From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

Dovgoshey, Oleksiy; Петров, Е. А.

doi:10.1134/s2070046618040052

Cited by 20 publications

(8 citation statements)

References 17 publications

(28 reference statements)

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“…Following the labeled rooted tree language used in [13], we provide a description of a weighted rooted tree representation of any finite ultrametric space.…”

Section: A Data Structure For Ultrametric Spaces and Implementation D...mentioning

confidence: 99%

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

Mémoli,

Smith,

Wan

2021

Preprint

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The Gromov-Hausdorff distance (d GH ) provides a natural way of quantifying the dissimilarity between two given metric spaces. It is known that computing d 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 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 (p) GH } p∈[1,∞] of distances defined in a flavor similar to the Gromov-Hausdorff distance on the collection of finite ultrametric spaces, and in particular d (1) GH = d GH . The extreme case when p = ∞, which we also denote by u GH , turns out to be an ultrametric on the collection of ultrametric spaces. Whereas for all p ∈ [1, ∞), d (p) GH yields NP-hard problems, we prove that surprisingly u GH can be computed in polynomial time. The proof is based on a structural theorem for u GH established in this paper; (2) inspired by the structural theorem for u GH , and by carefully leveraging properties of ultrametric spaces, we also establish a structural theorem for d GH when restricted to ultrametric spaces. This structural theorem allows us to identify special families of ultrametric spaces on which d 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 GH between ultrametric spaces. We believe ours is the first such algorithm to be identified.

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“…Following the labeled rooted tree language used in [13], we provide a description of a weighted rooted tree representation of any finite ultrametric space.…”

Section: A Data Structure For Ultrametric Spaces and Implementation D...mentioning

confidence: 99%

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

Mémoli,

Smith,

Wan

2021

Preprint

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“…Let T = T (r) be a rooted tree, let v ∈ V (T ) and let L(T ) be the set of all leaves of T . As in [7,[9][10][11][12][13][14], we will denote by δ…”

Section: Initial Definitions and Factsmentioning

confidence: 99%

“…Some other extremal properties of finite ultrametric spaces and related properties of monotone rooted trees have been found in [12]. The interconnections between the Gurvich-Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in [7] (see also [10,20,[22][23][24]).…”

Section: Introductionmentioning

confidence: 99%

Isomorphism of Trees and Isometry of Ultrametric Spaces

Dovgoshey¹

2020

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We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated by labelings given on the vertices of trees. The obtained results generalized some facts previously known for phylogenetic trees and for Gurvich-Vyalyi monotone trees. Keywords and phrases. Monotone tree, equidistant tree, phylogenetic tree, planted tree, finite ultrametric space, isometry of ultrametric spaces, isomorphism of trees, star.

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“…Some other extremal properties of finite ultrametric spaces and related them properties of monotone rooted trees have been found in [6]. The interconnections between the Gurvich-Vyalyi representation and the space of balls endowed with the Hausdorff metric are discussed in [7] (see also [8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

Labeled trees generating complete, compact, and discrete ultrametric spaces

Dovgoshey¹,

Küçükaslan²

2021

Preprint

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We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The labeled trees, which generate complete ultrametrics, totally bounded ultrametrics, and discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of labeled trees generating compact ultrametrics, and discrete totally bounded ultrametrics. It is also shown that every ultrametric space generated by labeled tree contains a dense discrete subspace.

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From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

Cited by 20 publications

References 17 publications

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

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

Isomorphism of Trees and Isometry of Ultrametric Spaces

Labeled trees generating complete, compact, and discrete ultrametric spaces

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