Distance matrices of subsets of the Hamming cube

doi:10.1016/j.indag.2021.01.004

Indagationes Mathematicae

Volume 32, Issue 3, May 2021, Pages 646-657

https://doi.org/10.1016/j.indag.2021.01.004 Get rights and content

Abstract

Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points ${x_{0}, x_{1}, \dots, x_{n}}$ in the Hamming cube $H_{n} = ({0, 1}^{n}, ℓ_{1})$ . In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points ${x_{0}, x_{1}, \dots, x_{m}}$ in $H_{n}$ . It follows from this more general formula that $det (D) \neq 0$ if and only if the vectors $x_{0}, x_{1}, \dots, x_{m}$ are affinely independent. Specializing to the case $m = n$ provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases $m < n$ and $m = n$ is noted. We also show that if $D$ is the distance matrix of an unweighted tree on $n + 1$ vertices, then $〈 D^{- 1} 1, 1 〉 = 2 ∕ n$ where $1$ is the column vector all of whose coordinates are 1. Finally, we derive a new proof of Murugan’s classification of the subsets of $H_{n}$ that have strict 1-negative type.

Section snippets

Introduction: Distance and Gram matrices

The global geometry of a finite metric space $(X, d) = ({x_{0}, x_{1}, \dots, x_{m}}, d)$ is completely encoded within its distance matrix $D = {(d (x_{i}, x_{j}))}_{i, j = 0}^{m}$ . A now much-studied problem concerns conditions on $D$ which ensure that $(X, d)$ embeds isometrically in a Euclidean space. We refer the reader to the surveys [2] and [10] for further details of the theory and applications of these so-called Euclidean distance matrices.

A class of finite metric spaces which rarely embeds in Euclidean space is the class of metric

Distance matrices of subsets of the Hamming cube

The hypercube ${0, 1}^{n}$ has a natural additive group structure given by elementwise addition modulo 2. It is easy to check that the $ℓ_{1}$ metric is invariant under translation in this group. In particular, if $X = {x_{0}, x_{1}, \dots, x_{m}}$ is a given subset of $H_{n}$ , then the map $Φ (x) = x - x_{0}$ is an isometric isomorphism from $X$ to the set $X^{'} = {0, x_{1} - x_{0}, \dots, x_{m} - x_{0}}$ (where $0$ denotes the zero vector in ${0, 1}^{n}$ ). Note that if we set $x_{i}^{'} = x_{i} - x_{0}$ , $0 \leq i \leq m$ , then obviously $G (x_{1} - x_{0}, \dots, x_{m} - x_{0}) = G (x_{1}^{'} - x_{0}^{'}, \dots, x_{m}^{'} - x_{0}^{'})$ and consequently the $m$

Applications to supremal negative type

The results of Section 2 afford a new analysis of negative type properties of subsets of the Hamming cube $H_{n}$ . In particular, we develop a new proof of Murugan’s [11] classification of the subsets of $H_{n}$ that have strict $1$ -negative type. In order to proceed we need to recall some classical definitions and related theorems.

Definition 3.1

Let $(X, d)$ be a metric space and suppose that $p \geq 0$ . Then:

(a)
$(X, d)$ has $p$ -negative type iff for each finite subset ${x_{0}, \dots, x_{m}}$ of $X$ and each choice of scalars $ξ_{0}, \dots, ξ_{m}$ such that $ξ_{0} + \dots + ξ_{m} = 0$

Affinely independent subsets of the Hamming cube

In the proof of Murugan’s result given in the previous section, it was shown that if ${x_{0}, x_{1}, \dots, x_{m}}$ is an affinely independent subset of $H_{n}$ , then $〈 D^{- 1} 1, 1 〉 > 0$ (where $D = D_{1}^{(1)} = {(d_{1} (x_{i}, x_{j}))}_{i, j = 0}^{m}$ ). However, there is actually more that can be said in this setting. As stated earlier, results of Hjorth et al. [8] and Murugan [11] imply that any unweighted metric tree $T$ on $n + 1$ vertices embeds isometrically into $H_{n}$ as an affinely independent set. For such embedded trees we may compute the precise value of $〈 D$

Acknowledgments

The work of the second and third authors was supported by the Research Training Program of the Department of Education and Training of the Australian Government .

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Distance matrices of subsets of the Hamming cube

Abstract

Section snippets

Introduction: Distance and Gram matrices

Distance matrices of subsets of the Hamming cube

Applications to supremal negative type

Affinely independent subsets of the Hamming cube

Acknowledgments

Adv. Math.

Linear Algebra Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Introduction To Calculus and Analysis II/1

Euclidean distance matrices: Essential theory, algorithms, and applications

IEEE Signal Process. Mag.

The Theory of Matrices, Vol. I

On the addressing problem for loop switching

Bell Syst. Tech. J.