TY - JOUR
T1 - OPTIMAL SETS OF QUESTIONS FOR TWENTY QUESTIONS
AU - Filmus, Yuval
AU - Mehalel, Idan
N1 - Publisher Copyright:
© 2024 Society for Industrial and Applied Mathematics.
PY - 2024
Y1 - 2024
N2 - In the distributional Twenty Questions game, Bob chooses a number x from 1 to n according to a distribution μ, and Alice (who knows μ) attempts to identify x using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for μ, yet this strategy could potentially uses all 2n possible questions. Dagan et al. constructed a set of 1.25n+o(n) questions which suffice to construct an optimal strategy for all μ, and showed that this number is optimal (up to subexponential factors) for infinitely many n. We determine the optimal size of such a set of questions for all n (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the d-ary setting, obtaining similar results with 1.25
AB - In the distributional Twenty Questions game, Bob chooses a number x from 1 to n according to a distribution μ, and Alice (who knows μ) attempts to identify x using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for μ, yet this strategy could potentially uses all 2n possible questions. Dagan et al. constructed a set of 1.25n+o(n) questions which suffice to construct an optimal strategy for all μ, and showed that this number is optimal (up to subexponential factors) for infinitely many n. We determine the optimal size of such a set of questions for all n (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the d-ary setting, obtaining similar results with 1.25
KW - combinatorial search
KW - games
KW - information theory
UR - http://www.scopus.com/inward/record.url?scp=85184857394&partnerID=8YFLogxK
U2 - 10.1137/21M1424494
DO - 10.1137/21M1424494
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AN - SCOPUS:85184857394
SN - 0895-4801
VL - 38
SP - 412
EP - 452
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
IS - 1
ER -