate" in reading and writing the maths. I have some learning disability I fear. Great imagination but short on discipline.

In general, there is a Mathematics that allows us to have "versions" of any binary data. I believe this is not limited to binary but that is to be considered later.
In short for the expense of a few bits we can have thousands of versions of any data. For more bits billions of versions exist ( for the example of 4096 bytes). The data matters not.

Now this Mathematics I have known and have been exploring for several years already. I am confident the Mathematics is valid.
I am using it in my "Number Generation" effort concerning Mark Nelson's challenge. In a few days I expect the last 90 (out of 83,049) 40-bit values will have a code. I plan to run program generating those MDF "matches" through the rest of this year and have a set of codes to 40-bit elements of the MDF in the millions of matches(codes) so as to offer as many choices for an "compressed" encoding as possible.
Hopefully, from that we may find an indexing that will satisfy Mark's challenge. The decoding of these codes is already well established.

I am thinking to verify all the (codes) data and release the data for all to 'compress." What the hell, we are bored with the plain old MDF are we not?

So to extend on the original post, or jumping back to another thought, I am now looking at what I can do with something I am calling eXpress() and trying to determine what are the effects different variables have on the generated "eXpressions" as it relates to "compression." Naturally having some guide is necessary as is the fundamental skills of maths and English; improved or "revisited."
That is the reason for my question. It's not important that we nail it down to a point but I have saved this in my notes to be guided by.

I never do things the easy way it would seem

With BWT I believe it is valid to study David Scott's S(.). I have Emailed David to say I wish to iterate his S(.) and see how that works. I am an experimenter after all. Is there code available? I didn't see it when I looked before. I'll look again after this post.

What is the Gem of David's work is, it costs nothing! By that I mean we do not need to spend for a cycle of S(.) That means I can employ it in iteration. What will that do I wonder?

So assuming I do have some math that offers "versions" of some data and that I can select for qualities and use David's S(.) well the idea is that I can construct a data that is "Compressible."

This mathematics only gets bigger so it's appropriate to share the news sooner rather than later but what do I select I ask myself. What makes it "better this way or that way" I must define.

I am also willing to work/study if some employment arrangement can be made. Now is the time to have access to new Mathematics and help a poor old welder go to J.C.
I have to have material needs met LOL :) It never hurts to network.

I will continue to monitor this thread and I welcome all to add to this.

Thank You once again

Ernst

[Ernst]

On Sunday, April 14, 2013 5:20:55 PM UTC-7, Ernst wrote:

--------------------

I found the link to David's work. My bad I missed it with these old eyes the first time.

[Thomas Richter]

On 19.04.2013 15:41, LawCounsels wrote:
> .... also in the article were mentioned beginning of various
> 'cracks' / paradoxes , the trend already started then re Kolmogorov
> NOT a correct logical measure of quantitative a.k.a Shannon
> information

What do you mean by "not correct"? Of course it is correct, this is how
it is defined. It may not be "what you expect", but this is a different
question.

The "paradox" is not that "information" is ill-defined. Kolmogorov
complexity (by the very definition) gives you the shortest way of how to
describe the information content of a message. It does not give you the
shortest way how to describe a "text directly interpretable text
suitable for human readers", which is quite a different problem. If we
want to abstract this szenario, I would rather say that in this case the
"Turing machine" U is "given". It is your brain. So the question is then
to find the shortest description of a text your brain can meaningfully
decipher. And voila, the paradox goes away.

The paradox on the images in the text is that the text confuses
information with meaning. A white noise image is not "decodable" by the
human brain because it does not fit into the model of natural images
your brain has. However, its "information" is high because if you want
your receiver to reproduce the image, you cannot simply say "it's
lena!", but you need to describe every single pixel - which is a
mindbogglingly boring task. So the "message size" to accurately describe
the image is long, even though - or because! - it does *not* decode to a
meaning in your brain.

