On Lipschitz Bijections Between Boolean Functions

  title={On Lipschitz Bijections Between Boolean Functions},
  author={Shravas Rao and Igor Shinkar},
  journal={Combinatorics, Probability and Computing},
  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… 

Lipschitz bijections between boolean functions

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

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.



Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

It is proved that ψ is “approximately local” in the sense that all but the last output bit of ψ are essentially determined by a single input bit.

Gaussian noise sensitivity and Fourier tails

It is shown that certain cases of MatIsoLie -- for the wide and widely studied classes of semi simple and abelian Lie algebras -- are equivalent to graph isomorphism and linear code equivalence, respectively.

Bounded-Depth Circuits Cannot Sample Good Codes

Any data structure for storing codewords of a good code requires redundancy Ω(log n), if each bit of the codeword can be retrieved by a small AC0 circuit; and for some choice of the underlying combinatorial designs, the output distribution of Nisan’s pseudorandom generator against AC0 circuits of depth d cannot be sampled by small AC 0 circuits ofdepth less than d.

Noise stability of functions with low influences: Invariance and optimality

An invariance principle for multilinear polynomials with low influences and bounded degree is proved; it shows that under mild conditions the distribution of such polynmials is essentially invariant for all product spaces.

On the distribution of the fourier spectrum of Boolean functions

A general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1}N is obtained.

On the set of divisors of a number

If z is a natural number and if z=pipfy —Pj is its factorization into primes, then the sum X/ + \2 + '" + \ " will be called the degree of z. Let m be a squarefree natural number of degree /?, i.e.,

The Complexity of Distributions

  • Emanuele Viola
  • Computer Science, Mathematics
    2010 IEEE 51st Annual Symposium on Foundations of Computer Science
  • 2010
The first lower bound for the membership problem of representing a set $S \subseteq [n]$ of size $\alpha n$ is obtained, in the case where $1/\alpha$ is a power of $2 and queries ``$i \in S$?'' are answered by non-adaptively probing $o(\log n)$ bits.

Reconfiguring a hypercube in the presence of faults

Algorithms for embedding an N/2-node hypercube in an N- node hypercube with faulty processors and ways to produce embeddings which allow for low delay simulations, as well as ways to use a faulty hypercube to efficiently simulate a completely functioning hypercube of the same size are described.