Shotgun assembly of random jigsaw puzzles

We consider the shotgun assembly problem for a random jigsaw puzzle, introduced by Mossel and Ross (2015). Their model consists of a puzzle—an n×n grid, where each vertex is viewed as a center of a piece. Each of the four edges adjacent to a vertex is assigned one of q colors (corresponding to “jigs,” or cut shapes) uniformly at random. Unique assembly refers to there being only one puzzle (the original one) that is consistent with the collection of individual pieces. We show that for any ε>0, if q ≥ n1+ε, then unique assembly holds with high probability. The proof uses an algorithm that assembles the puzzle in time nΘ(1/ε).22