Paper Info
Title
Twenty (simple) questions.
Abstract
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 π over the numbers {1,…,n}, and announces it to Bob. She then chooses a number x according to π, 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 π: Bob’s questions reveal the codeword for x bit by bit. This strategy finds x using fewer than H(π)+1 questions on average. However, the questions asked by Bob could be arbitrary. In this paper, we investigate the following question: *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 π, Bob has a strategy that uses only questions of the form \"x < c?\" and \"x = c?\", and uncovers x using at most H(π)+1 questions on average, matching the performance of Huffman codes in this sense. We also give a natural set of O(rn1/r) questions that achieve a performance of at most H(π)+r, and show that Ω(rn1/r) questions are required to achieve such a guarantee. Our second main result gives a set Q of 1.25n+o(n) questions such that for every distribution π, Bob can implement an optimal strategy for π using only questions from Q. We also show that 1.25n−o(n) questions are needed, for infinitely many n. If we allow a small slack of r over the optimal strategy, then roughly (rn)Θ(1/r) questions are necessary and sufficient.
Year
DOI
Venue
2017
10.1145/3055399.3055422
STOC
Keywords
DocType
Volume
combinatorial search theory,twenty questions game,binary decision tree,information theory,redundancy
Conference
abs/1611.01655
ISSN
Citations 
PageRank 
0737-8017
0
0.34
References 
Authors
23
4
Name
Order
Citations
PageRank
Yuval Dagan100.68
Yuval Filmus227527.33
Ariel Gabizon315613.97
Shay Moran46327.30