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Down the Rabbit Hole: Robust Proximity Search in Sublinear Space

Nirman Kumar and Sariel Har-Peled.


For a set of n points in Rd, and parameters k and µ, we present a data structure that answers (1+µ)-approximate k nearest neighbor queries in logarithmic time. Surprisingly, the space used by the data-structure is O(n /k); that is, the space used is sublinear in the input size if k is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: (i) sum of distances of k closest points to the query point, and (ii) sum of squared distances of k closest points to the query point. Our approach generalizes to other distance based estimation of densities of similar flavor.


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Last modified: Tue Nov 6 11:14:14 CST 2012