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Re: Another little puzzle

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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?

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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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