# 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

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 Title Author Date Clarification jarek Jun 22, 2012, 4:09 PM Devaluation of individuality Anonymous Jul 1, 2012, 8:45 AM Re: How much one's individuality cost? At least 2.77544 bits of information Anonymous Oct 26, 2012, 2:27 AM Re: How much one's individuality cost? At least 2.77544 bits of information Of Course Oct 26, 2012, 6:30 AM Re: How much one's individuality cost? At least 2.77544 bits of information jarek Oct 26, 2012, 6:35 PM