I noticed that in complexity classes, logspace class is defined but there is no logtime.

I was wondering how is that possible?

Normally, I would expect the opposite, It is possible to do a search query in an ordered list in time less than the time of iterating over all items (==> Logtime?). However you can't store a list in of size n in log(n) space...

So how come Logspace exists but not logtime? there's obviously something wrong with my reasoning.

Could someone please correct me?

Thanks in advance


1 Answer 1


Turing machines operating in logarithmic time cannot even read the entire input. This makes them rather uninteresting.

What you have in mind is not Turing machines, but random-access machines, for which logarithmic time does make sense. Indeed, the corresponding complexity class exists: DLOGTIME. It is most often used in the context of DLOGTIME-uniform circuits. A related complexity class is ALOGTIME, which is the uniform version of NC¹. Buss famously proved that the formula evaluation problem is ALOGTIME-complete. For comparison, the circuit evaluation problem is P-complete.

Algorithms running in $o(n)$ time on random-access machines are known as sublinear time algorithms. Such algorithms also abound in the area of data structures.

  • 4
    $\begingroup$ Since the question also shows some confusion about logspace, it might be worth mentioning that in the Turing machine model, the "logarithmic space" corresponds to the size of a working tape additionally to the input tape. $\endgroup$
    – ttnick
    Feb 4, 2021 at 15:24
  • $\begingroup$ @ttnick, the reason I did not accept this answer immediately was because it lacked the extra information you just mentioned! Now I'm good, I have my answer. Thank you both! $\endgroup$
    – user206904
    Feb 4, 2021 at 15:46

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