I have been reading https://en.wikipedia.org/wiki/Line_code, but it is not obvious to me why 4b5b and 8b10b are called line code.
What is the fundamental difference between line code and more sophisticated hamming code?
What is the fundamental difference between line code and more sophisticated hamming code?
They're both serving entirely different purposes.
A line code enables clock recovery on a single, serial bit stream and provides bit-level synchronization. It may also provide byte-level synchronization. On differential electrical cabling, the line code (often in combination with a scrambler) also needs to remove DC bias.
A hamming code adds redundant bits for forward error correction.
The fundamental difference is the purpose of the code.
A line code is used to condition the data on a transmission line. Generally to balance the number of 0 and 1 bits sent to balance the current (what both 4b5b and 8b10b codes so)
A hamming code is used for error correction -- to detect transmission errors and possibly correct for them.
Calculating offset of branch instruction:
Hey guys,
My professor sent us a study guide with answers. He never actually went over how he got the answers though. I've searched online but I did not have any luck finding an explanation so at this point I'm a bit desperate.
Does anyone know how the author arrived at those answers?
0xfffb is the 16-bit signed two's complement representation of -5. So in this machine, the offset is scaled by the (presumably fixed) instruction length to get a byte address. (It could have byte sized instructions, but that is not possible since the offset itself is 16-bits.) The architecture is such that by the time the branch is executed the PC is already incremented so a 0 offset is a NOP, a -1 offset branches to the branch itself, -2 branches to the instruction before the branch, etc. Count backwards until you get to loop:. (Either there is some more info attached to the question, or known context, giving the details of the architecture I have used in making the answer, or it is a fairly badly written question.)
For the cache question, you mostly just have to know the names used to describe cache architectures (or "cache geometry", "cache shape" etc.). "2-way set associative" means there are two places in the cache any given address can be placed. There are 128 sets, each of which can hold two blocks because it is 2-way associative, and each block is 32-bytes. (I usually call the 32-byte structure a "cache line", though it appears here the word "block" refers specifically to the data that is stored there and "line" also includes the valid bit and tag, etc.) You then break down the address starting from the least significant bit going outwards in the cache geometry.
It looks like this is an instruction cache so we're going to insist the bottom two bits are 0 and organize the cache in 32-bit items. The block is 32-bytes or 5-bits. 2 are the "byte offset" which probably should just be 0, and then 3 bits to complete the 5-bit part which gets called a "block offset" (really offset within the block). (This subdivison of the 5 low bits doesn't really change anything here.) 128 entries in the set gives a 7-bit "index." The rest of the address, 20 bits, has to be used to tag the block to make sure it holds the address being looked up. (I.e. to determine cache hit or miss.) Plus we need one more bit to say whether there's actually data in the block.
Then we just add it all up -- 32-bytes or 256 bits for the data, plus 20 bits of tag and 1 valid bit, multiply by 128 sets and 2 ways.
While reading about Elias Gamma coding on wikipedia, I see it mentions that:
"Gamma coding is used in applications where the largest encoded value is not known ahead of time."
and that:
"It is used most commonly when coding integers whose upper-bound cannot be determined beforehand."
I don't really understand what is meant by these sentences, because whenever this algorithm is coded, the largest value of the test data or range of the test data would be known before hand. Any help is appreciated!
As far as I'm acquainted with Elias-gamma/delta encoding, the first sentence simply states that these compression methods are global, which means that it does not rely on the input data to generate the code. In other words, these methods do not need to process the input before performing the compression (as local methods do); it compresses the data with a function that does not depend on information from the database.
As for the second sentence, it may be taken as a guarantee that, although there may be some very large integers, the encoding will still perform well (and will represent such values with feasible amount of bytes, i.e., it is a universal method). Notice that, if you knew the biggest integer, some approaches (like minimal hashes) could perform better.
As a last consideration, the same page you referred to also states that:
Gamma coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values.
This may be obtained by generating lists of differences from the original lists of integers, and passing such differences to be compressed instead. For example, in a list of increasing numbers, you could generate:
list: 1 5 29 32 35 36 37
diff: 1 4 24 3 3 1 1
Which will give you many more small numbers, and therefore a greater level of compression, than the first list.
I'm trying to write a basic digit counter (an integer is inputted and the number of digits of that integer is outputted) for positive integers. This is my general formula:
dig(x) := Math.floor(Math.log(x,10))
I tried implementing the equivalent of dig(x) in Ruby, and found that when I was computing dig(1000) I was getting 2 instead of 3 because Math.log was returning 2.9999999999999996 which would then be truncated down to 2. What is the proper way to handle this problem? (I'm assuming this problem can occur regardless of the language used to implement this approach, but if that's not the case then please explain that in your answer).
