“If you’re running out of memory, you can buy more. But if you’re running out of time, you’re screwed.”
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.
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.
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
There are other similar lightweight encoding schemes like RLE and delta and frame of reference encoding which all are good for different data distributions.
Alternately, have you considered 8 byte blocks?
If your block pointers are 8-byte addresses, you don't need to count on block sparsity, in fact, you don't even need to have the actual blocks.
A pointer type, that implements self-read and writes, with null allocations and deletes, is easy to implement incredibly efficiently in any decent type system. A true zero-cost abstraction, if I have ever seen one!
(On a more serious note, a memory heap and CPU that cooperated to interpret pointers with the top bit set, as a 63-bit linear-access/write self-storage "pointer", is an interesting thought.
It's basically how deduplication works in ZFS. And that's why it only makes sense when you store a lot of repetitive data, e.g. VM images.
Using an LLM and caching eg FAQs can save a lot of token credits
AI is basically solving a search problem and the models are just approximations of the data - like linear regression or fourier transforms.
The training is basically your precalculation. The key is that it precalculates a model with billions of parameters, not overfitting with an exact random set of answers hehe
Do LLM providers use caches for FAQs, without changing the number of tokens billed to customer?
Community-scale caching? That's basically what pre-compiled software distributions are. And one idea for addressing the programming language design balk "that would be a nice feature, but it's not known how to compile it efficiently, so you can't have it", is highly-parallel cloud compilation, paired with a community-scale compiler cache. You might not mind if something takes say a day to resolve, if the community only needs it run once per release.
Makes sense. Say you have a pattern (surrounded by empty space) that 'flickers': A-B-A-B-A... etc. Then as long as nothing intrudes, nth generation is the same pattern as in n+1000,000th generation. Similar for patterns that do a 3-cycle, 4-cycle etc.
All you'd need is a) a way to detect repeating patterns, and b) do some kind of collision detection between areas/patterns (there's a thing called 'lightspeed' in Life, that helps).
On my way to memoize your search history.
In time O(n) you can use O(n) cells on a tape, but there are O(2^n) possible configurations of symbols on a tape of length n (for an alphabet with 2 symbols), so you can do so much more with n space than with n time.
If intermediate results are entirely uncorrelated, with no overlap in the work at all, that would not hold - space will not help you. Edit: This kind of problem is very rare. Think of a cache with 0 percent hit rate - almost never happens.
And you can't really do it the other way around (at least not in current computing terms / concepts): you cannot use a single unit of time as a standin / proxy for hundreds of cells, since we don't quite have infinitely-broad SIMD architectures.
Expensive calculation, cheap storage → caching results helps.
Limited bandwidth / 'expensive' storage, simple calculation (see: today's hyper-fast CPU+L1 cache combo's) → better to re-compute some things on the fly as needed.
I suspect there's a lot of existing software (components) out there designed along the "save CPU cycles, burn storage" path, where in modern reality a "save storage, CPU cycles are cheap" would be more effective. CPU speeds have grown way way faster than main memory bandwidth (or even size?) over the last decades.
For a datacenter, supercomputer, embedded system, PC or some end-user's phone, the metrics will be different. But same principle applies.
I forget the name of the O(1) access model. Not UMA, something else.
(Quite aside from the practical "we build on the surface of the earth" consideration, heat dissipation considerations limit you to a 2 dimensional circuit in 3-space.)
EDIT: I am a dumbass and misremembered.
Something we all intuitively expected but someone finally figured out an obscure way to prove it.
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This is a really roundabout article that takes a meandering path to a total bombshell in the field of complexity theory. Sorry for spoiling but uhhh, you'd expect an article about P == PSPACE would get to the point faster....
More precisely, I think it is intuitive that the class of problems that can be solved in any time given O(n) space is far larger than the class of problems that can be solved in any space given O(n) time.
I did a lot of raster graphics programming, in my career, and graphics work makes heavy use of lookup tables.
