Portal de Periódicos da CAPES

Indistinguishable Sceneries on the Boolean Hypercube

2018; Cambridge University Press; Volume: 28; Issue: 1 Linguagem: Inglês

10.1017/s0963548318000305

1469-2163

Renan Gross, Uri Grupel,

Markov Chains and Monte Carlo Methods

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbors colored by $1$, and locally stable functions, in which every vertex has a constant fraction of neighbors colored by its own color. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for $\mathbb{Z}^n$ and other graphs, and offer several follow-up questions.