Separating Decision Tree Complexity from Subcube Partition Complexity

Authors Robin Kothari, David Racicot-Desloges, Miklos Santha

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Robin Kothari
David Racicot-Desloges
Miklos Santha

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Robin Kothari, David Racicot-Desloges, and Miklos Santha. Separating Decision Tree Complexity from Subcube Partition Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 915-930, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


The subcube partition model of computation is at least as powerful as decision trees but no separation between these models was known. We show that there exists a function whose deterministic subcube partition complexity is asymptotically smaller than its randomized decision tree complexity, resolving an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is based on the information-theoretic techniques first introduced to lower bound the randomized decision tree complexity of the recursive majority function. We also show that the public-coin partition bound, the best known lower bound method for randomized decision tree complexity subsuming other general techniques such as block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound, also lower bounds randomized subcube partition complexity. This shows that all these lower bound techniques cannot prove optimal lower bounds for randomized decision tree complexity, which answers an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014).
  • Decision tree complexity
  • query complexity
  • randomized algorithms
  • subcube partition complexity


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