Yesterday, I posted a rather simple tool I wrote[0]: a server that “frontloads” a set of polygons into a database, and then uses them, at query time. It’s fairly fast (but I’m sure it could be a lot faster). I wrote it in a few hours, and got pretty good performance, right out of the starting gate.
Pretty basic stuff. I doubt the pattern is unique, but it’s one that I’ve used for ages. It’s where I try to do as much “upfront” work as possible, and store “half-baked” results into memory.
Like I said, I always “just worked that way,” and never really thought about it. There’s been a lot of “rule of thumb” stuff in my work. Didn’t have an MIT professor to teach it, but it’s sort of gratifying to see that it wasn’t just “old wives” stuff.
There’s probably a ton of stuff that we do, every day, that we don’t think about. Some of it almost certainly originated from really smart folks like him, finding the best way (like the “winding number” algorithm, in that server[1]), and some of it also probably comes from “grug-brained programmers,” simply doing what makes sense.
[0] https://news.ycombinator.com/item?id=44046227
[1] https://github.com/LittleGreenViper/LGV_TZ_Lookup/blob/e247f...
What I mean is that traditionally I think peoples' ideas of lookup tables are things like statically baked arrays setup at compile time, or even first thing at runtime, and they never change. But if you loosen your adherence to that last idea a bit, where a lookup table can change slightly over time, you can get a ton of mileage out of a comparatively small amount of memory compared to wasting cycles every frame.
As for the right tool for the job, I've read tons of dev logs and research papers over the years about moving work to the GPU, but this last few months stint of ripping my game inside out has really made me see the light. It's not just lookup tables built at compile or early runtime, but lookup tables modified slightly over time, and sent to the GPU as textures and used there.
Follow this train of thought long enough, and now we're just calling memory writes and reads "lookup tables" when they aren't really that anymore, but whos to say where the barrier really lies?
If you take the example of a game, drawing sprites say. Drawing a single preloaded sprite of reasonable size is always cheap, so a slow frame must have an excessive number. It's very hard to construct a practical scenario of a large number of truly distinct sprites though. A level has a finite tile palette, a finite cast of characters, abilities, etc. It's hard to logistically get them all into a scene together, and even then it won't be that many. So the only scenario left where sprite drawing will be slow is drawing the same handful of sprites over and over again. By contrast that's super common: just spam a persistent projectile, tap the analog stick to generate dust particles, etc.
For example, each block of pixels might have some calculated characteristics that were accessed by a hash into a LUT, but the characteristics would change, as we went through the image.
We'd do a "triage" run, where we'd build the LUT, then a "detailed" run, where we'd apply the LUT to the pixels.
It could get pretty hairy.
And paper: https://people.csail.mit.edu/rrw/time-vs-space.pdf
If you expect N ones at the output, how can this machine be simulated in the space smaller than N?
This machine must decrement the digit N at the beginning of the tape, and move to the end of the tape to write "1", so it runs in time O(N^2), not O(N)? (as it takes N "trips" to the end of the tape, and each "trip" takes 1, 2, 3 .. N steps)
Since turing machines can not jump to any place on a tape in constant time (like computers can), does it have any impact on real computers?
But to answer your question: "space" here refers to working space, excluding the input and output.
EDIT: This makes sense because if you look at all problems with N outputs then that is just the same as "gluing together" N different decision problems (+ some epsilon of overhead)
This "reverse" relationship works with other parameters. Such as in sorting algorithms, where besides time and space requirements you have stability requirement. If you constrain the sorting algorithms by stability, you won't have any advantage (will likely to have some disadvantage) in performance over non constrained algorithms. And wee see that the known stable sorting algorithms are either slower or require more memory. Nobody has yet invented a stable sorting algorithm that has comparable performance (both time and space) to non stable sorting algorithms. That's it.
