Uniformity of Point Samples in Metric Spaces Using Gap Ratio

Teramoto et al. [22] defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from \(\mathcal S\), a bounded subset of \(\mathbb {R}^2\). We attempt to generalize the definition of this measure over all metric spaces. We solve optimization related questions about selecting uniform point samples from metric spaces; the uniformity is measured using gap ratio. We give lower bounds for specific metric spaces, prove hardness and approximation hardness results. We also give a general approximation algorithm framework giving different approximation ratios for different metric spaces and give a \(\left( 1+\epsilon \right) \)-approximation algorithm for a set of points in a Euclidean space.