arXiv 2110.031362021-10-07math.MGcs.CG

超度量空间之间的Gromov-Hausdorff距离：结构与计算

The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation

Facundo Mémoli, Zane Smith, Zhengchao Wan

摘要：

Gromov-Hausdorff距离（$d_\mathrm{GH}$）提供了一种量化两个给定度量空间之间相异性的自然方法。已知计算两个有限度量空间之间的$d_\mathrm{GH}$是NP困难的，即使在有限超度量空间的情况下也是如此，所述有限超度量空间是高度结构化的度量空间，在这个意义上它们满足所谓的\emph{strong triangle inequality}。超度量空间自然出现在许多应用中，如层次聚类，遗传学，基因组学，甚至语言学。利用超度量空间的特殊结构，（1）在有限超度量空间的集合上，特别是在$d_\mathrm{GH}^{(1)} =d_\mathrm{GH}$上，我们确定了一个单参数距离族$\{d_\mathrm{GH}^{(p)}\}_{p\in[1,\infty]}$，它的定义类似于Gromov-Hausdorff距离.当$p=\infty$（我们也记为$u_\mathrm{GH}$）是超度量空间集合上的超度量时的极端情况。而对于所有的$p\in[1,\infty)$，$d_\mathrm{GH}^{(p)}$产生NP-困难的问题，我们证明了令人惊讶的$u_\mathrm{GH}$可以计算在多项式时间。证明是基于本文建立的$u_\mathrm{GH}$的一个结构定理;（2）受$u_\mathrm{GH}$的结构定理的启发，通过仔细利用超度量空间的性质，我们还建立了$d_\mathrm{GH}$在限制于超度量空间时的一个结构定理.这个结构定理使我们能够识别特殊的超度量空间族，在这些超度量空间上$d_\mathrm{GH}$是可计算的。这些族由与度量空间的加倍常数有关的性质决定。基于这些族，我们设计了一个计算超度量空间之间$d_\mathrm{GH}$精确值的固定参数易处理（FPT）算法。我们相信我们的算法是第一个被识别的算法。

英文摘要：

The Gromov-Hausdorff distance ($d_\mathrm{GH}$) provides a natural way of quantifying the dissimilarity between two given metric spaces. It is known that computing $d_\mathrm{GH}$ between two finite metric spaces is NP-hard, even in the case of finite ultrametric spaces which are highly structured metric spaces in the sense that they satisfy the so-called \emph{strong triangle inequality}. Ultrametric spaces naturally arise in many applications such as hierarchical clustering, phylogenetics, genomics, and even linguistics. By exploiting the special structures of ultrametric spaces, (1) we identify a one parameter family $\{d_\mathrm{GH}^{(p)}\}_{p\in[1,\infty]}$ of distances defined in a flavor similar to the Gromov-Hausdorff distance on the collection of finite ultrametric spaces, and in particular $d_\mathrm{GH}^{(1)} =d_\mathrm{GH}$. The extreme case when $p=\infty$, which we also denote by $u_\mathrm{GH}$, turns out to be an ultrametric on the collection of ultrametric spaces. Whereas for all $p\in[1,\infty)$, $d_\mathrm{GH}^{(p)}$ yields NP-hard problems, we prove that surprisingly $u_\mathrm{GH}$ can be computed in polynomial time. The proof is based on a structural theorem for $u_\mathrm{GH}$ established in this paper; (2) inspired by the structural theorem for $u_\mathrm{GH}$, and by carefully leveraging properties of ultrametric spaces, we also establish a structural theorem for $d_\mathrm{GH}$ when restricted to ultrametric spaces. This structural theorem allows us to identify special families of ultrametric spaces on which $d_\mathrm{GH}$ is computationally tractable. These families are determined by properties related to the doubling constant of metric space. Based on these families, we devise a fixed-parameter tractable (FPT) algorithm for computing the exact value of $d_\mathrm{GH}$ between ultrametric spaces. We believe ours is the first such algorithm to be identified.

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