But the speed of light is the maximum space in the smallest time, which computationally would correspond to filling the largest amount of memory in the shortest time :facepalm: (and thanks for the links!)
Ok, but space is cheap, and we usually want to trade processing time for space.
I.e., the opposite.
He's not trying to please programmers.
Give operating system development organizations a lot more memory and they will find a way to worsen the response time.
More at eleven.
I don't think Quanta should be afraid of showing math to people. That's really their whole purpose. Even if I think they've made some egregious mistakes that make them untrustable...[2]
[0] https://en.wikipedia.org/wiki/PSPACE#/media/File:Complexity_...
[1] https://www.quantamagazine.org/june-huh-high-school-dropout-...
"I think it is very intuitive that more space beats the pants off of more time." (poster is absolutely right) The The article say "Until now, the only known algorithms for accomplishing certain tasks required an amount of space roughly proportional to their runtime, and researchers had long assumed there’s no way to do better.", which is interpreted as that there's a proportional relation between time and space. However, a quick look at the complexity hierarchy would never suggest such a thing. Reading more carefully, it says "known algorithms" for "certain tasks", but then where do you get a general intuition from such a particular statement?
My article was quite explicit in multiple places that the universal/comprehensive character of the result was that counterintuitive part:
- In the first paragraph: "memory was more powerful than computer scientists believed: A small amount would be as helpful as a lot of time in all conceivable computations."
- Further down in the introduction, in the passage you quoted: "Until now, the only known algorithms for accomplishing certain tasks required an amount of space roughly proportional to their runtime, and researchers had long assumed there’s no way to do better. Williams’ proof established a mathematical procedure for transforming any algorithm — no matter what it does — into a form that uses much less space.
- In the third section, I explicitly state that researchers do believe space is more powerful than time in the specific sense that you're criticizing my article for misrepresenting: "But complexity theorists suspect that PSPACE is a much larger class, containing many problems that aren’t in P. In other words, they believe that space is a far more powerful computational resource than time. This belief stems from the fact that algorithms can use the same small chunk of memory over and over, while time isn’t as forgiving — once it passes, you can’t get it back."
- In the fourth section, I explain why researchers didn't think the HPV75 result could be improved further, despite their intuition that space is more powerful than time in the above sense: "While many problems can be solved with much less space than time, some intuitively seemed like they’d need nearly as much space as time."
TCS (and complexity theory specifically) are complicated subjects. I spend a lot of time interviewing researchers and thinking about how to distill the results of my reporting into a form that is accessible to readers with widely varying levels of familiarity with the subject matter. You are of course well within your rights to critique my stylistic choices, the narrative aspects of the story, and the order in which I presented information, but I will push back against the claim that my article is spreading misinformation about complexity theory. You're referring to a misconception that arises, by your own admission, when you don't read carefully. If it's the headline you object to, you could lodge a similar complaint against the complexity theorist Lance Fortnow: https://blog.computationalcomplexity.org/2025/02/you-need-mu....
I can imagine it's tough to put that much effort into communicating something complex to a wide audience, only to have a bunch of very smart people here attempt to tear it apart.
All this accomplishes is discrediting science. Trading personal gains for eroding the very thing that they make their money off of. This is a major part of why Americans (and people) have such high distrust for science. News outlets, and in particular science focused news outlets, constantly spew inaccurate information. It really should be no wonder that so many people are confused about so many scientific topics, as unless they actually take the years it takes to become an expert in a field, they are going to have a difficult time distinguishing fact from fiction. And why shouldn't the average person expect to trust a source like Quanta? They're "full of experts", right? smh
[0] This is the earliest archive I see with the note. Press back one day and it should not be there. Article was published on Nov 30 2022, along with a youtube video https://web.archive.org/web/20230329191417/https://www.quant...
What a strange, sad way to think about this. Academia is perverse.
Either way, the week is not yet over, at least since the quanta article, so maybe no kicking will ensue.
