Shotgun Threshold for Sparse Erdos-Renyi Graphs

In the shotgun assembly problem for a graph,we are given the empirical profile for rooted neighborhoods of depth r(up to isomorphism) for somer >= 1 and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erdos-Renyi G(n,lambda / n), we show that the shotgun assembly threshold r(& lowast; )satisfies that r(& lowast;) approximate to logn / log(lambda(2)gamma lambda(-1 )where gamma(lambda )is the probability for two independent Poisson-Galton-Watson trees with parameter lambda to be rooted isomorphic with each other. Our result sharpens a constant factor in a previous work by Mossel and Ross (2019) and thus solves a question therein