To get an exact count of the number of digits in an integer, you can do the usual thing: (in C/C++, assuming n is non-negative)
int digits = 0;
while (n > 0) {
n = n / 10; // integer division, just drops the ones digit and shifts right
digits = digits + 1;
}
I'm not certain but I suspect running a built-in logarithm function won't be faster than this, and this will give you an exact answer.
I thought about it for a minute and couldn't come up with a way to make the logarithm-based approach work with any guarantees, and almost convinced myself that it is probably a doomed pursuit in the first place because of floating point rounding errors, etc.
From The Art of Computer Programming volume 2, we will eliminate one bit of error before the floor function is applied by adding that one bit back in.
Let x be the result of log and then do x += x / 0x10000000 for a single precision floating point number (C's float). Then pass the value into floor.
This is guaranteed to be the fastest (assuming you have the answer in numerical form) because it uses only a few floating point instructions.
Floating point is always subject to roundoff error; that's one of the hazards you need to be aware of, and actively manage, when working with it. The proper way to handle it, if you must use floats is to figure out what the expected amount of accumulated error is and allow for that in comparisons and printouts -- round off appropriately, compare for whether the difference is within that range rather than comparing for equality, etcetera.
There is no exact binary-floating-point representation of simple things like 1/10th, for example.
(As others have noted, you could rewrite the problem to avoid using the floating-point-based solution entirely, but since you asked specifically about working log() I wanted to address that question; apologies if I'm off target. Some of the other answers provide specific suggestions for how you might round off the result. That would "solve" this particular case, but as your floating operations get more complicated you'll have to continue to allow for roundoff accumulating at each step and either deal with the error at each step or deal with the cumulative error -- the latter being the more complicated but more accurate solution.)
If this is a serious problem for an application, folks sometimes use scaled fixed point instead (running financial computations in terms of pennies rather than dollars, for example). Or they use one of the "big number" packages which computes in decimal rather than in binary; those have their own round-off problems, but they round off more the way humans expect them to.
Just asked by my 5 year old kid: what is the biggest number in the computer?
We are not talking about max number for a specific data types, but the biggest number that a computer can represent.
Infinity is not allowed.
UPDATE my kid always wants to print as
well, so lets say the computer needs
to print this number and the kid to
know that its a big number. Of course,
in practice we won't print because
theres not enough trees.
This question is actually a very interesting one which mathematicians have devoted a fair bit of thought to. You can read about it in this article, which is a fascinating and accessible read.
Briefly, a guy named Tibor Rado set out to find some really big, but still well-defined, numbers by defining a sequence called the Busy Beaver numbers. He defined BB(n) to be the largest number of steps any Turing Machine could take before halting, given an input of n symbols. Note that this sequence is by its very nature not computable, so the numbers themselves, while well-defined, are very difficult to pin down. Here are the first few:
BB(1) = 1
BB(2) = 6
BB(3) = 21
BB(4) = 107
... wait for it ...
BB(5) >= 8,690,333,381,690,951
No one is sure how big exactly BB(5) is, but it is finite. And no one has any idea how big BB(6) and above are. But at least these numbers are completely well-defined mathematically, unlike "the largest number any human has ever thought of, plus one." ;)
So how about this:
The biggest number a computer can represent is the most instructions a program small enough to fit in its available memory can perform before halting.
Squared.
No, wait, cubed. No, raised to the power of itself!
Dammit!
Bits are not numbers. You, as a programmer, give them the meaning you want, possibly numbers.
Now, I decide that 1 represents "the biggest number ever thought by a human plus one".
Errr this is a five year old?
How about something along the lines of: "I'd love to tell you but the number is so big and would take so long to say, I'd die before I finished telling you".
// wait to see
for(;;)
{
printf("9");
}
roughly 2^AVAILABLE_MEMORY_IN_BITS
EDIT: The above is for actually storing a number and treats all media (RAM, HD, cloud etc.) as memory. Subtracting the OS footprint (measured in KB) doesn't make "roughly" less accurate...
If you want to "represent" a number in a meaningful way, then you probably want to go with what the CPU provides: unsigned 32 bit integers (roughly 4 Gigs) or unsigned 64 bit integers for most computers your kid will come into contact with.
NOTE for talking to 5-year-olds: Often, they just want a factoid. Give him a really big and very accurate number (lots of digits), like 4'294'967'295. Then, once the glazing leaves his eyes, try to see how far you can get with explaining how computers represent numbers.
EDIT #2: I once read this article: Who Can Name the Bigger Number that should provide a whole lot of interesting information for your kid. Obviously he's not your normal five-year-old. So this might get you started in a cool direction about numbers and computation.
The answer to life (and this kids question): 42
That depends on the datatype you use to represent it. The computer only stores bits (0/1). We, as developers, give the bits meaning. (65 can be a number or the letter A).
For example, I can define my datatype as 1^N where N is unsigned and represented by an array of bits of arbitrary size. The next person can come up with 10^N which would be ten times larger than my biggest number.
