Shotgun assembly of Erdős-Rényi random graphs

Abstract

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 Erdős-Rényi random graphs G(n, pn), where pn = 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 < 1 3 , but not reconstructable for 1 2 < α < 1. Given the collection of distance-2 neighborhoods, G is exactly reconstructable for α ∈ 0, 1 2 ∪ 1 2 , 3 5 , but not reconstructable for 3 4 < α < 1.

Shotgun assembly threshold for lattice labeling model

“…We learned the precise formulation and the general mathematical framework for shotgun assembly questions from the inspiring paper [11]. Since (the circulation of) [11], there has been extensive study on shotgun assembly questions including on random jigsaw problems [3,10,4,8], on random graph models [12,6,7,1], on random coloring model [15] and on some extension of DNA sequencing model [16].…”