Shotgun Assembly of Random Jigsaw Puzzles

In a recent work, Mossel and Ross considered the shotgun assembly problem for a random jigsaw puzzle. Their model consists of a puzzle - an $n\times n$ grid, where each vertex is viewed as a center of a piece. They assume that each of the four edges adjacent to a vertex, is assigned one of $q$ colors (corresponding to "jigs", or cut shapes) uniformly at random. Mossel and Ross asked: how large should $q = q(n)$ be so that with high probability the puzzle can be assembled uniquely given the collection of individual tiles? They showed that if $q = ω(n^2)$, then the puzzle can be assembled uniquely with high probability, while if $q = o(n^{2/3})$, then with high probability the puzzle cannot be uniquely assembled. Here we improve the upper bound and show that for any $\eps > 0$, the puzzle can be assembled uniquely with high probability if $q \geq n^{1+\eps}$. The proof uses an algorithm of $n^{Θ(1/\eps)}$ running time.