Metric Dimension and Resolvability of Jaccard Spaces

TL;DR

This paper investigates the metric dimension of Jaccard spaces formed by the power set of a finite set, providing probabilistic constructions of nearly minimal resolving sets and analyzing their size relative to the set's cardinality.

Contribution

It introduces a probabilistic method to construct nearly optimal resolving sets for Jaccard spaces and determines their size as proportional to |X|/ln|X|.

Findings

The metric dimension of (2^X, Jac) is Θ(|X|/ln|X|).

A smaller subset can resolve pairs of subsets of size up to √|X|/ln|X| with high probability.

Probabilistic and linear algebra techniques effectively construct resolving sets in Jaccard spaces.

Abstract

A subset of points in a metric space is said to resolve it if each point in the space is uniquely characterized by its distance to each point in the subset. In particular, resolving sets can be used to represent points in abstract metric spaces as Euclidean vectors. Importantly, due to the triangle inequality, points close by in the space are represented as vectors with similar coordinates, which may find applications in classification problems of symbolic objects under suitably chosen metrics. In this manuscript, we address the resolvability of Jaccard spaces, i.e., metric spaces of the form $(2^X,\text{Jac})$, where $2^X$ is the power set of a finite set $X$, and $\text{Jac}$ is the Jaccard distance between subsets of $X$. Specifically, for different $a,b\in 2^X$, $\text{Jac}(a,b)=|a\Delta b|/|a\cup b|$, where $|\cdot|$ denotes size (i.e., cardinality) and $\Delta$ denotes the symmetric difference of sets. We combine probabilistic and linear algebra arguments to construct highly likely but nearly optimal (i.e., of minimal size) resolving sets of $(2^X,\text{Jac})$. In particular, we show that the metric dimension of $(2^X,\text{Jac})$, i.e., the minimum size of a resolving set of this space, is $\Theta(|X|/\ln|X|)$. In addition, we show that a much smaller subset of $2^X$ suffices to resolve, with high probability, all different pairs of subsets of $X$ of cardinality at most $\sqrt{|X|}/\ln|X|$, up to a factor.