# On Lipschitz Bijections Between Boolean Functions

@article{Rao2017OnLB, title={On Lipschitz Bijections Between Boolean Functions}, author={Shravas Rao and Igor Shinkar}, journal={Combinatorics, Probability and Computing}, year={2017}, volume={27}, pages={411 - 426} }

Given two functions f,g : {0,1}n → {0,1}, a mapping ψ : {0,1}n → {0,1}n is said to be a mapping from f to g if it is a bijection and f(z) = g(ψ(z)) for every z ∈ {0,1}n. In this paper we study Lipschitz mappings between Boolean functions. Our first result gives a construction of a C-Lipschitz mapping from the Majority function to the Dictator function for some universal constant C. On the other hand, there is no n/2-Lipschitz mapping in the other direction, namely from the Dictator function to…

## 2 Citations

### Lipschitz bijections between boolean functions

- MathematicsComb. Probab. Comput.
- 2021

There is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(2) input bits, but a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound.

### On Mappings on the Hypercube with Small Average Stretch

- MathematicsArXiv
- 2019

The average stretch of $\phi$ is defined as ${\sf avgStretch}(\phi) = {\mathbb E}[{\sf dist}(x),\phi(x'))]$, where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate.

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