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Robust Proximity Search for Balls Using Sublinear Space

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Abstract

Given a set of n disjoint balls \(b_1, \dots , b_n\) in \(\mathrm{I\! R}^d\), we provide a data structure of near linear size that can answer \((1\pm {\varepsilon })\)-approximate kth-nearest neighbor queries on the balls in \(O(\log n + 1/{\varepsilon }^d)\) time, where k and \({\varepsilon }\) may be provided at query time. If k and \({\varepsilon }\) are provided in advance, we provide a data structure to answer such queries requiring O(n / k) space; that is, the data structure requires sublinear space if k is sufficiently large.

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Notes

  1. Our data structure and algorithm work for the more general case where the balls are interior disjoint, where we define the interior of a “point ball”, i.e., a ball of radius 0, as the point itself. This is not the usual topological definition.

  2. That is, intuitively, if the query point falls into one of the grid cells of \(\mathcal {I}\), we can answer a query in constant time.

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA

    Sariel Har-Peled

  2. Department of Computer Science, University of Memphis, Dunn Hall 375, Memphis, TN, 38152, USA

    Nirman Kumar

Authors
  1. Sariel Har-Peled
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  2. Nirman Kumar
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Corresponding author

Correspondence to Nirman Kumar.

Additional information

A preliminary version of this paper appeared in FSTTCS 2014 [15].

Work on this paper was partially support by NSF AF awards CCF-0915984, CCF-1421231, and CCF-1217462.

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Har-Peled, S., Kumar, N. Robust Proximity Search for Balls Using Sublinear Space. Algorithmica 80, 279–299 (2018). https://doi.org/10.1007/s00453-016-0254-4

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  • DOI: https://doi.org/10.1007/s00453-016-0254-4

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