A formula for the weight of a minimal filling of a finite metric space

2013 ◽  
Vol 204 (9) ◽  
pp. 1285-1306 ◽  
Author(s):  
A Yu Eremin
Keyword(s):  
Metric Space ◽  
Minimal Filling ◽  
Finite Metric Space

scholarly journals Lipschitz-free Spaces on Finite Metric Spaces

2019 ◽  
Vol 72 (3) ◽  
pp. 774-804 ◽  
Author(s):  
Stephen J. Dilworth ◽  
Denka Kutzarova ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Space ◽  
Free Space ◽  
Metric Spaces ◽  
Large Classes ◽  
Complemented Subspace ◽  
Finite Metric Space ◽  
Free Spaces ◽  
Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
Author(s):  
TIMOTHY FAVER ◽  
KATELYNN KOCHALSKI ◽  
MATHAV KISHORE MURUGAN ◽  
HEIDI VERHEGGEN ◽  
ELIZABETH WESSON ◽  
...  
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Independent Set ◽  
Ultrametric Space ◽  
Ultrametric Spaces ◽  
Negative Type ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Generalized Roundness

AbstractMotivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

scholarly journals Axioms for Consensus Functions on then-Cube

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
C. Garcia-Martinez ◽  
F. R. McMorris ◽  
O. Ortega ◽  
R. C. Powers
Keyword(s):  
Metric Space ◽  
Minimax Criterion ◽  
Finite Metric Space ◽  
Finite Sequences ◽  
Consensus Functions

Apvalue of a sequenceπ=(x1,x2,…,xk)of elements of a finite metric space(X,d)is an elementxfor which∑i=1kdp(x,xi)is minimum. Thelp–function with domain the set of all finite sequences onXand defined bylp(π)={x:  xis apvalue ofπ}is called thelp–function on(X,d). Thel1andl2functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of thelp–functions on then-cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions.

Distance Sets of Urysohn Metric Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 222-240 ◽  
Author(s):  
N.W. Sauer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Finite Metric Space ◽  
Distance Set ◽  
Distance Sets ◽  
Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

Axiomatization and the antimean function on paths

2014 ◽  
Vol 06 (04) ◽  
pp. 1450057
Author(s):  
Oscar Ortega ◽  
Y. Wang
Keyword(s):  
Metric Space ◽  
Finite Metric Space ◽  
Finite Sequences

An antimean of a sequence π = ( x1, x2, …, xk) of elements of a finite metric space (X, d) is an element x for which [Formula: see text] is maximum. The function antimean with domain the set of all finite sequences on X and defined by AMean (π) = {x : x is an antimean of π} is called the antimean function on X. In this paper, the antimean function on finite paths is characterized.

Optimal realizations and the block decomposition of a finite metric space

2021 ◽  
Vol 302 ◽  
pp. 103-113
Author(s):  
Katharina T. Huber ◽  
Vincent Moulton ◽  
Andreas Spillner
Keyword(s):  
Metric Space ◽  
Block Decomposition ◽  
Finite Metric Space ◽  
Optimal Realizations

scholarly journals Characterizing Cell-Decomposable Metrics

10.37236/882 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Katharina T. Huber ◽  
Jacobus Koolen ◽  
Vincent Moulton ◽  
Andreas Spillner
Keyword(s):  
Phylogenetic Analysis ◽  
Convex Hull ◽  
Metric Space ◽  
Molecular Data ◽  
Multicommodity Flow ◽  
Flow Problems ◽  
Canonical Metric ◽  
Tight Span ◽  
Finite Metric Space ◽  
A Cell

To a finite metric space $(X,d)$ one can associate the so called tight-span $T(d)$ of $d$, that is, a canonical metric space $(T(d),d_\infty)$ into which $(X,d)$ isometrically embeds and which may be thought of as the abstract convex hull of $(X,d)$. Amongst other applications, the tight-span of a finite metric space has been used to decompose and classify finite metrics, to solve instances of the server and multicommodity flow problems, and to perform evolutionary analyses of molecular data. To better understand the structure of $(T(d),d_\infty)$ the concept of a cell-decomposable metric was recently introduced, a metric whose associated tight-span can be decomposed into simpler tight-spans. Here we show that cell-decomposable metrics and totally split-decomposable metrics — a class of metrics commonly applied within phylogenetic analysis — are one and the same thing, and also provide some additional characterizations of such metrics.

scholarly journals All finite sets are Ramsey in the maximum norm

2021 ◽  
Vol 9 ◽  
Author(s):  
Andrey Kupavskii ◽  
Arsenii Sagdeev
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Metric Spaces ◽  
Upper Bounds ◽  
Maximum Norm ◽  
Lower And Upper Bounds ◽  
Finite Sets ◽  
Finite Metric Space ◽  
Monochromatic Copy

Abstract For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$ .

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