Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the -dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for , almost every -coloring can be reconstructed from the multiset of colorings of 1-balls.

中文翻译：

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超立方体中的霰弹枪重建

Mossel 和 Ross 提出了何时可以从局部信息重建图的随机着色的问题，即给定半径的球的着色（具有多重性）。在本文中，我们关注维超立方体顶点的随机 2-着色，或等效的随机布尔函数。在最坏的情况下，需要直径的球来重建。然而，随机着色的情况却截然不同：我们表明，几乎所有 2-着色都可以从半径为 2 的球的着色多重集重建。此外，我们表明，对于，几乎所有 -着色都可以从多重集重建1 球的着色。