散弹枪识别
Shotgun identification on groups
摘要：
我们考虑群体上模式的鸟枪鉴定问题，这扩展了以前关于DNA序列和标号图的鸟枪鉴定的工作。群$G$上的鸟枪识别问题由两个有限子集$C \subset G$和$K \subset G$以及一个有限字母表$\mathcal{A}$指定.在这样的问题中，存在"全局”模式$w \in \mathcal{A}^{CK}$，并且人们希望能够仅基于对以$C$的元素为中心的$w$的"局部”$K$形子模式（称为读段）的观察来识别该模式（直到翻译）。我们考虑一个渐近的制度，其中的大小$w$趋于无穷大和符号的$w$绘制在一个i.i.d.时尚.我们的第一个一般性结果建立了随机模式$w$从其读段中以概率趋于1可识别的充分条件，我们的第二个一般性结果建立了随机模式$w$以概率趋于1不可识别的充分条件。此外，我们说明了我们的主要结果，将其应用到几个家庭的例子。
英文摘要：
We consider the problem of shotgun identification of patterns on groups, which extends previous work on shotgun identification of DNA sequences and labeled graphs. A shotgun identification problem on a group $G$ is specified by two finite subsets $C \subset G$ and $K \subset G$ and a finite alphabet $\mathcal{A}$. In such problems, there is a ``global" pattern $w \in \mathcal{A}^{CK}$, and one would like to be able to identify this pattern (up to translation) based only on observation of the ``local" $K$-shaped subpatterns of $w$, called reads, centered at the elements of $C$. We consider an asymptotic regime in which the size of $w$ tends to infinity and the symbols of $w$ are drawn in an i.i.d. fashion. Our first general result establishes sufficient conditions under which the random pattern $w$ is identifiable from its reads with probability tending to one, and our second general result establishes sufficient conditions under which the random pattern $w$ is non-identifiable with probability tending to one. Additionally, we illustrate our main results by applying them to several families of examples.
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