TY - GEN
T1 - Twenty (Simple) questions
AU - Dagan, Yuval
AU - Filmus, Yuval
AU - Gabizon, Ariel
AU - Moran, Shay
N1 - Publisher Copyright:
© 2017 ACM.
PY - 2017/6/19
Y1 - 2017/6/19
N2 - 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 atmostH (π) + 1 questions on average, matching the performance of Huffman codes in this sense. We also give anatural set of O(rn1/r) questions that achieve aperformance 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.
AB - 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 atmostH (π) + 1 questions on average, matching the performance of Huffman codes in this sense. We also give anatural set of O(rn1/r) questions that achieve aperformance 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.
KW - Binary decision tree
KW - Combinatorial search theory
KW - Information theory
KW - Redundancy
KW - Twenty questions game
UR - http://www.scopus.com/inward/record.url?scp=85024384063&partnerID=8YFLogxK
U2 - 10.1145/3055399.3055422
DO - 10.1145/3055399.3055422
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AN - SCOPUS:85024384063
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 9
EP - 21
BT - STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
A2 - McKenzie, Pierre
A2 - King, Valerie
A2 - Hatami, Hamed
T2 - 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017
Y2 - 19 June 2017 through 23 June 2017
ER -