Spring 2017

Twenty (Simple) Questions (and Hitting Maximal Antichains)

Thursday, April 13th, 2017 4:00 pm4:30 pm

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A basic combinatorial interpretation of Shannon's entropy function is via the ``20 questions'' game. 
This cooperative game is played by two players, Alice and Bob: 
Alice picks a distribution $\pi$ over the numbers $\{1,\ldots,n\}$, 
and announces it to Bob.
 She then chooses a number $x$ according to $\pi$, 
and Bob attempts to identify $x$ using as few 
Yes/No queries as possible, on average.
An optimal strategy for the ``20 questions'' game is given by a Huffman code for $\pi$: 
Bob's questions reveal the codeword for $x$ bit by bit. 
This strategy finds $x$ using fewer than $H(\pi)+1$ questions on average. 
However, the questions asked by Bob could be arbitrary. 
In this paper, we investigate the following question:
\emph{Are there restricted sets of questions that match the performance of Huffman codes, either exactly or approximately?}
Our first main result shows that for every distribution $\pi$, 
Bob has a strategy that uses only questions of the form ``$x < c$?'' and ``$x = c$?'’, 
and uncovers $x$ using at most $H(\pi)+1$ questions on average, 
matching the performance of Huffman codes in this sense.
We also give a natural set of $O(rn^{1/r})$ questions that achieve a performance of at most $H(\pi)+r$, 
and show that $\Omega(rn^{1/r})$ questions are required to achieve such a guarantee.
Our second main result gives a set $\Q$ of $1.25^{n+o(n)}$ questions such that 
for every distribution $\pi$, Bob can implement an \emph{optimal} strategy for $\pi$ using only questions from $\cQ$. 
We also show that $1.25^{n-o(n)}$ questions are needed, for infinitely many $n$.
Here, the set $\Q$ is derived via a random construction.
If time will allow, we will discuss the challenge of derandomizing
this construction, and a connection with hitting maximal antichains.
Based on joint work with Yuval Dagan, Yuval Filmus, and Ariel Gabizon