Sure, there would be gaps but if you don't need them, that doesn't matter.
Therefore, the question is meaningless since it doesn't have context.
Well I had the same question earlier this day, so thought why not to make a little c++ codes to see where the computer gonna stop ...
But my laptop wasn't with me in class so I used another, well the number was to big but it never ends, i'll run it again for a night then i'll share the number
you can try the code is stupid
#include <stdlib.h>
#include <stdio.h>
int main() {
int i = 0;
for (i = 0; i <= i; i++) {
printf("%i\n", i);
i++;
}
}
And let it run till it stops ^^
The size will obviously be limited by the total size of hard drives you manage to put into your PC. After all, you can store a number in a text file occupying all disk space.
You can have 4x2Tb drives even in a simple box so around 8Tb available. if you store as binary, then the biggest number is 2 pow 64000000000000.
If your hard drive is 1 TB (8'000'000'000'000 bits), and you would print the number that fits on it on paper as hex digits (nobody would do that, but let's assume), that's 2,000,000,000,000 hex digits.
Each page would contain 4000 hex digits (40 x 100 digits). That's 500,000,000 pages.
Now stack the pages on top of each other (let's say each page is 0.004 inches / 0.1 mm thick), then the stack would be as 5 km (about 3 miles) tall.
I'll try to give a practical answer.
Common Lisp number crunching is particularly powerful. It has something called "bignums" which are integers that can be arbitrarily large, limited by the amount of available.
See: http://en.wikibooks.org/wiki/Common_Lisp/Advanced_topics/Numbers#Fixnums_and_Bignums
Don't know much about theory, but I far as I understood from your question, is: what is the largest number that the computer can represent (and I add: in a reasonable time, and not printing "9" until the Earth will "be eaten by the Sun"). And I put my PC to make one simple calculation (in PHP or whatever language): echo pow(2,1023) - resulting: 8.9884656743116E+307. So I guess this is the largest number that my PC can calculate. On the other side, I think the respresentation of the largest negative number can be: -0,(0)1
LE: That computed value was obataind through PHP, but I tried to figure out what's the largest number that my windows calculator can compute, and it is pow(2, 33219) = 8.2304951207588748764521361245002E+9999. Now I guess this is the largest number my PC can handle.
I think you should be very proud that your 5 year old is already asking questions like this.
And you should continue to promote that! This is truly amazing! With that said, I would say that saying Infinity does not
count is thinking incorrectly about what numbers mean in computer memory.
I feel like this way of thinking is a handicap.
Mathematicians will never be able to write out ALL the digits of pi or eulers number, BUT we FULLY understand it.
Pi, as an example, is perfectly represented by infinite this series: (Pi / 4) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - …
Just because you literally can’t go to inf. or print every single digit in a console means nothing.
You could have printed the symbol representing pi and therefore capturing the inf. series.
Computer Algebra Systems (CAS) represent numbers symbolically all the time. Pi, for instance,
may be a Symbolic object in memory (the binary in memory did not DIRECTLY represent the number. It represents an "mathematical algorithm" for producing the answer to arbitrary precision).
Then you do some math with it, transforming from one expression to the next.
At no point in time did we not represent the number COMPLETELY.
At the end, you can do 2 things with this:
A) Evaluate the expression, turning it into a number of some kind (or Matrix or whatever).
BUT this number could very well be an approximation (say like 20 digits of pi).
B) Keep it in its symbolic form for reference. Obviously we don’t like staring at symbols because we
need to eventually turn the nobs on the apparatii.
NOTE: sometimes you can get a finite (non-irrational) number perfectly represented in memory (like number 1)
by taking limits or going to inf. Not literally having an inf. number in memory, but symbolically representing it.
Just throw this in Wolfram alpha: Lim[Exp[-x], x --> Inf]; It gives you the number 0. Which is EXACT.
In short:
It was the HUMANS need to have some binary in memory that DIRECTLY represented the number that caused
the number to degrade. Symbolically it was perfectly represented. You could design some algorithm that
just continues to calculate the next digits of pi or eulers number giving you an arbitrary amount of precision (Now, this is obviously not practical of course).
I hope this was at least somewhat useful or interesting to you, even if you disagree =)
Depends on how much the computer can handle. Although there are some times when the computer can handle numbers greater than (2^(bits-1)-1)... For example:
My computer is 64 bit (9223372036854775807), however the calculator that comes with the computer itself can handle numbers of up to 10^9999.
Many other supercomputers can exceed these limits, and the one with the most memory (bits) might as well be the one with the record (current largest number that can be held by computers).
Or, if it comes to visually seeing it on computers, you can just make a program that, on monitor, repeats writing 9 and not skips that line to form an ever-growing bunch of 9. :P
go on chrome then go on three dots above and click them then go on tools and then go on developer tool click on console and type Number.MAX_VALUE