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Re: Sort one sequence by O(n) in time and O(1) in space

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On 10 February 2014 15:03, Sturla Molden <sturla.molden@gmail.com> wrote:
> Chris Angelico <rosuav@gmail.com> wrote:
>
>> That's assuming it really is a sort operation. The problem description
>> isn't entirely clear on this point, but if it's actually a zip, then
>> it can definitely be done in O(n).
>
> Ah, I didn't read it carefully enough. :-)
>
> Granted, a zip can be done in O(n) time and O(1) memory using a generator,
> which by the way is what itertools.izip does.

Yes but turning the generator into a list takes O(N) storage (for the
new list!). The OP wants to rearrange a list in-place.

Something like

    mylist[:] = reorder_generator(mylist)

won't work because the generator would need to access the data
non-sequentially (it would need to read elements after they were
overwritten).

The way to do this is to find the cycles of data movement i.e. the
sets of indices for which a permutation occurs. If you know the size
of the input then you can find these once and hard-code them.
Otherwise you need an algorithm that finds each cycle exactly once
using O(1) storage which is definitely not trivial.

You can see the top-level code that fftw uses for this here (I think
this code is very hard to follow without prior experience of their
code base - I certainly don't understand it):
https://github.com/FFTW/fftw3/blob/master/rdft/vrank3-transpose.c#L159

I'm not even sure if that really is O(1) storage though: it may be
something like O(MN/gcd(M, N)).

This page describes the general problem

    http://en.wikipedia.org/wiki/In-place_matrix_transposition

and mentions the existence of "more complicated" algorithms that can
use O(N+M) or O(log(MN)) storage.

So I don't think an O(1) storage O(N) operations solution exists for
the general M*N case although it may be possible for the
specialisation to 2*M. (I haven't tried this but if you're interested
see what cycles come up for different input sizes and whether there's
a pattern that can be predicted using O(1) storage).


Oscar

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Sort one sequence by O(n) in time and O(1) in space Wesley <nispray@gmail.com> - 2014-02-09 04:13 -0800
  Re: Sort one sequence by O(n) in time and O(1) in space Asaf Las <roegltd@gmail.com> - 2014-02-09 04:29 -0800
  Re: Sort one sequence by O(n) in time and O(1) in space Chris Angelico <rosuav@gmail.com> - 2014-02-09 23:39 +1100
    Re: Sort one sequence by O(n) in time and O(1) in space Wesley <nispray@gmail.com> - 2014-02-09 18:09 -0800
      Re: Sort one sequence by O(n) in time and O(1) in space Dave Angel <davea@davea.name> - 2014-02-09 22:00 -0500
  Re: Sort one sequence by O(n) in time and O(1) in space Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-09 13:41 +0000
  Re: Sort one sequence by O(n) in time and O(1) in space Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-09 14:30 +0000
  Re: Sort one sequence by O(n) in time and O(1) in space Ni Wesley <nispray@gmail.com> - 2014-02-09 22:40 +0800
  Re: Sort one sequence by O(n) in time and O(1) in space Roy Smith <roy@panix.com> - 2014-02-09 10:05 -0500
    Re: Sort one sequence by O(n) in time and O(1) in space Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-02-09 23:14 +0000
      Re: Sort one sequence by O(n) in time and O(1) in space Roy Smith <roy@panix.com> - 2014-02-09 19:15 -0500
      Re: Sort one sequence by O(n) in time and O(1) in space Wesley <nispray@gmail.com> - 2014-02-09 18:26 -0800
        Re: Sort one sequence by O(n) in time and O(1) in space Sturla Molden <sturla.molden@gmail.com> - 2014-02-10 10:20 +0000
          Re: Sort one sequence by O(n) in time and O(1) in space Steven D'Aprano <steve@pearwood.info> - 2014-02-11 05:20 +0000
            Re: Sort one sequence by O(n) in time and O(1) in space Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-12 11:56 +1300
          Re: Sort one sequence by O(n) in time and O(1) in space Tim Daneliuk <tundra@tundraware.com> - 2014-02-10 23:37 -0600
          Re: Sort one sequence by O(n) in time and O(1) in space Tim Daneliuk <tundra@tundraware.com> - 2014-02-10 23:37 -0600
        Re: Sort one sequence by O(n) in time and O(1) in space Chris Angelico <rosuav@gmail.com> - 2014-02-10 21:37 +1100
        Re: Sort one sequence by O(n) in time and O(1) in space Sturla Molden <sturla.molden@gmail.com> - 2014-02-10 15:03 +0000
        Re: Sort one sequence by O(n) in time and O(1) in space Chris Angelico <rosuav@gmail.com> - 2014-02-11 02:23 +1100
        Re: Sort one sequence by O(n) in time and O(1) in space Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-10 15:45 +0000
        Re: Sort one sequence by O(n) in time and O(1) in space Chris Angelico <rosuav@gmail.com> - 2014-02-11 02:52 +1100
          Re: Sort one sequence by O(n) in time and O(1) in space Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-11 13:19 +1300
        Re: Sort one sequence by O(n) in time and O(1) in space Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-10 16:41 +0000
  Re: Sort one sequence by O(n) in time and O(1) in space Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-09 15:48 +0000
    Re: Sort one sequence by O(n) in time and O(1) in space Wesley <nispray@gmail.com> - 2014-02-09 18:07 -0800
  Re: Sort one sequence by O(n) in time and O(1) in space Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-02-09 21:31 +0000

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