Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, December 17, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Chih-Hung Liu

Nearly Optimal Planar k Nearest Neighbors Queries under General Distance Functions

We study the k nearest neighbors problem in the plane for general, convex, pairwise disjoint objects of constant description complexity such as line segments, disks, and quadrilaterals and with respect to a general family of distance functions including the L_p-norms and additively weighted Euclidean distances. For point objects in the Euclidean metric, after four decades of effort, an optimal data structure has recently been developed with O(n) space, O(log n+k) query time, and O(n log n) preprocessing time. We compose nearly optimal data structures for the k nearest neighbors problem under general distance functions, for which we devise shallow cuttings of linear size. Shallow cuttings are a key technique for point objects in the Euclidean metric. Although shallow cuttings have been generalized to general distance functions, existing ones either could not be applied to the k nearest neighbors problem or have an extra polylog-factor in size. Our innovation is a new random sampling technique for the analysis of geometric structures.

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