For algorithms, a little memory outweighs a lot of time
quantamagazine.org343 points by makira 2 months ago
343 points by makira 2 months ago
Minus the fuzz: A multitape Turing machine running in time t can be simulated using O(sqrt(t log t)) space (and typically more than t time).
From the „Camel Book”, one of my favorite programming books (not because it was enlightening, but because it was entertaining); on the Perl philosophy:
“If you’re running out of memory, you can buy more. But if you’re running out of time, you’re screwed.”
This can work both ways. If the program needs more memory than the computer has, it can't run until you buy more. But if it takes twice as long, at least it runs at all.
Modern computers have so much memory it feels like it doesn't matter. Spending that memory on arrays for algorithms or things like a Garbage Collector just make sense. And, extra memory is worthless. You WANT the summation of all your programs to use all your memory. The processor, on the other hand, can context switch and do everything in it's power to make sure it stays busy.
The CPU is like an engine and memory is your gas tank. Idling the engine is bad, but leaving gas in the tank doesn't hurt, but it doesn't help either. I'm not gonna get to my destination faster because I have a full tank.
Only if running one such memory-hungry program at a time, which usually cannot be afforded. Multi-program workloads are much more common and the strategy of using as much ram as possible can't work in that environment.
The Camel book was written when Moore’s Law was trucking along. These days you can’t buy much more time but you used to be able to just fine. Now it’s horizontal scaling. Which is still more time.
Lookup tables with precalculated things for the win!
In fact I don’t think we would need processors anymore if we were centrally storing all of the operations ever done in our processors.
Now fast retrieval is another problem for another thread.
Reminds me of when I started working on storage systems as a young man and once suggested pre-computing every 4KB block once and just using pointers to the correct block as data is written, until someone pointed out that the number of unique 4KB blocks (2^32768) far exceeds the number of atoms in the universe.
It seems like you weren’t really that far off from implementing it, you just need a 4 KB pointer to point to the right block. And in fact, that is what all storage systems do!
Reminds me of when I imagined brute-forcing every possible small picture as simply 256 shades of gray for each pixel x (640 x 480 = 307200 pixels) = 78 million possible pictures.
Actually I don't have any intuition for why that's wrong, except that if we catenate the rows into one long row then the picture can be considered as a number 307200 digits long in base 256, and then I see that it could represent 256^307200 possible different values. Which is a lot: https://www.wolframalpha.com/input?i=256%5E307200
78 million is how many pixels would be in 256 different pictures with 307200 pixels each. You're only counting each pixel once for each possible value, but you actually need to count each possible value on each pixel once per possible combinations of all of the other pixels.
The number of possible pictures is indeed 256^307200, which is an unfathomably larger number than 78 million. (256 possible values for the first pixel * 256 possible values for the second pixel * 256 possi...).
Yeah I had a similar thought back in the 90s and made a program to iterate through all possible images at a fairly low res, I left it running while I was at school and got home after many hours to find it had hardly got past the first row of pixels! This was a huge eye-opener about how big a possibility-space digital images really exist in!
I has the same idea when I first learned about programming as a teenager. I wonder how many young programmers have had this exact same train of thought?
i think at some point you should have realized that there are obviously more than 78 million possible greyscale 640x480 pictures. theres a lot of intuitive examples but just think of this:
https://images.lsnglobal.com/ZFSJiK61WTql9okXV1N5XyGtCEc=/fi...
if there were only 78 million possible pictures, how could that portrait be so recongizably one specific person? wouldnt that mean that your entire picture space wouldnt even be able to fit a single portrait of everyone in Germany?
"At some point" I do realise it. What I don't have is an intuitive feel for why a number can be three digits 000 to 999 and each place has ten choices, but it's not 10 x 3 possibles. I tried to ask ChatGPT to give me an intuition for it, but all it does is go into an explanation of combinations. I know it's 10 x 10 x 10 meaning 10^3 I don't need that explanation again, what I'm looking for is an intuition for why it isn't 10x3.
