Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

Itai Benjamini, Gil Cohen, Igor Shinkar
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ : {0, 1} n → x ∈ {0, 1} n+1 : |x| > n/2 such that for every x = y ∈ {0, 1} n it holds that where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who
more » ... aised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions will require some new ideas beyond the sensitivity-based structural results of Boppana [IPL 97]. We study the mapping ψ further and show that it (and its inverse) are computable in DLOGTIME-uniform TC 0 , but not in AC 0 . Moreover, we prove 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.
doi:10.1109/focs.2014.17 dblp:conf/focs/BenjaminiCS14 fatcat:3sj2ftzgrfbafhxnnefuc5l3ba