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Comment by cubefox | Hacker News Reader

> 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?

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)

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?

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...

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.

In the best case

This is a worst-case result. In the best case, it is easy to see that the gap can be exponential. Imagine a program that increments a counter until it reaches value n. This "program" would only need O(logn) memory: mainly the counter itself.

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