We study the problem of circular seriation, where we are given a matrix of pairwise dissimilarities between $n$ objects, and the goal is to find a {\em circular order} of the objects in a manner that is consistent with their dissimilarity. This problem is a generalization of the classical {\em linear seriation} problem where the goal is to find a {\em linear order}, and for which optimal ${\cal O}(n^2)$ algorithms are known. Our contributions can be summarized as follows. First, we introduce {\em circular Robinson matrices} as the natural class of dissimilarity matrices for the circular seriation problem. Second, for the case of {\em strict circular Robinson dissimilarity matrices} we provide an optimal ${\cal O}(n^2)$ algorithm for the circular seriation problem. Finally, we propose a statistical model to analyze the well-posedness of the circular seriation problem for large $n$. In particular, we establish ${\cal O}(\log(n)/n)$ rates on the distance between any circular ordering found by solving the circular seriation problem to the underlying order of the model, in the Kendall-tau metric.

中文翻译：

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一种严格循环序列的最优算法

我们研究了循环序列化的问题，其中给定了 $n$ 个对象之间成对不同的矩阵，目标是以与它们的相异性一致的方式找到对象的 {\em 循环顺序}。这个问题是经典的 {\em 线性序列} 问题的推广，其目标是找到一个 {\em 线性顺序}，并且已知最优的 ${\cal O}(n^2)$ 算法。我们的贡献可以总结如下。首先，我们引入 {\em 循环罗宾逊矩阵}作为循环序列化问题的自然相异矩阵类。其次，对于{\em严格循环罗宾逊相异矩阵}的情况，我们为循环序列化问题提供了一个最优的${\cal O}(n^2)$算法。最后，我们提出了一个统计模型来分析大 $n$ 的循环序列问题的适定性。特别是，我们在 Kendall-tau 度量中通过求解循环序列化问题到模型的底层顺序找到的任何循环排序之间的距离建立 ${\cal O}(\log(n)/n)$ 比率.