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How much one's individuality cost? At least 2.77544 bits of information

June 22 2012 at 9:04 AM
jarek 

 

Imagine there is a population/database/dictionary and we would like to distinguish its elements.
So for each element, let us somehow encode its individual features (e.g. using a hash function) as a bit sequence - the most dense way is to use sequence of uncorrelated P(0)=P(1)=1/2 bits.
We can now create minimal prefix tree required to distinguish these sequences, like in the figure below.
For such ensemble of random trees of given number of leaves (n), we can calculate Shannon entropy (H_n) - the amount of information it contains.
It turns out that it asymptotically grows with at average 2.77544 bits per element (1/2+(1+\gamma)lg(e)).
The calculations can be found here:http://arxiv.org/abs/1206.4555

[linked image]

Is it the minimal cost of distinguishability/individuality?
How to understand it?

ps. related question: can anyone find D(n)=\sum_{i=0}^{\infty} 1-(1-2^{-i})^n ?

 
 Respond to this message   
Response TitleAuthorDate
 ClarificationjarekJun 22, 2012, 4:09 PM
 Devaluation of individualityAnonymousJul 1, 2012, 8:45 AM
 Re: How much one's individuality cost? At least 2.77544 bits of informationAnonymousOct 26, 2012, 2:27 AM
 Re: How much one's individuality cost? At least 2.77544 bits of informationOf CourseOct 26, 2012, 6:30 AM
 Re: How much one's individuality cost? At least 2.77544 bits of informationjarekOct 26, 2012, 6:35 PM
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