Twenty (Simple) estions

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 Hu man code for π : Bob’s questions reveal the codeword for x bit by bit. This strategy nds 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 Hu man codes, either exactly or approximately? Our rst 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 uncoversx using at mostH (π )+1 questions on average, matching the performance of Hu man codes in this sense. We also give a natural set ofO (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 in nitely many n. If we allow a small slack of r over the optimal strategy, then roughly (rn)Θ(1/r ) questions are necessary and su cient. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior speci c permission and/or a fee. Request permissions from [email protected]. STOC’17, Montreal, Canada © 2017 ACM. 978-1-4503-4528-6/17/06. . . $15.00 DOI: 10.1145/3055399.3055422 CCS CONCEPTS • Mathematics of computing → Combinatoric problems; Information theory;