随机正则图的 shotgun 组装
Shotgun assembly of random regular graphs
AI总结：
研究通过分析随机正则图的 shotgun 组装问题，证明了在特定半径 R 下，可以高效恢复图结构，并展示了该方法在图同构性检测中的应用。
摘要：
Mossel 和 Ross（2019）引入了随机图的 shotgun 组装问题：什么半径 R 能够确保随机图 G 可以从其根节点的 R 邻域列表中唯一恢复，以高概率？本文考虑了固定度数 d≥3 的随机正则图的情况。Bollobás（1982）的结果表明，以高概率可以在 R = (1 + ε) (1/2) log_{d-1} n 的情况下实现高效恢复——此外，这种恢复算法仅使用每个邻域中的距离总结。我们证明使用完整的邻域结构可以得到更紧的界：R = [log n + log log n]/[2 log(d-1)] + O(1)，我们证明该界在 O(1) 项内是紧的。我们的证明的一个结果是，如果 G 和 H 是独立图，其中 G 遵循随机正则分布，则以高概率 G 和 H 不是同构的；此外，这可以通过测试 H 的 R 邻域列表与 G 中一个敌对选择的顶点的 R 邻域的对比来高效验证。
英文摘要：
Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider this question for random regular graphs of fixed degree $d\ge3$. A result of Bollobás (1982) implies efficient recovery at $R = (1 + ε) \frac12 \log_{d-1}n$ with high probability -- moreover, this recovery algorithm uses only a summary of the distances in each neighborhood. We show that using the full neighborhood structure gives a sharper bound \[ R = \frac{\log n + \log\log n}{2\log(d-1)} + O(1)\,, \] which we prove is tight up to the $O(1)$ term. One consequence of our proof is that if $G,H$ are independent graphs where $G$ follows the random regular law, then with high probability the graphs are non-isomorphic; furthermore, this can be efficiently certified by testing the $R$-neighborhood list of $H$ against the $R$-neighborhood of a single adversarially chosen vertex of $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
