论文标题
Erdos-Renyi随机图的shot弹枪组装
Shotgun Assembly of Erdos-Renyi Random Graphs
论文作者
论文摘要
图shot弹枪装配是指从当地社区集合中重建图形的问题。在本文中，我们考虑了\ er随机图的shot弹枪汇编$ g（n，p_n）$，其中$ p_n = n^{ - α} $ for $ 0 <α<1 $。我们认为重建为同构和精确的重建（恢复顶点标签以及结构）。我们表明，鉴于距离的集合 - $ 1 $社区，$ g $完全可以重建，价格为$ 0 <α<\ frac {1} {3} $，但不能以$ \ frac {1} {2} {2}<α<α<1 $重建。给定距离的收集 - $ 2 $ neighbounkory，$ g $完全可以重建\ in \ in \ left（0，\ frac {1} {2} {2} {2} \ right）\ cup \ left（\ frac {1} {1} {2} {2}，\ frac {3} {3} {5} {5} {5} {5} {5} {5} {5} {5} {3} α<1 $。
Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of \ER random graphs $G(n, p_n)$, where $p_n = n^{-α}$ for $0 < α< 1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-$1$ neighborhoods, $G$ is exactly reconstructable for $0 < α< \frac{1}{3}$, but not reconstructable for $\frac{1}{2} < α< 1$. Given the collection of distance-$2$ neighborhoods, $G$ is exactly reconstructable for $α\in \left(0, \frac{1}{2}\right) \cup \left(\frac{1}{2}, \frac{3}{5}\right)$, but not reconstructable for $\frac{3}{4} < α< 1$.