> "if there were only 78 million possible pictures, how could that portrait be so recongizably one specific person? wouldnt that mean that your entire picture space wouldnt even be able to fit a single portrait of everyone in Germany?"
It's not intuitive that "a 640x480 computer picture must be able to fit a single portrait of everyone in Germany"; A human couldn't check it, a human couldn't remember 78 million distinct pictures, look through them, and see that they all look sufficiently distinct and at no point is it representing 50k people with one picture; human attention and memory isn't enough for that.
Try starting with a 2x2, then 3x3, etc. image and manually list all the possibilities.
That's focusing on the wrong thing; as I said, "I know it's 10 x 10 x 10 meaning 10^3 I don't need that explanation [for the correct combinations], what I'm looking for is an intuition for why it isn't 10x3".
ChatGPT might be able to explain combinatorics if you use the keyterm.
I’m fond of derangements and their relationship with permutations, which contain a factor of e.
I had friend who had the same idea to do it for pixel fonts with only two colors and 16x16 canvas. It was still 2^256. Watching that thing run and trying to estimate when it would finish made me understand encryption.
The other problem is that (if we take literally the absurd proposal of computing "every possible block" up front) you're not actually saving any space by doing this, since your "pointers" would be the same size as the blocks they point to.
If you don't do _actually_ every single block then you have Huffman Coding [1].
I imagine if you have a good idea of the data incoming you could probably do a similar encoding scheme where you use 7 bits to point to a ~512 bit blob and the 8th bit means the next 512 couldn't be compressed.
In some contexts, dictionary encoding (which is what you're suggesting, approximately) can actually work great. For example common values or null values (which is a common type of common value). It's just less efficient to try to do it with /every/ block. You have to make it "worth it", which is a factor of the frequency of occurrence of the value. Shorter values give you a worse compression ratio on one hand, but on the other hand it's often likelier that you'll find it in the data so it makes up for it, to a point.
There are other similar lightweight encoding schemes like RLE and delta and frame of reference encoding which all are good for different data distributions.
The idea is not too far off. You could compute a hash on an existing data block. Store the hash and data block mapping. Now you can use the hash in anywhere that data block resides, i.e. any duplicate data blocks can use the same hash. That's how storage deduplication works in the nutshell.
Except that there are collisions...
This might be completely naive but can a reversible time component be incorporated into distinguishing two hash calculations? Meaning when unpacked/extrapolated it is a unique signifier but when decomposed it folds back into the standard calculation - is this feasible?
Some hashes do have verification bits, that are used not just to verify intact hash, but one "identical" hash from another. However, they do tend to be slower hashes.
Do you have an example? That just sounds like a hash that is a few bits longer.
Mostly use of GCM (Galois/Counter Mode). Usually you tag the key, but you can also tag the value to check verification of collisions instead.
But as I said, slow.
hashes by definition are not reversible. you could store a timestamp together with a hash, and/or you could include a timestamp in the digested content, but the timestamp can’t be part of the hash.
> hashes by definition are not reversible.
Sure they are. You could generate every possible input, compute hash & compare with a given one.
Ok it might take infinite amount of compute (time/energy). But that's just a technicality, right?
Sure they are. You could generate every possible input
Depends entirely on what you mean by reversible. For every hash value, there are an infinite number of inputs that give that value. So while it is certainly possible to find some input that hashes to a given value, you cannot know which input I originally hashed to get that that value.
Can use cryptographic hashing.
How does that get around the pigeonhole principle?
I think you'd have to compare the data value before purging, and you can only do the deduplication (purge) if the block is actually the same, otherwise you have to keep the block (you can't replace it with the hash because the hash link in the pool points to different data)
The hash collision chance is extremely low.
For small amounts of data yeah. With growing data, the chance of a collision grows more than proportional. So in the context of working on storage systems (like s3 or so) that won't work unless customers actually accept the risk of a collission as okay. So for example, when storing media data (movies, photos), I could imagine that, but not for data in general.
Cryptographic hashing collisions are very very small, like end of universe in numerous times small. They're smaller than AWS being burnt down and all backups were lost leading to data loss.
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