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- W2355346348 abstract "In a recent work, Mossel and Ross considered the shotgun assembly problem for a random jigsaw puzzle. Their model consists of a puzzle - an $ntimes n$ 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 $q = omega(n^2)$, 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 $n^{Theta(1/eps)}$ running time." @default.
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- W2355346348 date "2016-05-10" @default.
- W2355346348 modified "2023-09-25" @default.
- W2355346348 title "Shotgun Assembly of Random Jigsaw Puzzles" @default.
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