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- W2223787892 abstract "In this note, we study the relation between the parity decision tree complexity of a boolean function $f$, denoted by $mathrm{D}_{oplus}(f)$, and the $k$-party number-in-hand multiparty communication complexity of the XOR functions $F(x_1,ldots, x_k)= f(x_1opluscdotsoplus x_k)$, denoted by $mathrm{CC}^{(k)}(F)$. It is known that $mathrm{CC}^{(k)}(F)leq kcdotmathrm{D}_{oplus}(f)$ because the players can simulate the parity decision tree that computes $f$. In this note, we show that [mathrm{D}_{oplus}(f)leq Obig(mathrm{CC}^{(4)}(F)^5big).] Our main tool is a recent result from additive combinatorics due to Sanders. As $mathrm{CC}^{(k)}(F)$ is non-decreasing as $k$ grows, the parity decision tree complexity of $f$ and the communication complexity of the corresponding $k$-argument XOR functions are polynomially equivalent whenever $kgeq 4$. Remark: After the first version of this paper was finished, we discovered that Hatami and Lovett had already discovered the same result a few years ago, without writing it up." @default.
- W2223787892 created "2016-06-24" @default.
- W2223787892 creator A5022043412 @default.
- W2223787892 date "2015-06-09" @default.
- W2223787892 modified "2023-09-26" @default.
- W2223787892 title "Parity Decision Tree Complexity and 4-Party Communication Complexity of XOR-functions Are Polynomially Equivalent" @default.
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- W2223787892 doi "https://doi.org/10.48550/arxiv.1506.02936" @default.
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