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 ≥ n^1+ε, then unique assembly holds with high probability. The proof uses an algorithm that assembles the puzzle in time n^Θ(1/ε).22

中文翻译：

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散弹拼图的弹枪组装

我们考虑由Mossel和Ross（2015）提出的随机拼图游戏的shot弹枪组装问题。他们的模型由一个谜题-一个n × n网格组成，其中每个顶点都被视为一块的中心。邻近顶点的四个边缘中的每一个均被随机均匀地分配q种颜色（对应于“锯齿”或切割形状）中的一种。独特的装配是指只有一个拼图（原始拼图）与单个零件的集合一致。我们表明，对于任何ε > 0，如果q  ≥  Ñ ^{1+ ε}，那么唯一组件以高概率成立。证明使用一种算法，该算法在时间n内组装拼图^{Θ（1 / ε）} 0.22