Twenty (short) questions

Yuval Dagan, Yuval Filmus, Ariel Gabizon and Shay Moran

dyadic distribution is one in which all probabilities are negative powers of 2. If $\mu$ is a dyadic distribution on a finite domain, then Huffman’s algorithm produces a code whose average codeword length is $H(\mu)$. We can interpret this code as a decision tree.

A dyadic distribution can have many different optimal decision trees. We are interested in the following question: given $n$, what is the smallest list of queries that suffices to implement optimal decision trees for all dyadic distributions on $n$ elements?

We show that $1.25^{n+o(1)}$ queries suffice, and moreover, $1.25^{n-o(1)}$ queries are necessary for infinitely many $n$.

We also discuss how the number of queries scales as we allow slight deviations from optimality.

This paper is extracted from a longer STOC paper.


 author = {Yuval Dagan and Yuval Filmus and Ariel Gabizon and Shay Moran},
 title = {Twenty (short) questions},
 journal = {Combinatorica},
 volume = {39},
 number = {3},
 pages = {597--626},
 year = {2019}
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