Thus, if you want to efficiently encode images, it is a useful strategy
to code those images more efficient that have a meaning, i.e. to code
natural images. That again means that you need to have a good model of
what a natural image is. The better this model, the better you can
encode natural images - and the worse you can encode all other images.
Of course, to find such models is hard, and the methods we have right
now are rather crude, but the principle works: We do not attempt to
"compress all images lossless" because that simply does not work.
Instead, we focus on compressing "meaningful images", or even "compress
them approximately", i.e. replace an image with another image "that has
the same meaning", i.e. is "visually lossless". Once that is understood,
we're at the heart of lossy compression.

... including latest re Wolfram's rule 30& 110

> 'randomness' not easy reconciled with Kolmogorov

It is completely trivial to reconcile with Kolmogorov. You just said it:
Wolfram's automata *are* Turing machines. So if you want to find the
shortest Turing machine that reproduces this output, there is at least a
good candidate for it: The machine itself! So where is the problem? The
output of those automata looks "random" to humans because, once again,
the output does not fit into the model a human brain has on what an
"ordered sequence is". But in fact, it is a completely ordered and
perfectly predictable sequence. That you are unable to detect this order
by "looking at the output" is not a problem of the Kolmogorov
complexity. It is a problem of the brain that is not adjusted to detect
such orders - due to the way how it (we) evolved.

[Ernst]

On Sunday, April 14, 2013 5:20:55 PM UTC-7, Ernst wrote:

-------------------------

I needed to access the Internet so I am having a cup of coffee here.

I was reading this:

"Kolmogorov structure function
From Wikipedia, the free encyclopedia
Jump to: navigation, search

In 1973 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let each data be a finite binary string and models be finite sets of binary strings. Consider model classes consisting of models of given maximal Kolmogorov complexity. The Kolmogorov structure function of an individual data string expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. The structure function determines all stochastic properties of the individual data string: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the true model is in the model class considered or not. In the classical case we talk about a set of data with a probability distribution, and the properties are those of the expectations. In contrast, here we deal with individual data strings and the properties of the individual string focussed on. In this setting, a property holds with certainty rather than with high probability as in the classical case. The Kolmogorov structure function precisely quantify the goodness-of-fit of an individual model with respect to individual data.

The Kolmogorov structure function is used in the algorithmic information theory, also known as the theory of Kolmogorov complexity, for describing the structure of a string by use of models of increasing complexity. "

-----------------

I am no where near educated on this but if I read this correctly I would wish to define a class and then have some scoring system to rate each string.
The "same-size" or finite string length I understand.


This suggests that if I use Scott's S(.) bwts then one of the qualities would be runs.
Perhaps a secondary rank could be having specific elements over others.

With this "Kolmogorov proposed non-probabilistic approach" having a static set of same length strings seem to qualify as input.

Am I heading in the right direction?

I'm sure I have a great deal to comprehend.

Again thanks..

[Ernst]

On Sunday, April 14, 2013 5:20:55 PM UTC-7, Ernst wrote:

=======================================


For those who like video. Here are a few links to youtube on various sorts.

https://www.youtube.com/watch?v=XaqR3G_NVoo
https://www.youtube.com/watch?v=CmPA7zE8mx0
https://www.youtube.com/watch?v=ywWBy6J5gz8

Who said learning cannot be fun?

[Ernst]

Looks like I will be studying Bwts()

I've been refitting David's archive to be an in-line version single function call for both encode and decode plus OpenMP functionality.

I didn't rework their basic design.

If you are wishing to have a copy of this refit that works under OpenMP or not let me know

qwerty...@gmail.com


As to what is compressible? I guess like the aspects of applying different processes for different effects I have to look at things in layers.
The beauty is so much work has been done already so it's reasonable to follow the works of those who came before.

Thanks to Mark Nelson and David Scott for this version of BWT and now perhaps I offer a version of that.

[Ernst]

I wanted to share a great site for those who like videos on data compression.

http://www.google.com/url?q=http://ocw.mit.edu/&sa=U&ei=aW6RUYSDEaWqigL5pYCgBA&ved=0CBkQFjAA&sig2=NoSYZt595Br8OPnXN-5SUw&usg=AFQjCNFIfcMBpVxQs37kak-2kgoQ9i6A5A

I am planning to attend Junior College so all I can do now to acclimate me makes sense to do it now.

I think one has to register but it's free.