Uniformity of point samples in metric spaces using gap ratio

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Conference Article

Publication Title

SIAM Journal on Discrete Mathematics


Teramoto et al. [IEICE Trans. Inform. Syst., 89-D (2006), pp. 2348-2356] defined a measure called the gap ratio that measures the uniformity of a finite point set sampled from S, a bounded subset of ℝ2. This definition of uniformity measure can be generalized over all metric spaces by appealing to covering and packing radius. 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. The definition of the 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 show some interesting connections of the gap ratio to Delaunay triangulation and packing and covering. Asano [Inform. Process. Lett., 109 (2008), pp. 5760] opined that the discrete version of the uniformity problem makes it amenable to pose combinatorial optimization related questions. The major focus of this work is on finding lower bounds and solving optimization related questions about selecting uniform point samples from metric spaces using the gap ratio. In deducing lower bounds on the gap ratio, we exploit its relation to packing and covering. We have been able to show existence of point configurations with certain cardinality obtained using farthest point insertion and characterized using a recurrence to achieve the lower bounds deduced. Apart from the lower bounds, we prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives dierent approximation ratios for di erent 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 (1 +)-approximation algorithm for uniform sampling from the Euclidean space.

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Open Access, Green

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