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The task is to find an infinite set of strings $a_1,a_2\ldots$, where $|a_{i+1}|>|a_i|$ and to find a compression algorithm $f$ for these strings, such that $|f(a_i)|=o(\log_2 |a_i|)$ with $i\to\infty$.

I have considered a set of strings with minimal entropy: $a_1=b, a_2=bb$, etc. The optimal compression algorithm for these strings is $f : a_i\mapsto |a_i|$, but this compresses each string only to $O(\log i)$ space.

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  • $\begingroup$ Try $b^{2^n}$, for example. $\endgroup$ Mar 8, 2020 at 18:00

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Hint:

Instead of $a_i = b^i$, try $a_i = b^{f(i)}$ for some function $f$.

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