Uniformity of point samples in metric spaces using gap ratio

TL;DR

This paper introduces a generalized gap ratio measure for uniformity of point samples in metric spaces, explores its theoretical properties, and develops algorithms for optimal sampling, including coresets and approximation methods.

Contribution

It generalizes the gap ratio measure to all metric spaces, establishes bounds and hardness results, and provides approximation algorithms and coresets for uniform sampling.

Findings

Generalized gap ratio to all metric spaces.

Proved lower bounds and hardness for uniform sampling.

Developed a $(1+ ext{epsilon})$-approximation algorithm using coresets.

Abstract

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+\epsilon \right)$-approximation algorithm for uniform sampling from the Euclidean space.