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New proof dramatically compresses space needed for computation

(www.scientificamerican.com)
190 points baruchel | 2 comments | 27 Jun 25 13:59 UTC | HN request time: 0s | source
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zerof1l ◴[30 Jun 25 10:05 UTC] No.44421424[source]▶
Here's the gist:

For nearly 50 years theorists believed that if solving a problem takes t steps, it should also need roughly t bits of memory: 100 steps - 100bits. To be exact t/log(t).

Ryan Williams found that any problem solvable in time t needs only about sqrt(t) bits of memory: a 100-step computation could be compressed and solved with something on the order of 10 bits.

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cubefox ◴[30 Jun 25 13:10 UTC] No.44422855[source]▶
> Ryan Williams found that any problem solvable in time t needs only about sqrt(t) bits of memory

At least or at most or on average?

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user070223 ◴[30 Jun 25 13:55 UTC] No.44423326[source]▶
I've skimmed the result couple of weeks ago, and if I remember it states that there from all machines which takes t time to accomplish something there is one such that sqrt(t) memory is used. so it's at most but not on avg nor amorotized[not sure if et even make sense it the space where the problem lies] (you take the least memory hungry machine)
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cma ◴[30 Jun 25 18:18 UTC] No.44426291[source]▶
There must be more constraints, what if the problem is copying a string? Or is it only for turing tape machines where it has to backtrack during the copy, increasing number of steps?
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1. jasperry ◴[30 Jun 25 19:47 UTC] No.44427098[source]▶
That's a good example! So the sqrt(n) space in the result must refer to space beyond that needed to write down the output--basically a working memory scratchpad. In which case, a string-copying function could use just a constant amount of scratch space.

I mean, for all I know, the result may have been proved in the model of decision problems where the output is always one bit. But I'd guess that it generalizes just fine to the case where you have to write longer outputs.

However, your question does make me wonder if the result still applies in the RAM model, where you can move to any spot on the tape in constant time. My theory knowledge is getting rusty...

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2. jasperry ◴[30 Jun 25 20:20 UTC] No.44427404[source]▶
Update after youtube: It was already known that a time-t function can be computed in sqrt(t) space on a single-tape Turing machine. It's exactly like @cma said: a Turing machine wastes a lot of time moving the head back and forth. Williams' result shows sqrt(t) space for multi-tape Turing machines. Those are much more time-efficient in regard to head movement, so until Williams' proof it was not believed to be possible to to make any algorithm use only sqrt(t) space.