From: tr.17687@z991.linuxsc.com   
      
   Ben Bacarisse  writes:   
      
   > Tim Rentsch  writes:   
   >   
   >> Ben Bacarisse  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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