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Re: Another little puzzle

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Tim Rentsch <tr.17687@z991.linuxsc.com> writes:

> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>
>> Tim Rentsch <tr.17687@z991.linuxsc.com> writes:
>>
>>> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
>>>
>>>> I'm sorry to be obtuse, but what is the "conventional average"?  The
>>>> name makes it sound trivial, but the quadratic time makes me certain
>>>> that is isn't.
>>>
>>> Sorry, I meant to refer to your formulation of average
>>>
>>>     A that minimizes { Sum_{i=1,n} difference(A, t(i))^2 }
>>>
>>> where 'difference' means the shorter arc length.  This formula
>>> matches the result for 'mean' on real numbers.
>>>
>>>> My "conventional average" algorithm (which is not well thought
>>>> out) was to (a) rotate the data set to avoid the 23/0 boundary
>>>> (not always possible), (b) take the arithmetic mean, and then (c)
>>>> rotate the result back.  E.g. [23,0,1] -> [0,1,2] by adding one,
>>>> and the average is mean[0,1,2] - 1 = 0.
>>>
>>> Yes, if you know where to split the cycle then the answer can be
>>> found in O(n) time.  But how can we figure out where to split the
>>> cycle?
>>
>> Well I handily stopped considering this at the stage where I assumed
>> there must be a simple way to spot the optimal rotation, so I never
>> thought it might have to be quadratic.  Presumably your algorithm
>> tries all the offsets and minimises the result.
>
> Right.
>
>> Looking at it a bit more I can't see a better way (but that might be
>> the Ratafia de Champagne).  It feels as if there /should/ be one.
>> In fact it feels as if it should be linear.
>
> My best so far is only O( n * log n ).  Probably that isn't
> optimal.  I don't see how to make it linear though.  Do you have
> any ideas?

No.  I should try with a clear head.

-- 
Ben.

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Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-21 13:17 -0800
  Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-22 04:10 +0000
    Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-24 06:21 -0800
      Re: Another little puzzle Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-12-24 16:53 +0000
        Re: Another little puzzle Richard Heathfield <rjh@cpax.org.uk> - 2022-12-24 17:14 +0000
        Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-24 22:30 +0000
          Re: Another little puzzle Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2022-12-25 00:02 +0000
            Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-25 01:30 +0000
            Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-25 01:55 +0000
        Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-25 00:37 -0800
      Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-25 00:41 +0000
        Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-25 01:05 -0800
          Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-25 09:30 -0800
          Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-25 21:44 +0000
            Re: Another little puzzle Richard Damon <Richard@Damon-Family.org> - 2022-12-25 17:59 -0500
              Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-25 18:39 -0800
              Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-26 03:18 +0000
            Re: Another little puzzle Tim Rentsch <tr.17687@z991.linuxsc.com> - 2022-12-25 18:35 -0800
              Re: Another little puzzle Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-12-26 03:49 +0000

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