  Let $V$ be a set of $n$ points in $\mathbb{R}^d$, and suppose that the
distance between each pair of points is revealed independently with probability
$p$. We study when this information is sufficient to reconstruct large subsets
of $V$, up to isometry.
  Strong results for $d=1$ have been obtained by Gir\~ao, Illingworth, Michel,
Powierski, and Scott. In this paper, we investigate higher dimensions, and show
that if $p>n^{-2/(d+4)}$, then we can reconstruct almost all of $V$ up to
isometry, with high probability. We do this by relating it to a polluted graph
bootstrap percolation result, for which we adapt the methods of Balogh,
Bollob\'as, and Morris.
