Abstract
In an influential paper, Kushilevitz and Mansour (1993) introduced a natural extension of Boolean decision trees called parity decision tree (PDT) where one may query the sum modulo \(2,\) i.e., the parity, of an arbitrary subset of variables. Although originally introduced in the context of learning, parity decision trees have recently regained interest in the context of communication complexity (cf. Shi and Zhang 2010) and property testing (cf. Bhrushundi, Chakraborty, and Kulkarni 2013). In this paper, we investigate the power of parity queries. In particular, we show that the parity queries can be replaced by ordinary ones at the cost of the total influence aka average sensitivity per query. Our simulation is tight as demonstrated by the parity function.
At the heart of our result lies a qualitative extension of the result of O’Donnell, Saks, Schramme, and Servedio (2005) titled: Every decision tree has an influential variable. Recently Jain and Zhang (2011) obtained an alternate proof of the same. Our main contribution in this paper is a simple but surprising observation that the query elimination method of Jain and Zhang can indeed be adapted to eliminate, seemingly much more powerful, parity queries. Moreover, we extend our result to linear queries for Boolean valued functions over arbitrary finite fields.
Raghav Kulkarni—Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation.
Xiaoming Sun—Part of this work was done while the author was visiting the Centre for Quantum Techologies, National University of Singapore. He is supported in part by the National Natural Science Foundation of China Grant 61170062, 61222202, 61433014 and the China National Program for support of Top-notch Young Professionals.
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Notes
- 1.
The \(O_\epsilon \) notation hides a multiplicative constant depending on \(\epsilon \) and the \(\tilde{O_\epsilon }\) notation hides a further poly-logarithmic multiplicative factor.
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Acknowledgements
We thank Rahul Jain, Supartha Poddar, Miklos Santha, and Avishay Tal for several helpful discussions. We also thank Ben vee Volk for pointing out that the super-linear separation in [27] works for PDTs as well.
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Kulkarni, R., Qiao, Y., Sun, X. (2015). On the Power of Parity Queries in Boolean Decision Trees. In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_10
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