A $k$-deck of a (coloured) graph is a multiset of its induced $k$-vertex subgraphs. Given a graph $G$, when is it possible to reconstruct with high probability a uniformly random colouring of its vertices in $r$ colours from its $k$-deck? In this paper, we study this question for grids and random graphs. Reconstruction of random colourings of $d$-dimensional $n$-grids from the deck of their $k$-subgrids is one of the most studied colour reconstruction questions. The 1-dimensional case is motivated by the problem of reconstructing DNA sequences from their `shotgunned' stretches. It was comprehensively studied and the above reconstruction question was completely answered in the '90s. In this paper, we get a very precise answer for higher $d$. For every $d\geq 2$ and every $r\geq 2$, we present an almost linear algorithm that reconstructs with high probability a random $r$-colouring of vertices of a $d$-dimensional $n$-grid from the deck of all its $k$-subgrids for every $k\geq(d\log_r n)^{1/d}+1/d+\varepsilon$ and prove that the random $r$-colouring is not reconstructible with high probability if $k\leq (d\log_r n)^{1/d}-\varepsilon$. This answers the question of Narayanan and Yap (that was asked for $d\geq 3$) on "two-point concentration" of the minimum $k$ so that $k$-subgrids determine the entire colouring. Next, we prove that with high probability a uniformly random $r$-colouring of vertices of a uniformly random graph $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+8$ and is not reconstructible with high probability if $k\leq\sqrt{2\log_2 n}$. We further show that the colour reconstruction algorithm for random graphs can be modified and used for graph reconstruction: we prove that with high probability $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+11$ while it is not reconstructible with high probability if $k\leq 2\sqrt{\log_2 n}$.

翻译：一个（着色）图的$k$-牌是其所有$k$顶点诱导子图的多重集。给定图$G$，何时能够以高概率从它的$k$-牌重构出顶点在$r$种颜色上的均匀随机着色？本文针对网格图和随机图研究该问题。从$k$-子网格的牌重构$d$维$n$-网格的随机着色是着色重构问题中研究最为深入的之一。1维情形源于从DNA序列的“鸟枪法”片段重构该序列的问题，该问题在20世纪90年代已被全面研究并得到完整解答。本文为更高维$d$给出了非常精确的答案。对于任意$d\geq 2$和任意$r\geq 2$，我们对满足$k\geq(d\log_r n)^{1/d}+1/d+\varepsilon$的所有$k$，提出了一种几乎线性的算法，能以高概率从所有$k$-子网格的牌重构$d$维$n$-网格顶点的随机$r$-着色，并证明当$k\leq (d\log_r n)^{1/d}-\varepsilon$时，该随机$r$-着色无法以高概率被重构。这回答了Narayanan和Yap（针对$d\geq 3$提出的）关于最小$k$的“两点集中”问题，即当$k$达到何种阈值时$k$-子网格能确定整个着色。其次，我们证明：对于均匀随机图$G(n,1/2)$，当$k\geq 2\log_2 n+8$时，其顶点的均匀随机$r$-着色能以高概率从完整$k$-牌中重构；而当$k\leq\sqrt{2\log_2 n}$时，该着色无法以高概率被重构。进一步地，我们表明随机图的着色重构算法可经修改后用于图的重构：若$k\geq 2\log_2 n+11$，则$G(n,1/2)$能以高概率从完整$k$-牌中重构；若$k\leq 2\sqrt{\log_2 n}$，则无法以高概率重构。