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arXiv · 1907.07250

Shotgun reconstruction in the hypercube

Michał Przykucki·Alexander Roberts·Alex Scott
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Abstract

Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given radius. In this paper, we are concerned with random $2$-colourings of the vertices of the $n$-dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter $\Omega(n)$ are required to reconstruct. However, the situation for random colourings is dramatically different: we show that almost every $2$-colouring can be reconstructed from the multiset of colourings of balls of radius $2$. Furthermore, we show that for $q \ge n^{2+\epsilon}$, almost every $q$-colouring can be reconstructed from the multiset of colourings of $1$-balls.

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Michał Przykucki, Alexander Roberts, Alex Scott. 2019-07-16. Shotgun reconstruction in the hypercube. https://arxiv.org/abs/1907.07250

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