Reporting neighbors in high-dimensional Euclidean space

Dror Aiger*, Haim Kaplan, Micha Sharir

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations


We consider the following problem, which arises in many database and web-based applications: Given a set P of n points in a high-dimensional space ℝd and a distance r, we want to report all pairs of points of P at Euclidean distance at most r. We present two randomized algorithms, one based on randomly shifted grids, and the other on randomly shifted and rotated grids. The running time of both algorithms is of the form C(d)(n + k) log n, where k is the output size and C(d) is a constant that depends on the dimension d. The logn factor is needed to guarantee, with high probability, that all neighbor pairs are reported, and can be dropped if it suffices to report, in expectation, an arbitrarily large fraction of the pairs. When only translations are used, C(d) is of the form (a√d)d, for some (small) absolute constant a ≈ 0.484; this bound is worst-case tight, up to an exponential factor of about 2d. When both rotations and translations are used, C(d) can be improved to roughly 6.74d, getting rid of the super-exponential factor √dd. When the input set (lies in a subset of d-space that) has low doubling dimension δ, the performance of the first algorithm improves to C(d,δ)(n + k) log n (or to C(d, δ)(n + k)), where C(d, δ) = O((ed/δ)δ), for δ ≤ √d. Otherwise, C(d, δ) = O (e√dδ. We also present experimental results on several large datasets, demonstrating that our algorithms run significantly faster than all the leading existing algorithms for reporting neighbors.

Original languageEnglish
Title of host publicationProceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
PublisherAssociation for Computing Machinery
Number of pages20
ISBN (Print)9781611972511
StatePublished - 2013
Event24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States
Duration: 6 Jan 20138 Jan 2013

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Country/TerritoryUnited States
CityNew Orleans, LA


Dive into the research topics of 'Reporting neighbors in high-dimensional Euclidean space'. Together they form a unique fingerprint.

Cite this